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This is to certify that the
dissertation entitled

ANALYTICAL AND QUANTITATIVE CHARACTERIZATION
OF WIRELESS SENSOR NETWORKS

presented by
MUHAMMAD USMAN ILYAS

has been accepted towards fulfillment
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ANALYTICAL AND QUANTITATIVE CHARACTERIZATION OF
' WIRELESS SENSOR NETWORKS

By

Muhammad Usman Ilyas

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY

Electrical Engineering

2009

ABSTRACT

ANALYTICAL AND QUANTITATIVE CHARACTERIZATION OF
WIRELESS SENSOR NETWORKS

By

Muhammad Usman Ilyas

In this thesis we characterize key properties of wireless sensor networks (WSN) by
analytical and quantitative methods. These include the link-layer bit error rate (BER)
process, network lifetime and topology.

For the analysis of the BER process, we collected a large set of packet traces
over IEEE 80215.4 links. Our packet traces distinguish themselves from other data
sets in that they record channel state information (CSI) as well as full and partial
packet erasures. A channel model, which is conditioned on observed CSI, is developed.
This conditional model reduces the variance of the BER’s distribution by one order
of magnitude. Packet traces are also analyzed to determine memory length of bit
errors. Correlation analysis of bit and symbol level traces reveals that memory length
of errors in all traces is 2 bits and 2 symbols, respectively. For packet-level traces
consisting of BER measurements of individual packets the traditional correlogram
based analysis fails and so we introduce relative mutual information (RMI) as a more
robust method for measuring channel memory. RMI based analysis of packet traces
shows that memory length of the BER ranges from 0 to 2.360.

The research on the network lifetime problem proposes joint minimization of mean
and variance of sensor power consumption rates as an alternative to the minimax
formulation of the lifetime problem in WSN. This proposed statistical optimization
objective better fits the vision of WSNs consisting of large numbers of inexpensive,
redundant, disposable sensors than the minimax formulation which focuses on the
top power consuming node. We formulate this problem in quadratic program (QP)

form. To avoid scalability issues of using a QP, an approximate dynamic program

(DP) formulation of lower complexity rooted in operational rate—distortion theory
is developed. For a randomly generated WSN of 100 nodes DP exhibits upto 44%
reduction in variance at the cost of 19% increase in its mean, with many intermediate
operating points of higher benefit / cost ratios to choose from.

The research on topological characteristics of WSNs explores the possibility of
building WSNS with small-world topologies that combine desirable properties of Eu-
clidean / lattice graphs with those of random graphs. An analytical model is developed
to explain the phase difference in characteristic path length and clustering coefficient
in lattice graphs when shortcut links of limited range are used. We test and im-
plement a software based system for commercial-off-the—shelf motes that increases
communication range of links in WSNs using cooperative communication and diver-
sity combining. A trace based implementation demonstrates proof-of—concept of its
ability to reduce the fraction of packets with errors on a channel from 20% down to
1% and reduce the BER of packets that cannot be corrected. This is followed by an
implementation on the Crossbow Imote2 sensor mote. Results from the mote based
implementation show an increase in packet reception rate from 22 -— 30% to 73 —- 76%.

Finally, we develop a centrality measure to identify well connected clusters of
central nodes for the placement of network resources. For mesh network topologies
that are characteristic of WSNs, eigenvector centrality (EVC) consistently fails to
identify more than a single, arbitrarily located cluster of nodes as the most central.
We introduce principal component centrality (PCC), a node centrality inspired by the
Karhunen Loéve transform/ principal component analysis. we demonstrate PCC’S
ability to identify a larger number of central hub nodes than EVC, depending on the

number of features used in its computation.

COPYRIGHT

Copyright by
Muhammad Usman Ilyas

2009

DEDICATION

To my wife, Ayesha

and my daughter, Haadiya.

ACKNOWLEDGMENT

I would like to acknowledge my advisor Professor Hayder Radha for his years of
support through the Ph.D. program, for his professional mentoring and counsel. I
would also like to thank Professor Subir K. Biswas, Professor Tongtong Li and Pro—
fessor Philip K. McKinley for the guidance and valuable feedback they provided as
members of my PhD. program committee.

I would like to acknowledge the Higher Education Commission of the government
of Pakistan, the National Science Foundation and Michigan State University for gen-
erously providing funding at different stages during my Ph.D. program.

Many thanks to my colleagues at the WAVES lab Khayam, Shirish, Kiran, Sauleh,
Sohraab, N ima, Rarni, Yongju, Moonseong, Ahmed and Aqeel for letting me bounce
off ideas, many, many technical discussions and their company on our coffee rounds
& all-nighters. I am also grateful to Khawar, Awais, Aparna, Keyur and Zubair for
their friendships.

None of this would have been possible without the constant support and encour-
agement of my wife Ayesha. I would like to acknowledge Ayesha’s parents without
whose active support and assistance I would not be at MSU. I am grateful to my
parents whose years of investment in my education allowed me to pursue graduate
studies. Finally, I want to acknowledge my lovely daughter Haadiya for being a reality

check and bring balance merely by her presence.

Muhammad Usman Ilyas

vi

As the area of your knowledge grows, so does the periphery of your ignorance.

Neil deGrasse Tyson

vii

TABLE OF CONTENTS

List of Tables .................................

List of Figures ................................

Introduction
1.1 Organization ...............................
1.2 Overview of Contributions ........................

Network Channel Capacity of IEEE 80215.4 Wireless Sensor Net-

works Under Reachback

2.1 Introduction ................................

2.2 System Model ...............................
2.2.1 Wireless Networking Standards for VVSNs ...........
2.2.2 Source & Channel Coding ....................
2.2.3 Power Consumption Model ....................

2.3 Channel Model ..............................
2.3.1 1-H0p Communication ......................
2.3.2 Overlay Network Communication ................
2.3.3 Sensor-To—Base Station Capacity ................

2.4 Results and Analysis ...........................

CSI Driven Model of BER Process on IEEE 802.15.4 Links
3.1 Introduction ................................
3.1.1 Previous Work ........................ ' . .
3.2 Trace Collection Setup ..........................
3.2.1 Experimental Setup ........................
3.2.2 Packet Payload ..........................
3.2.3 Trace Generation .........................
3.2.4 Channel State Information ....................
3.2.5 Spectral and Environmental Diversity ..............
3.3 Correlation Analysis of CSI Measures ..................
3.4 CSI Driven BER Model .........................
3.5 Model Evaluation .............................
3.5.1 Variance Reduction ........................
3.5.2 Dependence On Deployment Environment ...........
3.6 Conclusions ................................

xii

t0

O1QDQDN03

1
17
18
19
24
27
28

33
34
35
37
37
38
39
39
44
45
50
52
52
53
56

4 Memory Properties of the Link-level BER Process in IEEE 802.15.4

Links 58
4.1 Introduction ................................ 59
4.2 Memory Length Measurement By Correlation Analysis ........ 61
4.2.1 Correlation Function ....................... 61
4.2.2 Correlograms of Bit and Symbol-level Traces .......... 64
4.2.3 Correlograms of Packet-level Traces ............... 64

4.3 Hurst Analysis of Packet-level BER Process .............. 67
4.3.1 Observations For MC Trace Set ................. 69
4.3.2 Observations For ME Trace Set ................. 71

4.4 Memory Length Measurement By Relative Mutual Information . . . . 71
4.4.1 Shannon Information Measures ................. 71
4.4.2 Description: Relative Mutual Information ........... 72
4.4.3 Discussion ............................. 76

4.5 Conclusions ................................ 78

5 A Statistical Measure Of Network Lifetime For Wireless Sensor Net-
works 80
5.1 Introduction ................................ 81
5.2 Previous Work .............................. 82
5.3 Novelty Of Approach ........................... 83
5.4 Network Model .............................. 85
5.5 Quadratic Program Formulation ..................... 86
5.6 Results ................................... 91
5.7 Conclusions ................................ 92
6 A Dynamic Programming Approach to Maximizing Lifetime of Sen-

sor Networks 95
6.1 Introduction ................................ 96
6.2 Previous Work .............................. 98
6.3 Network Model .............................. 99
6.3.1 Device Model ........................... 99
6.3.2 Link Model ............................ 100

6.4 Problem Formulation ........................... 102
6.5 Route Discovery .............................. 104
6.5.1 Bottleneck Edge Disjoint Paths ................. 106
6.5.2 Bottleneck Node Disjoint Paths ................. 106
6.5.3 Edge Disjoint Paths ....................... 106
6.5.4 Node Disjoint Paths ....................... 108

6.6 Dynamic Programming .......................... 108
6.6.1 Theoretical Background ..................... 109

, 6.6.2 Dynamic Programming Algorithm ................ 110
6.6.3 Computational Complexity of Finding Optimal Solution . . . 114

6.7 Performance Analysis ........................... 115

ix

6.7.1 Mean-Variance Trade-off ..................... 116

6.7.2 Spatial Redistribution of Energy ................. 119
6.8 Conclusions ................................ 122
Mean-Field Solution of Small-World Wireless Sensor Network Mod-
els With Range Limited Shortcuts 127
7.1 Introduction ................................ 128
7.2 Background: Small-World Networks ................... 129
7.2.1 Characteristic Path Length ................... 131
7.2.2 Clustering Coefficient ....................... 131
7.2.3 Small World, Geometric and Random Graphs ......... 132
7.3 Small-World Topology Construction Methods for Wireless Networks . 133
7.3.1 Hybrid Sensor Network ...................... 133
7.3.2 Multi-radio Network ....................... 133
7.3.3 Receiver Side Cooperation .................... 134
7.4 Mean Field Analysis of Small-World W'ireless Networks ........ 135
7.4.1 Clustering Coefficient ....................... 135
7.4.2 Characteristic Path Length ................... 139
7.5 Observations ................................ 148
7.6 Conclusions ................................ 151

Enabling Cooperative Communication and Diversity Combination in
IEEE 802.15.4 Wireless Networks Using Off-the—shelf Sensor Mote3157

8.1 Introduction ................................ 158
8.2 Related Work ............................... 161
8.3 SIMO Diversity Combining Techniques ................. 163
8.3.1 Selection Diversity ........................ 164
8.3.2 Equal Gain Diversity ....................... 166
8.3.3 Maximal Ratio Diversity ..................... 167

8.4 gPMSS Protocol ............................. 168
8.4.1 gPMSS Cluster Creation ..................... 169
8.4.2 Error-free Reception by at Least One Recipient ........ 171
8.4.3 Erroneous Reception by All Recipients ............. 172

8.5 Trace Based Proof of Concept ...................... 173
8.5.1 Experimental Setup ........................ 173
Packet Payload .......................... 174

Tiace Generation ......................... 175

8.5.2 Channel State Information .................... 176
8.5.3 Implementation Results ..................... 176
PER and PLR Analysis ..................... 178

BER Analysis ........................... 180

8.6 gPMSS Protocol Implementation .................... 181
8.7 Results and Analysis ........................... 183
8.7.1 Packet Reception Rate ...................... 183

x

 

8.7.2 Energy Per Packet ........................ 184

8.7.3 Packet Transmission Attempts .................. 188
8.8 Conclusions ................................ 189
9 Principal Component Centrality as a Measure of Node Centrality in

Communication Networks 191
9.1 Introduction ................................ 192
9.2 Background ................................ 194
9.2.1 Degree Centrality ......................... 195
9.2.2 Closeness Centrality ....................... 195
9.2.3 Betweenness Centrality ...................... 196
9.2.4 Eigenvector Centrality ...................... 196
9.2.5 The Need for a New Centrality Measure ............ 198

9.3 Principal Component Centrality ..................... 200
9.4 Evaluation ................................. 203
9.4.1 Interpretation of Eigenvalues ................... 203
9.4.2 Interpretation of Eigenvectors .................. 205
9.4.3 Graphical Interpretation of PCC ................ 208
9.4.4 Effect of Number of Features on PCC .............. 211

9.5 Conclusions ................................ 212
10 Conclusions 214
10.1 Channel Modeling ............................. 215
10.2 Network Lifetime ............................. 216
10.3 WSN Topology .............................. 218
Bibliography ..................................................... 221

xi

2.1

3.1
3.2
3.3

3.4

3.6

6.1

6.2

8.1

8.2

8.3

LIST OF TABLES

Tabular listing of relevant features of the three wireless networking
technologies under consideration for use in WSNs ............

Traffic information and statistics collected in various studies. . . . . .
State space variables, symbols and values. ...............
Collection environments of various traces in the ME trace set.

Error rates in trace sets. .........................
Cross correlation of different random processes. ............

Expected value of variance when using different combinations of R831

and LQI as CSI ...............................
Symbols and notation. ..........................

MLE parameters of marginal histograms generated from applying DPA
with BED, BND, ED and ND route discovery algorithms to 100 ran-
domly generated network topologies. ..................

Packet counts ................................

PRR of individual nodes without gPMSS, PRR with gPMSS protocol,
PRR gain for individual receivers R1, R2 and R3 due to selection
diversity, and the PRR gain due to diversity combining .........

Energy consumed by transmissions at transmitter and receiver side
per error-free received packet. Columns (1) and (2) correspond to
the baseline scenario using only retransmissions. Columns (3) and
(4) correspond to the case when only selection diversity is used by
receivers. Columns (5) and (6) corresponds to the case where a full
implementation of gPMSS is used that employs diversity combining
(equal gain or maximal—ratio) in addition with selection diversity.

xii

11

36

184

185

186

2.1

2.2

2.3

2.4

2.5

2.6

2.7

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

LIST OF FIGURES

Physical layout & network topology of WSN ............... 8
Relationship between 6DEC—CL’ 6DEC—0Ni dDEC—ON—EZE

and 6DEC—ON—S2BS .......................... 14
Slepian—VVolf coding in cluster—level communication. .......... 14

End-to—End channel between a transmitter and receiver on a multihop
wireless network. ............................. 19

E PS spent across the network for varying degrees of spatial correlation

and number of CLHs ............................ 29
E PS spent across the network for varying degrees of spatial correlation

and number of CLHs ............................ 30
E PS spent across the network for varying degrees of spatial correlation

and number of CLHs ............................ 31
Equipment setup for trace collection. .................. 40
CC2420 MAC frame format used for experiments. ........... 40
Office deployment environment. ..................... 41
Residential deployment environment. .................. 41
IEEE 802.11b and IEEE 802.15.4 channels in the ISM band. ..... 47
PDF of B conditioned on A and (I) = 1, 2 ................. 50
PDF of B conditioned on P and (I) = 1,2. ............... 51
Values of b for various (A, p) ........................ 52

xiii

3.9 The PDFs of the BER obtained from the actual traces for various LQI
measurements at an RSSI of 88dBm, pB(B[A, p = 88dBm, d) = 1, 2). . 53

3.10 The PDFs of the BER for the same range of LQI measurements at RSSI
of 88dBm as modeled by a discretized exponential PDF, 6303])“ p =
88dBm, a = 1,2). ............................. 54

3.11 Histogram of KDD values for a) Model from ‘Residential’ trace set and
‘Lab’ traces and b) Model from ‘Hallway’ trace set and ‘Outdoor’ traces. 57

4.1 Auto—correlation functions for bit level traces of the MC and ME trace
sets. .................................... 62

4.2 Auto-correlation functions for symbol level traces of the MC and ME

trace sets. ................................. 63
4.3 Auto-correlation functions for traces of the MC trace sets. ...... 66
4.4 Auto-correlation functions for traces of the ME trace sets. ...... 67

4.5 BER process of trace RIC—25 after filtering by 600 point averaging filter. 68

4.6 Plots of estimates of the Hurst parameter obtained using various tech—
niques along with their average BERs, PERs and PLRs. ....... 70

4.7 For traces MC-ll, RIC—12, MC-13, MC—14 and MC-15 each subfig-
ure, (from top to bottom): [Top] RAJ I B(1,m) of BER process ob-
served in MO traces for lag m varying from 1 through 40. [Middle]
ARM I B(1, m) of BER process for the same channel traces. [Bottom]
The memory length MB plotted as a function of (i, the increments in
RM I B(1, m). ............................... 73

4.8 For traces l\=IC-16, RIC-17, MC-18, MC-19 and MC—20 each subfig-
ure, (from top to bottom): [Top] RM I B(1,m) of BER process ob-
served in MC traces for lag m varying from 1 through 40. [Middle]
ARM I B(1, m) of BER process for the same channel traces. [Bottom]
The memory length MB plotted as a function of (5, the increments in
RMIB(1,m). ............................... 74

4.9 For traces MC-21, MC—22, MC-23, MC—24, MC-25 and MC-26 each
subfigure, (from top to bottom): [Top] RM I B(1, m) of BER process
observed in MO traces for lag m varying from 1 through 40. [Middle]
ARM I 3(1, m) of BER process for the same channel traces. [Bottom]
The memory length MB plotted as a function of (5, the increments in

RMIB(1,m). ............................... 75

xiv

4.10 For traces ME-l, ME—2, .\«IE-3, ME-4, ME-5 and ME-6 each subfig-

4.11

5.1

5.2

5.3

5.4

6.1

6.2

6.3

6.4

6.5

6.6

6.7

6.8

6.9

ure, (from top to bottom): [Top] RM I B(1,m) of BER process ob-
served in MC traces for lag m varying from 1 through 40. [Middle]
ARM 13(1, m) of BER process for the same channel traces. [Bottom]
The memory length 1MB plotted as a function of (5, the increments in

RA/{IB(1,m) ................................

For traces ME-7, l\IE-8, ME-9, l\'IE-10, ME—ll and ME—12 each sub-
figure, (from top to bottom): [Top] RAMBO, m) of BER process ob-
served in MC traces for lag m varying from 1 through 40. [Middle]
ARA! 13(1, m) of BER process for the same channel traces. [Bottom]
The memory length A! B plotted as a function of (5, the increments in

RAJIB(1,772.) ................................

The law of conservation of flow requires that the sum of incoming

flows qj z' and data Q,- generated at node n,- must equal the sum of all
9

outgoing flows (1.1-k. ............................

Tradeoff of Var[P] versus E [P] for a network with N = 10 .......
Tradeoff of Var[P] versus E [P] for a network with N = 15 .......
Tradeoff of Var[P] versus E [P] for a network with N = 20 .......
Paths from n99 to no. ..........................
N lists of routes sorted in ascending order of path energies. . . . . . .
Selection of next optimal point in the l\=’IV—plane by DPA ........

Mean-Variance tradeoffs offered by BED, BND, ED and ND paths.

Plots of percent decrease in variance against percent increase in mean.

Marginal histograms of percent increase )1 A ll E for the scatter plots in
figures 6.5a through 6.5d ..........................

Marginal histograms of percent decrease #A0'2 for the scatter plots in

E
figures 6.5a through 6.5d ..........................

Diffusion plot of energy consumption rates averaged over all 100 net—
works under SPF routing. ........................

Differential diffusion plots of energy consumption rates averaged over
100 networks using BED paths and DPA .................

XV

78

87

92

93

94

109

111

114

120

121

122

124

6.10 Differential diffusion plots of energy consumption rates averaged over
100 networks using BND paths and DPA ................. 124

6.11 Differential diffusion plots of energy consumption rates averaged over
100 networks using ED paths and DPA. ................ 125

6.12 Differential diffusion plots of energy consumption rates averaged over
100 networks using ND paths and DPA. ................ 125

7.1 Illustrated examples for three different classes of graphs; a) Geometric
graph, b) Random graph, and c) Small world graph ........... 131

7.2 Overlapping communication regions of two communicating sensor nodes
in a WSN .................................. 137

7.3 Geometry of WSN deployment and regions within it with respect to

an individual sensor a,- ........................... 141

7.4 Plots of clustering coefficient C [left] and characteristic path length L,
[right] as functions of ,u. for different values of R. Network parameters
that remain fixed are A = 10000, kglobal = 4, p = 10, g = 3 and nc = 1.148
7.5 Plots clustering coefficient C [left] and characteristic path length L,
[right] as a function of ,a for different. ratios of global scale link to local
scale link communication range 5. Network parameters remain fixed
at A = 10000, kglobal = 4, ,0 = 10, ”C = 1 and R = 4 .......... 149
7.6 Plots clustering coefficient C [left] and characteristic path length L,
[right] as a function of ,u for different values of kglobal' Network pa-

rameters remain fixed at A = 10000, p = 10, E = 3, R = 3 and nc = 3. 150
7.7 Graphical representation of integration term of d(F(v)) ......... 155

8.1 Application of generalized gPMSS in a wireless sensor network with
mesh topology. Path from transmitter T to receiver R1 marks the mul—
tihop path that would be taken in a network without gPMSS. Dashed
line links between T and receivers R1, R2 and R3 denote the longer

range but high loss links that are used under Generalized gPMSS. . . 160

8.2 Illustration of logical functioning of selection diversity. ........ 168

8.3 Illustration of logical functioning of equal gain diversity ......... 168

8.4 Illustration of logical functioning of maximal-ratio gain diversity. . . . 169

8.5 gPMSS protocol operations. ....................... 170
xvi

8.6

8.7

8.8

8.9

8.10

8.11

8.12

8.13

8.14

9.1

9.2

9.3

9.4

9.5

Equipment setup for trace collection. ..................
CC2420 MAC frame format used for experiments. ...........

PDF of BER experienced by receivers R1, R2 and R3 (p 308 = 0) is
cropped out for better view of non—zero range. .............

PDF of LQI experienced by receivers R1, R2 and R3 ..........
PDF of RSSI experienced by receivers R1, R2 and R3. ........

PER, PLR and PER+PLR experienced by receivers R1, R2 and R3
without gPMSS diversity combining and with selection, equal gain,
and maximal ratio diversity combining ..................

Histogram of BERs observed by receivers R1, R2 and R3 without
gPMSS diversity combining and with selection, equal gain, and maxi-
mal ratio diversity combining. ......................

The energy in )aJ consumed by transmitter and receivers per success-
fully delivered packet ............................

Maximum number of transmission attempts m versus delivery guaran-
tee g(%) ...................................

This figure shows a graph on the lower plane, overlayed with another
plane of the interpolated surface plot of node centrality scores. The
centrality planes typically exhibit a number of peaks or local maxima.

A spatial graph of 200 nodes. Node colors are indicative of the range
in which their EVC falls ..........................

[Top] Histogram of eigenvalues of adjacency matrix and Laplacian ma-
trix A of network in figure 9.2; [Bottom] Cumulative sum of the se-
quence of eigenvalues of adjacency matrix and Laplacian matrix of
network in figure 9.2 when sorted in descending order of magnitudes.
In both figures the lines plotted in red color are averages of 50 networks
generated randomly with the same parameters. ............

Reconstructed topologies of the graph from figure 9.2 using only the
first 1, 2, 3, 5, 10, 15, 50 and all 200 eigenvectors. ...........

Spectral drawing of graph in three dimensions using entries of x1, x2,

and X3 for the three coordinate axes. Nodes are colored according to
their C15 FCC ...............................

xvii

173

174

177

178

179

180

181

185

188

193

197

200

201

205

9.6

9.7

9.8

9.9

9.10

PCC of nodes in network of figure 9.2 when computed using first (a)
l and (b) 2 eigenvectors. The histograms accompanying each graph
plot show the distribution of PCC of their nodes. The lineplot in
the histogram represents the average PCC histograms of 50 randomly
generated networks with the same parameters as the network in figure
9.2 ......................................

PCC of nodes in network of figure 9.2 when computed using first (a)
3 and (b) 5 eigenvectors. The histograms accompanying each graph
plot show the distribution of PCC of their nodes. The lineplot in
the histogram represents the average PCC histograms of 50 randomly
generated networks with the same parameters as the network in figure

9.2 ......................................

PCC of nodes in network of figure 9.2 when computed using first (a)
10 and (b) 15 eigenvectors. The histograms accompanying each graph
plot Show the distribution of PCC of their nodes. The lineplot in
the histogram represents the average PCC histograms of 50 randomly
generated networks with the same parameters as the network in figure

9.2 ......................................

PCC of nodes in network of figure 9.2 when computed using first (a)
50 and (b) all 200 eigenvectors. The histograms accompanying each
graph plot show the distribution of PCC of their nodes .........

Plot of phase angles (1) (in radians) of PCC vectors with the EVC vector
for the graph in figures 9.6, 9.7, 9.8 and 9.9. ..............

xviii

207

208

209

Chapter 1

Introduction

1.1 Organization

This research characterizes, analytically and quantitatively, the bit error rate (BER)
process, the network lifetime and the topological preperties of wireless sensor net-
works (WSN) [2], [59]. The following section describes the contributions made in this

dissertation.

1.2 Overview of Contributions

Chapter 2 provides an analytical model for the end-to-end capacity between nodes in
a W SN and the base station. This study distinguishes itself from previous studies of
wireless network capacity in that it assumes a many-to—one data flow (in place of an
all-to-all flow) producing what is called the funneling or reachback effect. The model
also takes into account in-network data fusion/ compression. We consider scenarios
in which there is a) no compression, b) opportunistic compression and c) perfect com-
pression. The model shows that the distribution of power consumption rates of nodes
is skewed due to the uneven traffic load flowing through nodes. Power consumption
is lessened in systems in which sensor measurements are highly correlated and take
advantage of it by employing aggressive data compression like Slepian-Wolf coding
[114].

Chapters 3 and 4 relate to modeling of the BER process in IEEE 802.15.4 chan-
nels low-rate wireless personal area networks (LR-VV PAN). Chapter 3 describes the
experimental setup used to collect a large set of residual packet traces. These traces
distinguish themselves from traces collected by other studies in that they also cap-
I3111‘e channel state information (CSI) measurements alongside every packet and events
such as packet losses and truncated packets. The traces are used to establish the de-

31‘ 66 of correlation between CSI measurements and the BER process that packets

are subjected to. After establishing the usefulness of CSI for estimating BER, we
develop CSI-driven models of the BER process for sets of traces collected in different
environments. Divergence measures between the models derived from different traces
show that there is very small average divergence between different models leading
to a CSI-driven BER model that is independent of physical environment. Thus CSI
measurements of traffic received over a particular link can be used to estimate the
BER and assign it a link cost. Chapter 4 analyzes the memory properties of errors
in the traces. Correlation based analysis on bit and symbol level traces reveals a
constant memory length of 2 bits and 2 symbols, respectively. Furthermore, we high-
light the inability of correlation analysis to provide a clear measure of memory length
of the BER process when a channel is subjected to periodic interference and arbi-
trary selection of a significance threshold for the correlation function. To address this
shortcoming of the correlation function based analysis we introduce relative mutual
information (RMI) [56], a normalized form of Shannon’s mutual information (MI)
[32] that provides a more robust means of measuring memory length.

Chapters 5 and 6 contain our work on the network lifetime problem in WSNs.
The lifetime of a WSN is typically defined as the time until the first node runs out of
power. The corresponding optimization problem can formulated as a minimax linear
program (LP) ([18]). However, the vision of the future of WSN calls for a network
consisting of a very large number of devices that are inexpensive and redundant. The
fact that WSN applications by their very nature impose a many-to-one data flow
that produces a funneling effect (see Milgram [82]) leads to large variations in the
Power consumption rates in nodes. Chapter 5 proposes the redistribution of traffic
rOuted to a VVSN’s base station by the joint minimization of the mean and variance
0f power consumption rates. Such a statistical objective function cannot be held
hOstage by the life of an individual sensor but takes a more global view of network

lifGtime. This chapter provides a formulation of the optimization problem in the

3

form of a quadratic program (QP) ([18]). The quadratic nature of the optimization
problem puts restrictions on the scalability of the QP approach. Chapter 6 provides
an approximate formulation of the lifetime problem in the form of a dynamic program
(DP) ([18]). Experiments show that in W SN as large as 100 nodes there is a visible
difference in the rate of reduction of variance and rate of increase increase in mean of
power consumption rates as shorter routes are traded-off in favor of routes that are
less efficient in the greedy, shortest-path—first (SPF) sense.

Chapters 7, 8 and 9 describe topological properties of WSNs, particularly the
small-world effect. Chapter 7 describes the place small-worlds networks [128] oc-
cupy on the spectrum of networks with node connections varying between order and
randomness. In this chapter we describe the potential benefits of having wireless
networks with small-world topologies and previous approaches to achieve this goal.
Small-world networks are characterized by a phase difference between the drops in
characteristic path length and clustering coefficient of networks as the fraction of
links in the network that are shortcuts is increased. Prior models of characteristic
path length and clustering coefficient in various types of graphs did not consider the
addition of range limited links. This chapter presents analytical models for both
these quantities for graph topologies that are subject to the constraints of wireless
networks. The model developed in this chapter can be parameterized to accommo—
date any of the existing approaches to building small—worlds in WSNs. Chapter 8
describes a method based on single-input multiple-output (SIMO) principles, called
the generalized ‘Poor-Man’s-SIMO-System’ (gPMSS), a practical implementation of
Shortcut links in WSNs by leveraging cooperative communication and diversity combi-
nation principles. The diversity combining methods explored include selection diver-
Sity, equal gain and maximal-ratio gain diversity combining, where the weight factors
Used in maximal-ratio combining are provided by the CSI-driven channel model of

the BER developed earlier in chapter 3. gPMSS enables the construction of small-

4

world networks in wireless networks by alternative means. Its chief advantage over
existing approaches lies in the fact that it is not based on any modifications to the
hardware of sensor motes. Proof of concept is provided by performing detailed anal-
ysis on a set of wireless channel packet error traces and also implemented on the
Crossbow Imote2 sensor mote. The work on gPMSS is also relevant to the research
on the network lifetime problem. Chapter 9 attempts to identify the most central
nodes/regions in a wireless network. It highlights the problems demonstrates the
shortcoming of pre-existing centrality measures (degree, closeness, betweenness and
eigenvector centrality). A new measure of node centrality called principal component
centrality (PCC) is proposed which addresses the shortcoming of eigenvector central-
ity in identifying well connected hubs within graphs characteristic of wireless mobile
ad-hoc networks and WSNs. The hubs in a network identified by PCC can be used
for the placement of shared network resources, e. g. endpoints of shortcut links.

Finally, chapter 10 concludes this dissertation by summing up our finding.

Chapter 2

Network Channel Capacity of
IEEE 802.15.4 Wireless Sensor
Networks Under Reachback

2.1 Introduction

Over the past several years different assumptions have been made about the structure
and capabilities of wireless sensor networks (WSN) and the devices they are consti-
tuted of. In [43] Gupta and Kumar studied the scalability of wireless networks with
randomly chosen source destination pairs. Their conclusions offer two solutions to the
scalability problem; 1) Design smaller networks, or 2) localize communication by clus-
tering nodes. The idea of a WSN consisting of homogeneous devices gradually gave
way to that of a network consisting of a homogeneous mix of nodes with non-uniform
device capabilities. In this newly emerged view of W SNs, sensors are grouped into
clusters. Each cluster of sensor nodes elects from among its members a clusterhead
I10de (CLH) that acts as a gateway for all incoming and outgoing communications.
Consequently, WSNs can be thought of as hierarchical networks with 5 levels.
However, since most previous work on WSNS distinguishes only between two roles
for devices, sensors and CLHs, we assume a two-level hierarchy with s = 2. This
assumption is also supported by the IEEE 802.15.4 low rate-wireless personal area
networks (LR-VVPAN) draft standard. The standard refers to a hardware device with
ITlOre resources with a more complex implementation as a full function device (FFD)
and a device with fewer resources and a simpler, less expensive implementation as a
r educed function device (RFD). For the remainder of this chapter we will assume a
VVSN built out of devices compliant with the IEEE 802.15.4 physical layer (PHY)/

Inedium access control (MAC) specifications.

Level 1, the lower level, refers to the network formed by a CLH and its associated
Sensors. Typically, all devices within a cluster are capable of communicating with
ea~Ch other directly. Therefore, communication between sensors and their CLH are
a'SSurned to take place over a single hop. We will use the terms cluster, intra-cluster

or Cluster-level communication interchangeably to refer to the exchange of messages

7

10

x1
x

x

x

 

 

O 2 4 6 8 10

Figure 2.1: Physical layout & network topology of W SN.

between nodes belonging to the same cluster. Communication within a particular
Cluster proceeds at a common frequency, i.e. there is a potential for interference
between transmissions of different sensors of a cluster. Different clusters may or may
not use differing frequencies. l\"‘Ioreover, the network topology within a cluster is

r eStricted to a star topology with the CLH at the center.

Level 2, the upper layer, refers to the network formed by the CLHs of all clusters
in the WSN and the base station. We will also refer to this network as the overlay net-
WOrk (ON). We assume that the CLHs participating in the ON are capable of routing
and relaying their own and other clusters’ packets towards the base station. More-
Over, since the most widely used WSN routing algorithms like destination-sequenced
disfiance vector (DSDV) [99], directed source routing (DSR) [61], ad—hoc on-demand

distance vector (AODV) [100], directed diffusion [58] etc., are different. forms of short-

8

est path routing algorithms, we assume that at any given time the routes from CLHs

to base station in the ON form a tree rooted at the base station. We are discounting

the possibility of using bifurcated routing, i.e. multiple paths from source to desti-
nation. We will use the terms overlay, overlay network or CLH-level communication
interchangeably to refer to the exchange of messages between nodes belonging to
the ON. Moreover, it is assumed that message exchanges between CLHs in the ON
take place at one frequency, i.e. like sensors in a cluster, CLHs in the ON have the
potential to produce interference for each other. However, this frequency channel is
assumed to be free of interference from cluster-level communication. Moreover, the
topology of the ON is a mesh topology.

As 2.1 shows, as traffic generated by the CLHs situated farther away approaches
the base station, the expected volume of traffic carried by a link increases. This
leads to a capacity bottleneck around the base station that subsequently limits the
rate at which CLHs, and ultimately sensors, can inject data into the network. This is
Called the reachback problem [9]. Thus source coding is used to alleviate the effects of
the reachback channel. Besides entropy coding methods, a very popular compression
nlethod in WSNs is Slepian-VVolf coding [114] which assumes prior knowledge of the

degree of correlation between sensors.

2 . 2 System Model

2~2.1 Wireless Networking Standards for WSNs

In Our attempt to formulate a generalized expression for the end-to-end capacity of a
Channel between an arbitrary sensor and the base station we remain open to the pos-
Sibility of a number of different wireless networking technologies. Besides proprietary

radio interfaces, the most commonly encountered standardized wireless networking

technologies found in early implementations of WSNs are the IEEE 802.11x wireless
local area network (WLAN) standard [51], the IEEE 802.15.1/ Bluetooth standard [4]
and the IEEE 802.15.4 standard [5]. The distinguishing features of these three net-
working standards are many. Table 2.1 lists only the features that we were concerned
with in our work, i.e. the operating frequencies and the types of MAC protocols.
As we will show in a later section, our interpretation of how the end-to-end capac-
ity of a wireless channel can be computed requires us to assume a pathloss model to
model the physical channel. Pathlosses depend on numerous environmental factors
whose effects are generally too complicated to predict. Two parameters that influ-
ence pathlosses in a major way are the frequency and transmitter-receiver separation.
We assume that all transmissions are taking place in the 2.4 — 2.4835GHz industrial
scientific and medical (ISM) band which is used in all three wireless networking stan-
dards under consideration. This will allow us to use a single pathloss model that will
be applicable to all three networking standards under consideration. Therefore, our
end-to—end channel capacity expression will not be applicable to WSNs using IEEE
802.15.4 networks operating in the 868—868.6M H z or 902—928M H 2 bands. In addi-
tiOn, we are allowing for both collision avoiding carrier sense medium access/ collision
aVOidance (CSMA/ CA) and collision—free time division multiple access(TDMA), type
IVIAC protocols. The choice of the MAC protocol mode will affect parameters in the
model.
Let N be the total number of sensors in a WSN of M clusters. Let Cit/2' E
{1, 2,3, . . . , M} denote an individual cluster consisting of one CLH and N,- sensors
(therefore; N = 2911 Ni): Sensors are addressed using a 2-dimensional address
SCheme in which the ith cluster’s jth sensor node is labeled ni(j)Vi E {1, 2, 3, . . . , M},
VJ E {1,2,3, . . . ,Ni}, with viz-(0) denoting its CLH, and n0(0) denoting the base
StEltion. We also define a frequency function f (n,- (j )) that returns the frequency at

‘which the device passed in the argument communicates within a cluster. This way

10

Table 2.1: Tabular listing of relevant features of the three wireless networking tech-
nologies under consideration for use in W SNs.

 

 

Feature IEEE 802.15.4 IEEE 802.11b IEEE 802.15.1/
Bluetooth
Frequencies 2.4-2.4835 GHz 2.4-2.4835 GHz
1. W
2. 992—9218544459

3. 2.4-2.4835GHz

 

 

 

 

 

 

A-lACtype Polling
1. TDMA in GT8 1. CSMA/CA in
, DCF
2. CSMA/CA in
CTS 2. Polling in
PCF
Topologies Meshed Star Star Star/Meshed Star
(DCF) /Meshed (Piconet)
Star (PCF)
\
11

f (712(0)) denotes the intra—cluster communication frequency of cluster Ci- We use
fi‘v’i E {1, 2, 3, . . . ,M} as a shortened form to denote the cluster frequencies of all M
clusters. Similarly, we abbreviate f (n0(0)) by f0 to denote the frequency used by
CLHs for communicating in the ON. The probability of n k(l) making an interfering
transmission at the same time as n.,-(j) is making its transmission is denoted by

an (l) ("i(j))- We also define a frequency indicator function in equation 2.1;

0; f (ndflHWUi) . (2.1)

I ('rti(j),nk(l)) =
f 1; f (712:ij = f (7119(1))

\Nith the definitions of symbols describing the network in place we now come to
the terms that will quantize the channel conditions, i.e. capacity and probability of
error terms. First off we define the probability of a bit error or the bit error rate
(BER) denoted by p. The 802.15.4 standard uses binary phase shift keying (BPSK)
Illodulation at 20 and 40kbps and offset-quaternary phase shift keying (O—QPSK)
mOdulation at 250kbps. Both BPSK and QPSK modulators make hard decisions
ba-Sed on the received symbol and output either a ’0’ or a ’1’. Therefore, assuming
an additive white Gaussian noise (AWGN) channel both BPSK and QPSK receivers
Can be modeled by binary symmetric channel (BSC) with probability of error p. p is
defined as the fraction of transmitted bits that are received without errors. Hence, this

term models the channel between the Physical layers of the 081 model transmitter

and receiver.

_ # of bits received without errors (2 2)
# of bits transmitted '

Frames in 802.15.4 are protected only by a 16 bit frame check sequence (FCS)
With no error correction capability. This implies that whenever even a single bit in an

entire frame is received in error the receiver discards the entire frame. Packetization

12

affects net throughput. We denote the fraction of transmitted bits that are received
without errors by 6. 6 will be referred to as the packet error rate (PER). Since packets
are either received correctly and forwarded to the next higher layer or discarded due
to F CS failure, the PER is equivalent to the probability of error 6 of a binary erasure
channel (BEC). A BBC is used to model the channel between the MAC Layers of the

081 model transmitter and receiver.

_ #z of packets received without errors
— # of packets transmitted

 

a (2.3)

Finally, since we are assuming the use of Slepian-VVolf coding we have take into
account the dependencies that introduced in data streams. As a result of these
dependencies packets that will be received error free will at times not be decodable due
to losses in transmissions from other sources. As a result, packets that are received
free of errors can end up being undecodable due errors in another transmission it
(hipends on. Therefore, we define the PER with SIepian—VVolf coding as the fraction

Of transmitted packets that are decodable by the receiver denoted by 6 D EC'

# of decodable packets received

" = 2.4
ODEC # of packets transmitted ( )

 

p, 6 and (5 D EC for links within a cluster are denoted by pCL’ 6C L and (5 D EC—C L'
Sirnilarly, the corresponding quantities for links on the ON are abbreviated by pON’
50 N and 6 D EC—O N: Since, the ON channel between an arbitrary CLH and the
baSe station may comprise of multiple hops we need to define additional quantities for
the end-to—end channel in the ON pON—E2Ei 60N—E2E and 5DEC—ON-E2E'
FiIlally, the channel formed between each individual sensor and the base station

IS Characterized by a combination of the above computed values and denoted by

pON—S2BS’ 60N—S2BS and 6DEC—ON—S2BS' To clarify, figure 2.2 shows the
relationship between 5DEC—CLi (Sosa—0N: 5DEC—ON—E2E

13

6 6DEC—CL
DEC—ON—E2E

 

    

 

6DEC—ON—S2BS
* Base station I Clusterhead . Sensor

Figure 2.2: Relationship between 6DEC—CL3 (EDEC—ON’ éDEC—ON-E2E and
5DEC-0N—S2Bs-

H(X1|X2) H(X3|X2, X1)
0 O O --- O

\

H(Xn|Xn-1, ,X1)

H(X1)

V

   
 

Figure 2.3: Slepian-Wolf coding in cluster-level communication.

and 5DEC-ON—SQBS'

14

 

2.2.2 Source &: Channel Coding

\Ne consider three options for source coding; 1) No source coding, 2) opportunistic
compression [71], and 3) Slepian- Wolf coding. The first option does not perform
any compression at all. Data is merely collected at each CLH, concatenated, packe-
tized and routed to the base station. Opportunistic Compression assumes correlation
between different sensor readings. Each CLH compresses data it receives from down-
stream based on other data available to it from other sources. Slepian-Wolf coding
was first proposed by Slepian and Wolf in [114]. However, we make some simplifying
assumptions about Slepian-Wolf coding as it applies to W SNs. These assumptions
have been taken from Marco and Neuhoff in [80] and [81]. Slepian—Wolf coding also ex-
plOits the spatial correlation of sensor readings in neighboring sensors and is employed
as a. means to compress data before transmitting it to the base station. Slepian-Wolf
COding is based on equation 2.5. However, Slepian-Wolf assumes that the degree
Of correlation is known prior to transmission, whereas opportunistic compression as-
Sumes no prior knowledge. Consider a cluster i as in figure 2.3 consisting of N, sensors
and a CLH in which the first sensor 711(1) produces reading X Ilifl)’ the second sensor

712' (2) produces reading X , 2 and so on.
”A )

(2.5)

The size of all N,- sensor readings after lossless compression is lower-bounded by

t1leir joint entropy. As an example, Slepian-Wolf coding within a cluster proceeds as

fOllows.

1. Sensors transmit their readings to their CLH. For the purpose of simplifying

15

analytical expressions, let us assume that the transmissions to the CLH are

scheduled in the ascending order of sensor nodes’ ID numbers.

2- The first transmission Xn'(1) by nz-(l) is not compressed (see first term on
2,

left-hand side of ( 1))

3. The second sensor nl-(2) compresses its sensor reading to H (Xni(2)[Xni(1))

based on the side information of n,- (1)’s transmission of size H (X (1)). There-

ft

rni

f a 't d ‘ 1 't I t»- -. t: 't at (X . .

ore, 1e ] no e in cus er 2 ransnn s IS '1 a as H ni(9)IXni(]‘1)

"Xn-(1)) bits. This way the total volume of all transmissions approaches
2

the joint entropy as shown in (1) and also depicted in figure 2.3.

However, this coding scheme has one major disadvantage. While the savings
in transmissions can be substantial, depending on the spatial distribution of the
phenomena being sensed, the failure of the CLH to receive the kth transmission
results in its inability to reconstruct all subsequent transmissions k + 1 through N,-
for that round. The high sensitivity of a receiver’s ability to decode packets to packet
losses, and the fact that bit and packet error rates in single hop wireless networks are
Orders of magnitude higher than those in multi-hop wired networks makes the use of
IV'IAC layer channel codes imperative. Even without any dependencies between data
Streams the BERs of end-to-end channels representative of multi-hop paths in wireless
n(itworks can often times be large enough to impose unrealistically large overheads
On end-to—end channel codes.

The PERs of various links will depend on the size of packets. When Opportunistic
Compression or Slepian-Wolf coding are employed that will depend on the degree of
COrrelation in data readings. To model the lossless compression of data and determine
the message sizes Lm (n, ( j )) we use the model proposed by Pattern, Krishnamachari
a11d Govindan in [97]. The use of Slepian-Wolf coding forces an order on the devices in

the WSN which is also described in detail in [97]. The number assigned to a particular

16

device ni(j) in this order is returned by the order function O(-) which returns a value

in the range 1 S O (ni(j)) S N.

2.2.3 Power Consumption Model

The total energy E (n,(j)) consumed in mote ni(j) in equation 2.6 is modeled
as the sum of energies consumed in communication ET X (n,- (J)) and processing

EPR (7120))-

E ("Nil = ECOMM (MUD + EPR (”Mil - (2-6)

ECO M M (ni(j)) in equation 2.7 can be further broken down into energy con-
sumed in transmission ET X (ni(j)) and reception E R X (nZ-(j)).

ECOMM (in (1')) = ETX (”103) + ERX W31) - (2-7)

The total energy spent by all devices in a network divided by the uncompressed
number of bits of data communicated to the base station yields the energy per symbol

(EPS) of the network configuration in equation 2.8.

Ni
2531 531:1 Enifj)
N,- '
2,1312,le <an0))

The processing cost incorporates the cost of performing source and channel coding

EPS =

 

(2.8)

and decoding operations encoding and decoding. Since several algorithms of varying
degrees of complexity are available for both source and channel coding we model the
COmplexity of these operations at the transmitter and receiver by terms cngn1 and

(21,332, respectively. The cost is dependent on the message size L-m on which the

Operations are performed.

17

2. 3 Channel Model

In this section we derive a channel model for the end—to—end channel between a trans-
mitter and receiver communicating in a multi-hop 802.15.4 wireless network. The
first step consists of identifying a suitable pathloss model for the 2.4 - 2.4835CH2
IS‘NI band. We use the model put forward by the physical channel modeling sub-
group of the IEEE 802.15 taskgroup 4 in [86] for this purpose. Figure 2.4 depicts
a transmitter and receiver communicating over a multi-hop wireless channel. The
transmitter and receiver are depicted by the two protocol stacks of the Open Sys-
tems Interconnect (081) model. At the Physical layer of the 081 model we assume
a discrete memoryless channel (DMC). The pathloss model is an abstraction of the
Physical channel between a single transmitter and receiver. In terms of the 081
model, the pathloss model represents an abstraction of everything that falls under
the Physical layer. From the pathloss model we can determine a bit error model that -
Will be representative of everything below the data-link layer. The ultimate goal here
iS to determine a model that is capable of abstracting everything down from layer 5
0f the 081 model. In the next step, we add another layer of abstraction to the bit
error model by using it to obtain a packet error model. The packet error model is an
abstraction of the network stack from the Network layer down. Note that the packet
eI‘ror model can serve as an end-to—end model in single-hop networks. However, for
IIlulti-hop networks such as the ones we are considering we will have to modify the
packet error model to obtain the desirable end—to—end packet error model that will
abstract everything from the session layer down. This successive abstraction of the
Channel between transmitter and receiver based on the next lower model is depicted
in figure 2.4. The layers of the 081 model inside a dashed box represent the layers

encapsulated in the corresponding model.

18

Transmitter Relay Receiver

 

 

Higher

 

 

 

 

Lay” $323;
(Applifatlon (Application+
' Presentation Packet Error Model
Presentation . .
_ +Session) _ _ _ _ _L _ _ _ +Sessmn) (Elt'd-to-End)
Transport Transport
Network

    

 

 

 

 

 

 

 

 

 

 

 

Pathloss Model Packet Error Model Bit Error Model
(1 Hop)

Figure 2.4: End-to—End channel between a transmitter and receiver on a multihop
Wireless network.

2.3.1 1-Hop Communication

The expressions we derive in this section are equally applicable to all l-hop commu—
Ilication links in the WSN, irrespective of whether they are in the ON or cluster.
Therefore, we will refrain from using the CL and ON at the end of the subscripts
0f the sought quantities, i.e. received signal power, SINR. BER and PER. From the
pathloss model in [86] we can obtain the expression in equation 2.9 to obtain the

Dower P (Viz-(j) —) nk(l)) of a signal transmitted by ni(j) at receiver nk(l).

19

P (WU) —> "19(0) =1f (f (nail) ,f (Tl/ill)» KOPTX—amp"lTX-a72.t"lRX—a'n.t
PLO

X (d(”z‘(j(]6nk(l)))2 (79210.2

 

 

(2.9)
Here, PTX—a-mp (typically 17an for IEEE 802.15.4 compliant devices) is the
Signal power at the transmitter after amplification before it is passed to the antenna,
"TX—ant is the transmitter antenna efficiency. 77 R X—ant is the receiver antenna
Efficiency and K, PLO, and K0 are environmental parameters that depends on the
Operating environment (see [86]). Since we are assuming that a WSN consists of de-
vices with identical radio interfaces transmitting at a fixed power we consider these
terms to be constant across the entire network. We also define two reference param-
eters fc and do for this pathloss model. For the sets of parameters provided in [86],
f c is set to 5GHz and do is set to 1m. This leaves the expression dependent on the
transmission frequency f and the distance between devices 72.,(3') and n k(l) that is
rGturned by the distance function d (n,(j), nk(l)).
In order to obtain the BER for a DMC we need an expression for the signal-to-
interference 8 noise ratio (SINR). The general expression for the SINR is given in
equation 2.10. Using the expression for the pathloss model described above we can

arrive at an expression for the SINR in the terms of known quantities.

 

PTX
SINR = , (2.10)
PA 'I' Z Pint

where PT X is the power of the received signal for which the SIN R is being com-
puted and Pint is the power of interfering signals caused by undesired concurrent

transmissions elsewhere in the network. P A is the ambient noise power of interfer-

20

ence produced by sources that are not part of the WSN and are not modeled by
any terms in Pint: Sources of ambient noise power may include but are not limited
to other networks that are co—located with the WSN under consideration [136] or
devices or appliances (microwave ovens, cordless phones) that operate in the same
frequency band. Since at the cluster-level all transmissions are directed from sensors
to their respective CLH, the term S I NR (til-(j) —-> n All» represents the SINR of the
transmission from nZ-(j) to n k(l). In the general case, all sensors and CLHs can be
considered potential sources of interference. Obviously, we assign qni (j) (712(3)) 2 0.
The interference power 23 Pint in equation 2.10 is the sum of all other signals that
are received originating from sources other than n11 ( j). This term is computed by
Summing the interference power of all sensors. In order for nk(l) to contribute to
the power of the interference signal for a transmission originating from n,,-(j), nk(l)
must be 1) operating in the same frequency band and 2) transmitting at the same
time as n.z'(j). Transmitters operating at different frequencies make effectively no
contribution to interference. Therefore, the indicator function defined earlier is used.
The probability that a sensor will produce an interfering signal at the same time as

712(3) transmits is accounted for by multiplying the term further by anm (”2'ij

SINR (711(7) —-) 7tk(l))
If (n,(j),nk(l)P (n,,j(j) —> nk(l)))

PA + 2:2le 23:0 Ifrnaj). ff'nnifillllqnmm)(WUNP (nme) a nae)’

(2.11)

Then equation 2.11 is the final expression for SINR(ni(j) —) nk(l)). In the
Ilext step we use the SINR and Q-function to obtain the probability of error for a

BSC. The use of the Q—function to arrive at the BER using the SINR is described

21

by Rappaport in [104]. If Q(a: =\/12_7? ffo. eIZLdu, then the probability of receiving
a bit error at nA(l) in a transmission originating at nZ-(j) is called the bit error rate
(BER) p(n. -]( ) ——> nA(l)). IEEE 802.15.4 uses two different modulation techniques.
802.15.4 uses Binary Phase-Shift Keying (BPSK) at the 20 and 40hbps data rates
in the 868 — 868.6MHz and 902 — 928MHz bands, respectively. At 250kbps in
the 2.400 — 2.4835GHz band it uses offset-quadrature phase-shift keying (O-QPSK).
In either case the BER is obtained by using the Q—function. Equation 2.12 shows
the relationship between SIN R and BER for a QPSK receiver. Equation 2.13 is the
corresponding equation for a BPSK receiver. This can be thought of as the probability
Of error in a binary symmetric channel (BSC). The corresponding channel capacity
in terms of the BER is obtained from equation 2.14, where Hb(-) is a function that

returns the entropy of a Bernoulli random variable with the parameter provided in

the argument.

 

p(n,'(j) —+ nA(l =Q(\/ZSIR( we )—> nA(z)))

 

 

 

_u (2.12)
= ))_/00 e Ydu.
W27? \/2SINR(nl-(j)——>nA(l))
p(n,~(j) ——> nA.(l)) =Q(n(\/SINR ,(j )—+ nA(l)))
_u (2.13)
= —/00 e gdu
fl? \/SINR(nl-(j)——>nA.(l))
CBERln Z-j()——>nA.(l))=1-—Hb(n i(j)—+nA(l)) (2.14)

From the BER we now determine expression 2.17 for obtaining the probability
of a packet loss or the packet error rate (PER) 6 (‘71.,(3') —+ nA(l)) for a transmission
fI‘om n ,(j) to nA(l). The PER corresponds to the probability of error of a binary
eI‘asure channel (BEC). We assume that witl'iout channel coding a received packet is

discarded if a single bit is in error, a valid assumption considering our choice of wireless

22

standards. L h is the per sample header size in bits. The size of the sample making up
the message depends, of course, on the entropy reduction method employed. When no
source coding is used Lm (n3(t)) is simply [ H (X 7115(0)] , where H () is the entropy
function. When opportunistic compression is used the size of a message originating
at n3(t) is Lm(n3(t)) is H [H (Xn5(t)|Xn5-(1)"'"X715(t——1))-|' Fiom 2.15 we
trivially obtain expression 2.16 for the channel capacity C p E R (n2(j) —> nA(l)) in
terms of 6 (72.10) ——> nA(l)). Note that for the case of Slepian-Wolf coding, 2.15 and
2.16 do not yet take into account the additional losses due to a receiver’s inability
to decode a received message caused by losses in transmissions of other messages the

received message is dependent on.

5 (71,10) —-> nA(l)) = 1 — [1 — p (12,-(3') —> nA.(l))]Lh+le("5(‘))l (2.15)

CPER("i(j)_"nk(l)) =1— p (Til-(j) —> nA(l)) (2.16)

Now we consider the case where Slepian-Wolf coding is employed and a further
expression for the PER of a message can be obtained which takes into account losses
in streams that the receiver depends on to decode the message. Since cluster-level
communication is single hop only and the ON is essentially a multi—hop network the
expressions for (5 D EC—C L and (5 D EC—ON are significantly different and we will

depart from our practice of finding general expressions that we followed to this point.

6DEC—CL (me) —+ 71210)) = 1— [1— 22 ("1(1) —> nA:(0))] Lh+Lm (2.17)

23

where,

W (”2‘0” 2 lH (Xnmfl/‘ll
A=XA,va:1gagO(n,,(j)) (2.18)

CPER-SW—CL (”2'03 -> nzt(0>) = 1 - 5DEC—CL (7120) -> MUD)
2.3.2 Overlay Network Communication

We now turn our attention to the end—to-end capacity of the channel between an
arbitrary CLH and the base station communicating over a multi—hop wireless network.
The case in which all CLHs are directly communicating with the base station in the
ON becomes a special case of the more general case of a multi—hop ON. Like for the
l-hop case, we start from the expression for SINR, this time for a signal transmitted
by a CLH TIA-(0) to its upstream neighbor. This is given in 2.19.

Before proceeding further we define a set of new functions that will subsequently
be used in this section. lining-(0)) returns the immediate upstream neighbor of CLH
nZ-(O), where upstream denotes the direction towards the base station in the network
topology. R1i(n.i(0)) returns the set of CLHs that is 1 hop downstream from 711(0),
where the term downstream refers to the direction away from the base station in

the network topology. RT(n.A(0)) returns the set of all CLHs that are upstream from

nA-(O). Similarly, R‘L(ni(0)) returns the set of all CLHs that are downstream from

712(0).

 

24

Note that for a TDMA protocol in the ON anm) = 0V1: , and hence 2.19 simplifies
to 2.20.

P (”110) —+ Elfin-MOD)

SINRON(nA(0)) = PON A

 

(2.20)

Applying the Q—function leads us to similar expressions 2.21 and 2.22 for the BER

as before.

 

pON(n,-(0)) = Q (\/2SINRON(nz-(0)))

 

 

 

1 00 _
= —— e 2 u
«5? /\/2SINRON(n,-(0)) d (2'21)
pow-x0» = Q (\/SINRON(nA-<0)>)
1 00 _u
: _ e 2 A
m /\/51N120N(n,(0)) d (2'22)

From 2.21 and 2.22 we can obtain equation 2.23 a recursive definition for the BER

of the multi-hop, end-to—end channel between a CLH and the base station.

pomggEmA-w» =PON("2‘(0)) [1— mum (R1T<n.<0>>)]

+pON—E2E (R1T(n,§(0))) [1 — p0,7\r(n,-(0))] . (2.23)

The expression for the PER for a packet originating at 7227(0) on a link between

nA(0) and its upstream neighbor R1T(nA(0)) is then provided by equation 2.24.

25

60N_Az.k<0)<nA-(0>> = 1— (1 - meme»)LhtLMWW”, (2.24)

where (50N_nk(0)(nj(0)) = 1 if ‘nA(0) (A? RT(n]-(0)) and the message size origi-

nating from nA-(O) is,

[Vi
LmrnA-(on = Z Hm). (2%)
i=0

This leads us to the final expression 2.26 for the end-to-end PER from an arbitrary
CLH ni(0) to the base station. Equation 2.27 gives the corresponding expression for

the end—to-end capacity.

(SON__E2E(n,-(O)) = 1 — an(0)ERT(ni (0)) (1 — 6ON__nkA0)(n,-(O))) (2.26)

CPER—O]V—E2E(”i(0)) = 1 - (SON—EZEl'H/iiol) (2.27)

For Opportunistic compression and Slepian-Wolf coding the derivations for expres-
sions in equations 2.19 through 2.27 proceed in exactly the same manner, equation
2.25 being the exception. Opportunistic compression and Slepian-Wolf coding replace

equation 2.25 by equation 2.28 and 2.30, respectively.

Lm(‘n.,,;(0)) = [H(an,(0) . . . X7Li(Ni)|A)l (2.28)

A = {vxa : a e Rl(n,(0))} (2.29)

26

Lm(n,(0)) = Z Lm(a) (2.30)
aeRi(n,-(0))

2.3.3 Sensor-To-Base Station Capacity

Using the above results the sensor-to—base station BER/ PER for any arbitrary sen-
sor nZ-(j) can be obtained by multiplying the cluster-level BER p(n,(j)) in 2.12/
PER 602.2(3)) in 2.17 with the end—to—end BER pON—E2E(ni(0)) in 2.23/ PER
(50 N— E2 E(n,(0)) in equation 2.26, respectively. This yields an expression for the
sensor-to—base station BER in equation 2.31 and a corresponding sensor-to—base sta-
tion PER expression in equation 2.32. From 2.31 and 2.32 we can obtain the corre-

sponding capacity expressions in equations 2.33 and 2.34.

11521350120 ))
= p0N_EzE(ni(0)) [1 - M71100] + 290120)) [1 — P0N_E2E(”z'(0))] (231)

652350121» = 1 - [1 — 6mm») [1 — (SON—EQEinz‘iODJ (2.32)

CBER—SQBSiniijl) = 1 - Hb (Psggsmiljm (2-33)

CPER_SQBs(’Il.i(j)) :1" 60120))

(2.34)
= [1 — we») [1 - éoiv_E25<'rz..<o>>]

27

2.4 Results and Analysis

For the following experiments we use an 802.15.4 network with its MAC operating
in TDMA/ GTS enabled mode. We use the PHY channel model for residential
environments in [86]. The WSN consists of N = 150 sensor nodes randomly placed
over a square region of dimensions 10 x 10 according to a uniform random distribution.
To create longer multi-hOp routes in the ON we place the base station at coordinates
(0,0). We assume a set of 15 available frequencies for cluster-level communication
in addition to one frequency reserved for communication between CLHs in the ON.
Figure 2.1 depicts a sample WSN. Circles denote the positions of CLHs while the
x-marks denote the positions of sensors. Sensors in closest proximity of the means
obtained by the k-means clustering algorithm [44] are assigned the role of the CLH
for that cluster. Solid lines depict the topology of the ON while broken lines indicate
a sensor’s association with its respective CLH.

Values for the correlation factor p are varied through ,0 = 0.1, 1, 10, 20, 30. Increase
in p can alternatively be viewed as an increase in the node density. The number of
CLHs N1 is also varied through M = 5, 10,20, 30,40. We assume the use of near
capacity achieving channel codes at the MAC layer and evaluate the EPS for the

cases when,
1. No source coding,
2. Opportunistic compression, and
3. Slepian-Wolf coding are employed.

For each of these cases we assume two sets of complexity vectors c1, (1'1,c2,c12,
1) {5,1,10,1}, and 2) {5, 2, 10, 2}. The constant coefficient at the receiver side c2 is
assigned a higher value than the corresponding transmitter side coefficient c1 because

channel decoding is typically more complex than encoding. This produces six options

28

 

 

Correlation Number of CLHS

(a) No source coding 01 = 1, 02 = 1, c1 = 5, and c2 = 10.

 

 

Correlation Number of CLHs

(b) No source coding in = 2, (i2 = 2, C1 = 5, and c2 = 10.

Figure 2.5: EPS spent across the network for varying degrees of spatial correlation

and number of CLHs.

to evaluate. Obviously, a low is desirable. Furthermore, based on the measurements
reported by Polastre, Szewczyk and Culler in [101] and Madden, Franklin, Hellerstein

and Hong in [79] we take the ratio of per bit energy consumption for transmission,

29

 

Correlation Number of CLHs

(a) Opportunistic compression a1 = l, 012 = 1, c1 2 5, and c2 = 10.

 

Correlation Number of CLHs

(b) Opportunistic compression m = 2. Hg 2 2, cl = 5, and Cg : 10.

Figure 2.6: EPS spent across the network for varying degrees of spatial correlation
and number of CLHs.
reception, and instruction execution to be 1 : 0.6 : 1 / 800.

Figures 2.5a, 2.5b. 2.6a, 2.6b, 2.7a and 2.7b plot EPS against the degree of cor-

relation and the number of CLHs. Figures 2.5a and 2.5b show a decrease in the EPS

30

40s

 

 

Correlation Number of CLHs

(a) Slepian-Wolf coding (11 = 1, a2 = 1, c1 = 5, and c2 = 10.

300s

200s

EPS

100s

 

 

Correlation Number of CLHs
(b) Slepian-Wolf coding (21 = 2, (Y2 = 2, c1 = 5, and C2 = 10.

Figure 2.7: EPS spent across the network for varying degrees of spatial correlation
and number of CLHs.

with increasing number of CLHs and no variation with increasing values of p. The
second observation is according to expectations because we are not using any source

coding to leverage the correlation in the data. Figure 2.6a, 2.6b, 2.7a and 2.7b ex-

31

hibit similar trends but on very different scales. EPS increases with increasing M
and decreases with increasing ,0. The greater exploitation of the inherent correlation
in the data means a smaller scale of the EPS for figure 2.7a and 2.7b relative to
figures 2.6a and 2.6b. The sudden increase at high M and p can be attributed to
outlier values. These results lead us to conclude that weakly correlated data favors
the omission of any source coding scheme and a higher number of M. For highly
correlated data on the other hand Opportunistic Compression and Slepian-Wolf cod-
ing are the better choice. At high values of p the value of M plays an increasingly
negligible role. The choice between Opportunistic compression and Slepian-Wolf will
depend on the complexity of the available implementations of the two, i.e. at similar
complexities Slepian-Wolf outperforms Opportunistic compression but in a compar-
iSOII between Slepian—Wolf with complexity vector {5,2,10,2} with Opportunistic

Compression with complexity {5, 1, 10, 1} the latter outperforms the former.

32

Chapter 3

CSI Driven Model of BER Process
011 IEEE 802.15.4 Links

33

3. 1 Introduction

The bit errors and packet losses that are observed at the wireless receiver’s medium
access control (MAC) layer are modeled by a random process that is commonly re-
ferred to as the error process. An understanding of the error process is of fundamental
importance for a wide variety of reasons, e.g. design of high level (network layer and

above) protocols, retransmission strategies, error correction and concealment strate-

gies etc.

The IEEE 802.15.4 low rate—wireless personal area network (LR-VVPAN) standard
[5] is of particular interest to the wireless sensor network (WSN) research community
because it is the first wireless communication standard built around devices with
severe constraints on power consumption rates. Thus it is widely anticipated that
IEEE 802.15.4 will play a major role in WSN applications. This chapter analyzes
the performance and contributes to the understanding of IEEE 802.15.4 based LR-

VVPANS.

The objective of this empirical study is it to gain better insight into the time
Varying error process. We begin our analysis with a correlation analysis of the bit
Error rate (BER), link quality indication (LQI) and received signal strength indica-
tion (RSSI) processes and establish that if a packet’s BER is known to be non-zero,

18‘ it has failed the cyclic redundancy check (CRC) test, it is correlated with its
LQI and RSSI measurements. Based on the knowledge and the empirical data set
We Come up with a model for the BER’s probability density function (PDF) driven
by Channel state information (CSI) measurements. We evaluate the utility of this
model in different environments by dividing the data set along lines of different col-
leCtions environments, generating models for each of them and, using traces from

Other environments as test data, measuring divergence between models and test data.

We further analyze the amount of memory in IEEE 802.15.4 LR—VV PAN links at the

34

packet, symbol and bit level.

3.1. 1 Previous Work

To develop a better understanding of wireless channels’ error and loss processes,
several recent data trace collection efforts have targeted a variety of wireless networks,
including 3G networks [70], WaveLAN and 802.11x WLANs [93],[67],[66],[133],[105],
CCllOO based MICA2 networks [137] and 802.15.4 LR-WPANS [77],[116],[115],[64].
All of these efforts involve the collection of received data while offering different levels
of insight and resolution (e.g., bit.-, byte—, and / or packet-level) into the error process
on wireless channels. These studies usually focus on what is referred to as the residual
error process. In general, residual errors are bit-level or packet—level errors that are
not corrected by the PHY-layer and hence appear at the MAC-layer. Such errors
(usually) cause packet drops in traditional wireless MAC protocols such as IEEE
802.11 and IEEE 802.15.4. Most the error trace collection efforts restrict the error
process resolution to packet—level information. The most that can be extracted from

packet-level traces are statistics such as

0 Packet reception rates (PRR), the fraction of transmitted packets that are re—

ceived error-free and pass the CRC.

. Packet error rates (PER), the fraction of transmitted packets that are received

with at least one bit in error and fail the CRC test.

- Packet loss rates (PLR), the fraction of transmitted packets that are not received

at all.

By our definition of these terms.

PRR = 1 — PER — PLR (3.1)

35

Table 3.1: Traffic information and statistics collected in various studies.

 

 

Network PRR PLR/ Bit Truncated CSI
PER Errors Packets Available
[93] WaveLan J
[66],[67] IEEE 802.11b J J
[77] IEEE 802.15.4 J J
[116],[115] IEEE 802.15.4 J J
[133] RFM/MICA J
[137] CC1100/ MICA2 J J
[105] IEEE 802.11x J J
[64] IEEE 802.11g J J J
This Work IEEE 802.15.4 J J J J J

 

 

 

 

 

 

 

 

 

Khayam, et al. work in [67] was arguably the first bit-level residual error trace
collection efforts for 802.11b/g W’LANs. Meanwhile, there have been other trace
collection efforts for the more recent IEEE 802.15.4 LR-VVPAN, but like most 802.11
trace collections these too are limited to the observation of packets that pass the CRC

test. Table 3.1 tabulates the statistics and observable parameters in various works.

The collection of data packets for traces is often augmented by recording of addi-
tional information with each packet. Various wireless networking standards require
measurement of physical layer channel conditions. For example, 802.11b/ g requires
the measurements of signal-to—noise ratio (SNR) and background traffic noise level.
Sirnilarly, 802.15.4 mandates the measurement of RSSI and LQI of each received
paCket. We refer to all such measurements by the umbrella term CSI. As the side-by-
Side comparison in Table 3.1 shows, our error traces are by far the most detailed in
terfns of data collection and the recording of CSI. Our traces distinguish themselves
in that they log are he only ones to provide bit-level residual error traces for IEEE

802.15.4, the positions of lost packets, as well as partially lost packets, a phenomenon,

36

which is probably unique to 802.15.4, and whose cause is explained a little later in
this chapter. Further details regarding the meaning and definition of these parameters

will also be provided.

The remainder of this chapter is organized as follows. Section 3.2 provides a de-
tailed description of the setup used for trace data collection. Section 3.3 establishes
the degree to which CSI measures provide information about the BER process. Sec-
tion 3.4 uses maximum likelihood estimation (MLE) to arrive at a CSI driven model
of the BER process. Section 3.5 evaluates the CSI driven BER model. Section 3.6

concludes this chapter.

3.2 Trace Collection Setup

To our knowledge, this is the first detailed trace collection effort for IEEE 802.15.4.
These traces differ from previously collected ones in quantitative and qualitative as—
pects. We collected error traces of approximately 10 million packets in a way that
Provides, to the authors’ best knowledge, an unprecedented level of insight into the

effects of the wireless channel state on the level of corruption of packets.

3-2-1 Experimental Setup

The trace-collection setup is depicted in Figure 3.1 and consists of a Crossbow
MI)1R2400 MICA2 mote [34] transmitter and another MICA2 mote mounted on a
Crossbow MIB600 Ethernet gateway [33] as receiver. The gateway is connected to

a host PC running an application that continuously retrieves data from the receiver

and logs it.

37

3.2.2 Packet Payload

TinyOS [75] is one of the most widely used open source operating system in WSN
devices. TinyOS v1.1 allows various packet. formats to be transmitted. We suitably
modified code to enable the standard 802.15.4 frame format which TinyOS v1.1 labels
CC2420 Frame Format (after the Chipcon CC2420 chipset [120] used in MICA2
devices). Strictly speaking, the term packet refers to the protocol data unit (PDU)
exchanged between network layers of the transmitter and receiver while the term
frame is used for PDU’S exchanged between MAC layers. However, since our analysis

is restricted to the MAC layer alone there is little cause for confusion and we will
be using these two terms interchangeably to refer to MAC layer PDUs. The exact
MAC frame format used is shown in Figure 3.2. The size of the frame is 41 bytes and
comprises of a 1 byte Length Field, 2 byte frame control field (FCF), 1 byte sequence
number, 2 byte destination PAN ID, 2 byte destination address, 1 byte type field, 1
byte group field, 29 bytes of data/ payload followed by a 2 byte frame check sequence

( F CS) containing a CRC. The contents of the payload field are of our own choosing
and consist of 3 unused bytes, the Source Address, the Destination Address and 6
CODies of a 32 bit sequence number. The sequence number in the data/ payload is
Used to keep track of lost packets. If the sequence number between two consecutively
r eceived packets skips one or more numbers that is indicative of a packet loss. The
Sequ ence number field alone proves insufficient for this task in the face of long fades.
A130 , a single bit error in the 1 byte counter could easily become a source of ambiguity
(did we just lose two long sequences of packets or receive bit errors in the sequence
nuTuber field?) Note that transmitted packets differ only in the 1 byte sequence
nuTuber in the header and the six 32 bit sequence numbers in the payload, and the
CRC. For a particular trace all remaining bits remain unchanged. However, since

the wireless channel will introduce bit errors the copies of the sequence number used

38

 

to track packet losses in the received packet may differ. For this purpose we use a
majority vote of the received sequence numbers to determine the transmitted sequence
number. From this we reconstruct the contents of the Data/ Payload field and hence

the transmitted packet.

3.2.3 Trace Generation

Bit-level error traces can be generated by comparing a transmitted packet with its
received version. A simple bit-wise XOR operation on the transmitted and received
packets yields a bit pattern in which a zero (‘0’) signifies a bit that is received without
error while a one (‘1’) represents an inverted bit. We observe that in some cases the
length of the received packet is shorter than the transmitted packets. This constitutes
a partial loss and we use the term truncated packets to refer to such packets. An erased
bit in a received packet will be denoted by a two (‘2’) in the error trace. Truncated
packets are logged when bits in the MAC header’s length field are inverted and the
receiver stops listening to the wireless channel prematurely. It has also been observed
that if bits in the length field are inverted in such a way that the length of the incoming
PaCket appears longer than actual the length of the logged packet still equals that
Of the transmission. Although the Length field in the received packet may falsely
indi cate a longer packet, the absence of a carrier signal allows the receiver to detect

the end of the transmission.

3-2 -4 Channel State Information

Each received packet’s logged entry is accompanied with three pieces of packet level
CSI parameters. The first is the FCS status of the packet modeled by random variable
(1’ \Vith the nth packet’s F CS status is represented by d)[n].

Ordinarily receivers only distinguish between two states, i.e. F CS Pass (denoted

39

Receiver

MICA2 Mote
Ethernet Gateway

\ Host PC

 

 

 

 

<——Dat
Transmitter
MICAz Mote ]

IEEE 802.15.4
(2.4 GHz ISM)

 

 

 

 

Figure 3.1: Equipment setup for trace collection.

 

 

 

 

 

 

 

 

 

 

 

 

 

Octets:
1 2 1 2 2 1 1 29 2
Frame Sq Best Best Data /
Le“ Control No PAN ID Addr TV“ (3'9 Payload FCS
2 1 1 4 4 4 4 4 4 1
0x8 Src Dst SeqNo SeqNo SeqNo SeqNo SeqNo SeqNo 0x
401 Adr Adr (1) (2) (3) (4) (5) (6) 00

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.2: CC2420 MAC frame format used for experiments.

by <1")[7L] = 0) if the CRC value in the F CS field matches the CRC of the received
Packet, and FCS Fail if does not. Since we have knowledge of packet erasures and
Size of transmitted packets we extend the definition of FCS status to accommodate
the reason for failure. We restrict the definition of F CS Fail BE (denoted by (l)[n] = 1)
to IIlean that the size of a received packet matches the size of the transmitted packet
and the CRC failure is due to bit errors (BE). We further define two additional states,
FCS Fail PL (denoted by gb[n] = 2) and F CS Fail CL (denoted by ¢[n] = 3), where

PL and CL are abbreviations for partial loss and complete loss respectively. Packets

40

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.4: Residential deployment environment.

that. are partially lost cannot pass the CRC test and are marked FCS Fail PL. Packets
that are not received at all, i.e. when the decoded Sequence Number at receiver skips,
are marked FCS Fail CL.
The Crossbow MICAz sensor mote uses a TI Chipcon CC2420 transceiver chip
[120] for its communication subsystem. The CC2420 is an IEEE 802.15.4 compliant

radio interface. In accordance with the standard, the receiver measures RSSI. The

41

RSSI constitutes the second piece of our CSI parameters and is modeled by random
variable P and the nth packet’s RSSI denoted by p['n]. The RSSI is recorded as an 8
bit, signed 2’s complement value.

Technically, the CC2420 does not measure the LQI directly. Instead, it measures
tl‘le correlation CORR between the first 8 received symbols (of the PHY header)
a11d a corresponding 8 known symbols (preamble). IEEE 802.15.4 uses 16-ary Offset.-
Quadrature Phase Shift Keying (O-QPSK) modulation which encodes 4 bits in one
s}' rnbol. The first 8 symbols, 4 bytes, of the PHY header comprise of the Preamble

s<2<111cnce consisting of 32 binary zeros. The LQI is then defined as

LQI = (CORR — c1) - c2. (32)

where the two constants c1 and c2 are functions of the Packet Error Rate (PER)
measured over an extended period of time and are determined experimentally. c1
and c2 serve to scale the 7 bit value of the correlation to the full range of an 8
bit 11111111381”. Since 8.2 is merely a linear transformation of the measured CORR we
Sirnply take C1 = 0 and ('2 = 1. The LQI is modeled by the random variable A and
the ’nth’ packet’s LQI denoted by Mn]. Although the BER process is not considered
a CSI measure we are including it here nevertheless. The BER process is modeled
by random variable B and the nth packet’s BER is denoted by Mn]. It must be
mentioned here that the term BER is not used in its strict traditional sense where it
denotes the long term average probability of a bit error, such as in a binary symmetric
Channel (BSC). Instead the BER is computed over each individually received packet.

Thus, for the nth received )acket it is defined as,
I

Number of inverted bits in. nth received packet

BER = mm] =

 

(3-3)
Length in bits of Tim received packet

42

 

Table 3.2: State space variables, symbols and values.

 

 

 

Quantity Symbol of Symbol of Real- Valid Values Assigned to
Packet Level ization of Random Random Processes
Random Pro- Process
cess
FCSStatus (I) ab[n] qb =
r
0, F CS Pass
4 1, FCS Fail BE
2, FCS Fail PE
A 3, FCS Fail CL
RSSI P p[n] p E Z /\ —128 S p 3127
LQI A A[n] AEZ+AOSA_<_127
BER B sin] BeR+Aogsgi

 

 

 

 

 

 

The value of B[n] associated with the nth packet is the instantaneous measure of
the BER over that packet. Note that at this point we do not make any assumptions
about the distribution of the inverted bits within a packet. Completely lost packets,

with q) = 3, are assigned [1 = —128, /\ = 0, and l3 = 1'

Readers might argue that transmitter receiver separation could have been included
as another dimension of the phase space. However, numerous previous works like
1101] and ones listed in Table 3.1 have already established the tenuous nature of
the relationship between CSI and distance. Hence, transmitter receiver separation

is excluded from the phase space. Thus each received packet is characterized by its
FCS Status, LQI, RSSI and BER processes. Together they constitute four state space
variables of our system. Table 3.2 summarizes this notation and lists the range of

possible values each may assume.

43

3.2.5 Spectral and Environmental Diversity

The traces collected in this study are divided into two sets, a multi-channel (MC) trace
set, and a multi-environment (ME) trace set. In order to obtain sets of traces rich
in environmental diversity, traces in the ME trace set were collected over a period of
months, at different times of the day, in office, residential and outdoor environments.
All ME traces were collected while operating in channel 26 in the 2.48OCHz band.
The reason for choosing channel 26 was the fact that it is the channel in the frequency

spectrum that is the farthest removed from all 802.11bg frequency channels.

Each trace collection is characterized by the locations of the transmitter and
receiver, separation between them, packet transmission rate to in packets per sec-
ond (pps) and whether communication was line-of—sight (LOS) or non-line-of-sight
(NLOS). For all our traces the transmission power was kept constant at the default
OdB which corresponds to 1777.11". Figure 3.3 and Figure 3.4 depict floor plans of the
environments in which traces were collected. The circles labeled T1 through T18
denote transmitter locations. The corresponding locations of receivers are marked by
R1 through R18. For the remainder of the chapter, we refer to individual traces TRl
through TR19. Traces collected in the same location are collectively referred to by
the name of the trace collection environment. This way T R1 through T R8 are collec-
tively referred to as the ‘Hallway’ trace set, TR9 through TR15 as the ‘Lab’ trace set,
TR16 and TR17 as the ‘Residential’ trace set and TR18 and TR19 as the ‘Outdoor’
trace set. With the exception of the ‘Hallway’ traces none of the environments had
any significant WLAN interference sources. Traces are subject to interference from
cordless phones and microwave ovens. Table 3.4 briefly characterizes the various trace

Sets by providing PER, PLR and PRR of each.

The MC set consists of 16 traces, labeled MC-ll through MC-26, all collected

in the same residential, non-line—of—sight (NLOS) setting with 15 feet transmitter-

44

receiver separation. The transmitter is operated at full transmission power. Each
MC trace is collected at one of the 16 channels numbered 11 through 26 specified by
IEEE 802.15.4. The residential environment used for the collection of the MC set is
subject to interference from multiple IEEE 802.11x W LANs. At the time the MC set
was collected there were two networks in WLAN channel 1, two networks in channel 6,
one network in channel 10 and 2 networks in channel 11 with vary'ing activity levels.
Figure 4 of [56] depicts the population of the 2.4GHz ISM band and what IEEE
802.11b/g WLAN channels interfere with which IEEE 802.15.4 LR—WPAN channels.

Furthermore, the transmit power of IEEE 802.11b/ g devices is 30mW which is
significantly higher than the 1mW of 802.15.4. As Srinivasan, Dutta, Tavakoli and
Levis reported in [115], to a co-located 802.11 WLAN sharing the same spectrum
traffic from IEEE 802.15.4 devices appears as noise. One might argue that since
interference scenarios are part of our evaluation, other interference sources occupying
the ISM band (see figure 3.5) such as Bluetooth/ IEEE 80215.1 [4], microwave ovens
and cordless phones should also have been part of our traces. However, as Sikora
and Groza have shown in their study [112] on the coexistence of 802.15.4 with other
systems in the 2.4GHz ISM band, co—located Bluetooth networks and microwave
ovens have no discernible effect on the operation of IEEE 802.15.4 networks. The
Effects of 802.11b/ g networks on the other hand are significant and have been the
Subject of many studies ( [112], [48], [132], [102]), which is why we have only included

802.11b/ g networks as interference sources.

3.3 Correlation Analysis of CSI Measures

A primary objective of this study is to model the probability density function (PDF)
of the BER process B conditioned on CSI. It is important to highlight that although

we were able to observe the BER process in our trace-collection study, in an actual

45

Table 3.3: Collection environments of various traces in the ME trace set.

 

 

 

 

 

 

 

 

 

 

 

Trace Environment Interference Distance LOS/
Sources (feet) NLOS
ME—l Office bldg corridor 802.11x 40 LOS
(strong)
ME—2 Office bldg corridor 802.11x 60 LOS
(strong)
ME—3 Office bldg corridor 802.11x 70 LOS
(strong)
ME—4 Office 802.11x 2O NLOS
ME—5 Office 802.11x 20 NLOS
ME—6 Residence 802.11x (multi- 15 NLOS
ple networks)
ME-7 Office bldg corridor 802.11.x (low) 20 LOS
ME—8 Office bldg corridor 802.11.x (low) 50 LOS
ME-9 Office bldg corridor 802.11.x (low) 100 LOS
ME—IO Office bldg corridor 802.11.x (low) 80 LOS
ME-ll Outdoors - 100 LOS
ME-12 Office bldg corridor 802.11x (low) 120 LOS
Table 3.4: Error rates in trace sets.
TraceSets PER PLR PRR
Hallway 0.1877 0.0762 0.7361
Lab 0.0275 0.0161 0.9564
Residential 0.0993 0.0420 0.8587
Outdoors 0.0005 0.0030 0.9965

 

 

 

 

46

 

 

2.412 GHz 5 MHz 22 MHz

\ \ 9.9“?“ N \ .1555...

2.405 GHZ 5 MHZ 2 MHZ
Figure 3.5: IEEE 802.11b and IEEE 802.15.4 channels in the ISM band.

 

 

IEEE
802.15.4

MK

24 25 26

.3
.h

11

network application the error process can only be estimated; and hence )3 [n] is not
observable in actual networks. However, all other processes (cf)[n], {)[n], and /\[n]) are
observable. Hence, the problem of determining p([3[n]), the PDF of the error process
5[n], for the nth received packet based on observable CSI measures /\[n], p[n] and ¢[n].
Note that this is different from prior uses of CSI such as in [116] where it is shown
that the average PLR/ PER of a link is correlated with the average RSSI. When gb[n]
this indicates that the received packet passed the CRC and can be considered free
0f errors, i.e. [3 [n] == 0 with certainty. When at = 3 the packet is erased completely
and there is no data and no CSI available. This leaves us to focus on the cases when

4) = 1, 2, i.e. the received packet contains errors and/or is partially lost, i.e.

Alr‘1],plnllcbl'nl = 1, 2 -+ Jot/5171]).

47

Reducing the uncertainty surrounding the PDF of ,S[n] of a packet received with some
bits in error on the basis of its LQI and RSSI implies that these CSI measurements
are somehow correlated with p(1’3[n]). We begin by establishing that CSI parameters
are indeed highly correlated with the actual BERs of packets by means of correlation
based analysis. The uncertainty in the knowledge about p(,13[n]) is measured in terms

of its variance.

Packets with low BER are expected to have high RSSI and LQI, and vice versa.
Thus, to qualify as ‘useful’ RSSI and LQI must both be negatively correlated with
BER. Let T 27(1'3 ) denote a signal consisting of the BERs observed for packets captured
in trace TRi. If X and Y are two random variables with means E [X] and E [Y], then

the correlation coefficient ny(t) is defined as,

C0v[X, YA]
\/Var[X] Var[Yt]
EllX - Ele)(Yt — ElYtlll

_ (bier?) - E21Xl \/Em21 —— E21121

 

 

RXYU) =

 

 

Where Yt is the t time delayed version of Y. When correlation is computed for a
range of time delays the set of correlation coefficients so obtained will be referred to
as a correlation function. For every trace we treat the observed values of state Space
Variables B, /\ and p as time series signals and compute the correlation coefficients
between them. Ideally we would like to see R B p(0). the cross-correlation coefficient of
B and P, and RBA(t), the cross-correlation coefficient of B and A, to be very close
130 —1. For sake of completeness we also computed R p A(t), the cross—correlation
Coefficient of P and A. These values are computed separately for different trace sets
by concatenating signals of state space variables from all traces contained in a trace

set. Strictly speaking though, correlation coefficients should be computed separately

48

Table 3.5: Cross correlation of different random processes.

 

 

RBAU) RBPU) RAP“)
Hallway -0.4662 -O.2938 0.8414
Lab -0.2745 -0.0919 0.5978
Residential -0.2071 —0.0091 0.5346
Outdoors —0.3262 -0.1613 0.8443

 

 

 

 

 

 

for each trace. However, the results shown in this subsection change very little across
traces in the same trace set. Therefore, for brevity reasons we are presenting results
for the set of concatenated traces. As Table 3.5 shows, for packets observed at the
receiver (packets with (b = 0,1,2), both RSSI and LQI have exhibit a moderate
to strong negative correlation with the BER. Furthermore, RSSI and LQI are also

strongly correlated.

This trend is also observable in a visual inspection of the data. We are able to
obtain the joint distribution pBAPCI)(1B1A1p1d’) and from it any conditional distribu-
tion, such as pB(/3]/\, <25 = 1, 2) and pB(/3|p, (b = 1, 2), by means of the error traces we
collected. Figures 3.6 and 3.7 show two conditional distributions p 3(5 |/\, (D = 1, 2)
and p 3(8] p, qb = 1, 2), respectively. The axis along the left hand side of the horizontal
plane is the BER axis. The axis along the right hand side of the horizontal plane is
the CSI measure (LQI in figure 3.6 and RSSI in dBm scale in figure 3.7). Thus for
each value of LQI and R881 the figures depict a valid PDF of BER. Figure 3.6 depicts
how variance and average BER of packets in the low LQI range increases which is
in accordance with our previous observations. The same trend is visible for RSSI in
figure 3.7. Thus, the figures depict inverse relationships of the BER with both CSI
measures. Our analysis so far leads us to conclude that both RSSI as well as LQI
measurements made with each received transmission have the potential to serve as

good indicators of the BER for that particular packet.

49

 

PDF of BER conditioned on LQI

pB(BIX.¢=1.2)

 

 

100

Figure 3.6: PDF of B conditioned on A and (I) = 1, 2.

3.4 CSI Driven BER Model

In this section we employ maximum likelihood estimation (MLE) to determine the
parameters opr(B|)\, p, (b = 1, 2), an exponential PDF modeling pB(B|)\, p, (b = 1,2).
From collected traces we observe that the BER PDFs are always decreasing functions.
We will be modeling PDFs of BER by a discretized exponential PDF. The shape of

the exponential PDF is determined by a single parameter b as shown in 3.6,

—a:
e7; :1: 2 0. (3-6)

o-IH

f(:13;b) =

The MLE estimates of parameter b can be obtained from a data set by,

b = qu. (3.7)

50

PDF of BER conditioned on RSSI

pBelp.¢=1.2)

 

 

0.3 ‘82

Figure 3.7: PDF of B conditioned on P and <I> = 1, 2.

The MLE of b for all observed combinations of values of (A, p) constitutes a model of
the BER process. b(A, p) are depicted in Figure 3.8. Depending on the objective of the
application seeking the channel state estimate, pB(,B|A, p, (b = 1, 2) can now either
be used as is or mapped to a single numerical value representative of the channel
state. For lack of a better graphical representation, figure 3.9 shows pB(b’|/\,p =
88dBm,d = 1,2), the PDFs of the BER process observed over the range of LQI
measurements with RSSI fixed at 88dBm (chosen arbitrarily for illustration purposes).
Figure 3.10 displays the same PDFs modeled by MLE exponentials. The above model

leaves us with two very important questions.

Q1: How useful is this model in terms of reducing uncertainty or variance
of the BER’s PDF?

51

Parameters of exponential PDF at different LQI and RSSI

b(?».p) = ElpB(BI?~,p)l

 

     
  
 

-84 '82

A. 60 —90 ‘88 ‘86P (dBm)

Figure 3.8: Values of b for various (A, p).

Q2: How universally applicable is this CSI driven BER model especially
when we consider that wireless channels behave very differently in different
environments?

We will address these two questions in the following section where we evaluate

this model.

3.5 Model Evaluation

3.5.1 Variance Reduction

Taking variance as a measure of uncertainty in stochastic systems we evaluate the
expected value of variance in estimates of the BER. In other words, we contrast
the performance of LQI and RSSI used individually and when used in conjunction
by evaluating the expected value of the variance of estimates of the BER’s PDFs,
i.e. E[VAR[pB(/3[,\,p,¢ = 1,2)]] = E[VARB|AP], E[VAR[pB(,B|A,¢> = 1,2)]] =

52

Actual pB(|3|}t,p=88dBm) from collected traces

pB(l3l7t.p=88dBm)

 

 

Figure 3.9: The PDFs of the BER obtained from the actual traces for various LQI
measurements at an RSSI of 88dBm, pB(_B|/\, p = 88dBm, ab = 1,2).

E[VARBIA], and E[VAR[]BB(,8]p, (,5 = 1,2)]] = Ell/ARBIP]. These expected values
are computed over the joint PDF pAP(/\, p|¢ = 1, 2) of LQIS and RSSIS with which
packets with errors are received, also obtained from our traces. Table 3.6 tabulates
the expected value of variance with which we estimate the BER process. Clearly,
once again we see that using LQI and RSSI together is more beneficial and estimates

the error process with lesser uncertainty.

3.5.2 Dependence On Deployment Environment

So far we have established that for every error-prone packet with g") = 1, 2 it is possible
to obtain a better estimate of the PDF of its BER based on measurements of RSSI
and LQI. Ideally, we would like to make this estimate dependent solely on the RSSI

and LQI and would like to see little or no dependence on the physical environment.

53

Model of pB(|3|?t,p=88dBm) based on MLE of exponential PDF

=88dBm)

pB(l3l7~.p

 

 

0.1 100

Figure 3.10: The PDFs of the BER for the same range of LQI measurements at RSSI of
88dBm as modeled by a discretized exponential PDF, pB(i3|/\, p = 88dBm. (0 = 1,2).

To evaluate the application of the CSI driven BER model in different environment
we partition the data set according to collection environments, i.e. ’Hallway’, ’Lab’,
’Residential’, and ’Outdoor’, and use'cach to arrive at a CSI driven BER model
independently. Ideally the four models so obtained should be identical. As we already
know, each model is a set of PDFs. If /\ and p vary in the ranges )‘mz'n S /\ g Amaa:

and pmin _<_ p S pmaay, respectively, that means each model will consist of (at most)

Table 3.6: Expected value of variance when using different combinations of RSSI and

LQI as CSI.

 

Expected Variance

 

E[VARB]; CSI = None 1.7000 x 10—3
ElVARBIA]; CSI = LQI 8.0863 x10"4
E[VARBIP]; CSI 2 RSSI 8.1242 x10—4

 

 

 

E[VARBMP]; CSI = LQI + RSSI 5.7524 X10—4

 

54

(Amag; — )‘min + 1) x (pmaj; — pmz’n + 1) number of PDFS pB(B|)\,p,¢ = 1,2),
one PDF of B for every combination of possible (Aha) tuples, we Show that the
PDFS pB(fi|/\, p, 05 = 1, 2) obtained from one trace set closely approximate the PDFS
p BM} I/\, p, 45 = 1, 2) of a different race set. for the same values of (A, p). This way
traces the CSI driven BER model derived using data from one environment will
be verified against test data collected in a different environment. At this point. we
require a measure that quantifies the degree of similarity between two PDFS. A well
established divergence measure is the Kullback-Lez’bler Divergence (KLD), also known
as relative entropy. The KLD of two PDFS p X and py is denoted by D(p X H py)
and defined as,

 

Dex II 103/) = Z pxmzogf

(3.8)

Sp X denotes the region of support of PDF p Xv or the elements in the domain
0f PX for which pX(:I:) > 0. Note that according to the definition of D(pX H py),
if Sp X Q SPY then for any :1: g2 Sp X the KLD is 00. Although this should not
happen in a large data set consisting of a comprehensive set of traces from various
10Cations, trace packets in individual partitions of the data set might not exhibit
SUfficient diversity across the entire spectrum of possible values of (A, p) to avoid such
a result. To circumnavigate this pitfall in our analysis we use a modified form of
the KLD called the K-directed divergence (KDD) introduced by Lin in [76] which is
defined.

 

PX (1") )
K = E a? lo . ~
(I? X H Py) $6 p px( ) 9 (%p (‘17) $1) (y) (3.9)

The KDD’S most significant feature relevant to our application is that the denomina-
t01‘ of its fractional term will never be zero, and hence the KDD cannot evaluate to co.

Thlis, to measure the similarity between PDF 158— X03 |/\, p, 09 = 1,2) derived from

55

traces collected in an environment X and a PDF pB_Y(,8|A,p, cf) 2 1,2) of traces
collected in an environment Y we employ the KDD. A complete comparison of two
environments will require the computation of KDDS for each (A, ,0) pair. This yields
an entire set of KDD measures. Ideally, all KDDs in this set should be close to zero.
In figure 3.11a we plot the histogram of the set of KDDS obtained from comparing
the CSI driven BER model derived from the ’Residential’ set with the PDF of the
data collected from the ’Office’ trace set. As the histogram shows, the vast majority
of KDDs are zero, with very few that are non-zero but nevertheless close to zero.
Similarly, figure 3.11b is the same histogram plot for PDFS from ’Hallway" and
’Outdoor’ traces. A particular reason why we chose this example for illustration is
that. it uses the ’Outdoor’ trace set which exhibits and significantly different from that
of the other three sets (see table 3.1). Those differences are manifesting themselves
in a less obvious way in the histogram of KDD values in figure 3.11b. The histogram
rises to a non—zero entry at the bin centered at 1 which, however, remains insignificant
in Comparison to the number of entries in the bin centered at O. For the vast majority
0f Valid (A, ,0) pairs the KDD between different trace sets is zero. we conclude that
the CSI driven model for the PDF of the BER process is applicable in a variety of

environments.

3 - 6 Conclusions

7 . . .
VA e collected and analyzed an extcns1ve and diverse set. of res1dual error traces from

802.15.4 links. Listed below are our conclusions.

1. LQI and RSSI exhibit moderate negative correlation with the BER process (and

strong positive correlation with each other).

2. LQI and RSSI can be used to reduce uncertainty regarding in the BER individ-

56

80,000

60,000-

# of KDDS

20,000*

80,000

60,000

40,000

# of KDDS

20,000-

Figure 3.11: Histogram of KDD values for a) Model from ‘Residential’ trace set and
‘Lab’ traces and b) Model from ‘Hallway’ trace set and ‘Outdoor’ traces.

 

40,000"

 

' Residence vs Lab j

 

0T2 (it 06 0:8

K(pB-Residence“pB-Lab)

(a)

 

j

 

'Hallway vs Orutdoor'

 

0:2 0:4 ofe oia

K(DB—Hallwayl"DB-Outdoor)

(b)

ual packets are subjected to.

3. The CSI driven BER model remains valid across a variety of physical environ-

ments.

57

 

 

Chapter 4

Memory Properties of the
Link-level BER Process in IEEE

802.15.4 Links

58

4. 1 Introduction

Communication channels in the real world are not perfect and are prone to introduce
errors into transmissions. Errors are orders of magnitude more frequent in wireless
channels than in wired channels. Design of network protocols and other architectural
components of wirelessly networked communication systems entail a better under-
standing of the error process that affects transmissions. The memory length of the
error process is an important parameter of interest that has to be taken into con-
sideration in the formulation of channel models. In particular, channel models that
take into account the persistence of errors in wireless channels, such as those based
on Markov chains, require information about the memory length of the error process.

Previous works ( [68], [66], [67]) measuring memory length in 802.11b channels
were restricted to the bit-level error process and relied on the correlation function
based analysis, which sometimes worked well enough at that scale. We were the first
to perform bit-level analysis for IEEE 802.15.4 low rate-wireless personal area network
(LR~WPAN) [5] channels and extend it to symbol and packet—level error processes in
[56]. In this chapter we are extending our previous analysis by using a much larger
data set that spans not only the set of all 16 channels in which 802.15.4 operates,
bUt also a variety of physical environments. The failure of traditional correlation
analysis is explained using results from [54]. At the bit and symbol level the error
Processes are modeled by binomial processes (a bit / symbol is either received correctly
or incorrectly). The packet-level process is called the bit error rate (BER) process
and is denoted by B. The BER process B is defined as a series of measurements of

h

the rate at which each packet has been subjected to bit-errors. The n.t measurement

in a realization of B is denoted by Mn] and is computed as shown in equation 8.1.

# of bits in nth packet received with. error (4 1)

 

[i n =
j J # of bits received in nth packet

59

We are concerned with determining the memory length of errors due to the effects
of slow and fast fading. As we will demonstrate, the determination of this memory
length is complicated by several factors which include slow fading, non-stationarity
due to long stretches of interference, and periodic interference. A way around the ef-
fects of slow fading and interference is to preprocess the BER process by de-trending
and normalization [131]. However, this is complicated by their non—periodic, unpre—

dictable nature.

Like some prior works that measured memory length of errors, this research too
depends on bit-level, binary signals, called residual bit-error traces (short: error traces
0r traces), representing the positions of bit—errors in received transmissions. The basic
concept of error traces is very simple and is explained easily enough. An error trace
is a map of bit positions of all packets collected in the course of a trace collection
session that reached the receiver with errors. Conceptually, for a single packet such
a bitmap of errors is obtained by comparing a transmitted packet (free of errors)
with its received version (may contain errors and failed the cyclic redundancy check).
This way the sequence of BERs computed by equation 8.1 for each packet in a trace
constitutes the BER process. The complications involved in working with traces arise
due to lack of access to low level drivers and / or firmware that need to be modified to

gain access to packets received with errors, otherwise discarded by receivers.

The remaining chapter is organized as follows. Section 4.2 demonstrates the use of
correlation coefficient in measuring the memory length of bit, symbol and packet-level
error processes. Section 4.3 presents the results of Hurst analysis on the packet-level
BER process. Section 4.4 describes relative mutual information (RMI), an alternative
means of measuring memory of in measuring the memory length of the BER process

observed of IEEE 802.15.4 LR—WPANS channels. Section 4.5 concludes the chapter.

60

4.2 Memory Length Measurement By Correlation

Analysis

4.2.1 Correlation Function

Let X be a random process consisting of random variables [X 1, X2, . . . , X N, . . .]. The
corresponding measurements in a random process are denoted by [51:1,:1:2,. ..,:1:N,

...] X (m) is an m time unit delayed (right-shifted) version of X. To measure
memory length we compute the correlation coefficient of X and Xim) for a range of
values of m. Thus, the correlation coefficients denoted by R X(m) are a function of

m and are defined as,

 

 

UXUX(m) 0X0X(m) . (4'2)

Here Cov (X , X (m)) denotes the covariance of X and X (m), o X the standard
deviation of X, p X the mean of X, and E [] the expectation function. This function of
m is also known as the correlation function or correlogram. The value of m after which
the correlation function RX(m) becomes ”insignificant” and drops below threshold
Rt is the memory length [W X of process X. There is no clear consensus on what
the value of Rt should be. In the simplest of cases the smallest value of m for which
the correlation function is 0 or close to it is taken as a measure of M X' There
is a 95% significance level that is frequently used as a. rule of thumb to determine
whether M’ X is zero (successive measurements in X are independent) or non-zero
(successive measurements in X are not independent). For a signal of N consecutive

2

measurements the 95% significance range is defined as TW' If 95% of correlation

coefficients for m in the range 1 S m 5 14V— lie within this specified range, then

61

 

(b) All bit-level traces of the ME trace set.

Figure 4.1: Auto-correlation functions for bit level traces of the MC and ME trace
sets.

consecutive measurements are deemed independent, leaving M X = 0.

62

 

(a) All symbol-level traces of the MC trace set.

 

(b) All symbol-level traces of the ME trace set.

Figure 4.2: Auto—correlation functions for symbol level traces of the MC and ME

trace sets.

(4.3)

min (Rx(7n) < Rt) .

m

MX=

63

4.2.2 Correlograms of Bit and Symbol-level Traces

For analysis of bit and symbol-level error processes we compute Rb(k) and R3(k)

defined as,

R 1.: =ER Am? ,Rsk =ER5k,i.
b() lb( )l (l l ( )l (4.4)

h packet’s bit and

Rb(k,i) and R3(k,i) are the correlation functions of the it
symbol traces with its kth‘ following packet, respectively. The correlation functions
Rb(k) and 123(k) of a particular trace are computed as the expectation function
over i of Rb(k,i) and R3(k,i), respectively. If for a k > mb,m3 the correlation
functions Rb(k) and 123(k) drop to a value very close to zero, then mb and 7713 are
the bit and symbol-level channel memory of the channel. Figures 4.1a and 4.1b plot
the auto—correlation functions or correlograms of all bit-level traces in the MC and
ME trace sets, respectively. What is apparent from this figure is that. in spite of
the wide variation in trace collection parameters, we consistently observe a bit-level
channel memory mb of at most 2 bits across all traces regardless of channel frequency
and environmental differences. Similarly, figures 4.2a and 4.2b are the correlograms
of the symbol-level traces for the MC and ME trace sets, respectively. Horn both
correlograms it is clearly visible that symbol level memory ms is also (at most) 2.
Therefore, there is no further need to check for long range dependence (LRD) and we

conclude that the bit and symbol-level error processes in 802.15.4 wireless channels

have constant memory of (at most) 2 bits and 2 symbols, respectively.

4.2.3 Correlograms of Packet-level Traces

Figure 4.3 depicts the correlogram functions for all 16 traces in the MC trace set,

and figure 4.4 for the 12 traces in the ME trace set. For the computation of the

64

autocorrelation function, the length of each of the traces is truncated to a uniform
N = 100,000 points, making the 95% significance range [—0.0063,+0.0063]. By
this measure it appears as if all channel traces exhibit memory to some degree. To
determine just how much memory several rules of thumb have been used in literature.

These include;

Rt = min,- (RB(i) = 0) : Correlation coefficient falls close to insignificance/zero. In

all our analyzed traces this has rarely been the case.

Rt = 0.1 x RB(1) : Correlation coefficient. drops to less than 10% of the coefficient
at lag 1. By this standard, the memory length of the BER process will range

in the tens to thousands of seconds.

Rt = min, (IRB(i) — RB(j)| < 6) : Vj > i, 6 ——> 0 : Correlation coefficient becomes
steady and subsequent changes between consecutive values are within a very
small value 6. Periodic interference, most likely from IEEE 802.11b beacon
frames, causes periodic spikes in the correlogram functions rendering this cri—
terion useless. Furthermore, there is no clear interpretation of the value of
correlation coefficients and the degree of predictability of one measurement on

another.

According to all these selection criteria for Rt, memory lengths of all these traces
lie in the range of tens to hundreds of seconds. Clearly, that rules out correlogram
analysis to measure memory length of fast fades. The failure of the correlogram anal-
ysis is explained by the non-stationary nature of the BER processes captured by the
traces. Figure 4.5 plots the BER process observed in MC-25, the trace of the MC
trace set that was collected with devices operating in channel 25. For clearer visibil-
ity, the BER process was pre-processed by passing it through a 600 point averaging

filter. There is an initial period of about 14()Osec in which the channel experiences

65

 

0.5 -

0.4 f

QAAAA

A

H :II". 1'

" 03Hl:'1:'.:‘-AAA,A,AAAAA

' o',n',,'a‘ ‘ \l A
mo- 1l-z1::: A A AAA ‘A’A’AAA 23

My

 

 

 

 

    

35 4 45

Figure 4.3: Auto-correlation functions for traces of the MC trace sets.

a moderate BER which then increases to a much higher value for about 9003ec. It
then reduces to a very low value and remains so for the remainder of the trace. We
attribute these changes in channel conditions to interference from co-located devices
sharing the spectrum in the 2.4GHz ISM band. When the correlation function is
computed over a trace which experiences a change in channel conditions like the one
in figure 4.5, the correlation function is expected to maintain a significant value for

an extended period of time.

66

 

ME-1
ME-Z
ME-3
ME-4
ME-5
ME-6
-~- ME?
—0— ME-8
—El- ME-9
—A— ME-10
--+-- ME-1‘l
-----x ME-12

0.6 Q

I

iiiill

 

 

 

 

 

 

 

 

A... a "."
.'\‘ ’ '5'“ ,

3‘ a '\ 7"..‘rr ,-H~. _. ‘0' ‘ I
. ---«' -n..-a..-.v.'.1*'. .% n ......- - ' n-

1.5 2 2.5

   

Figure 4.4: Auto-correlation functions for traces of the ME trace sets.
4.3 Hurst Analysis of Packet-level BER Process

The correlation function alone did not allow us to make a definitive conclusion about

memory length. For this reason we employ the Hurst parameter H for which several

estimators exist in literature [10]. We use the Hurst parameter, (or Hurst exponent)

as a means of determining whether a process is LRD or not. A stationary process is

said to be LRD if there exists a real number 01 E (0, 1) and constant c]; > 0 for which,
R(k)

lim 2 1. (4.5)
lit—>00 Cpk_a

 

The Hurst parameter is defined as H = 1 — 7%. A process is determined to be

LRD if 0.5 < a < 1. Several methods exist for estimating the Hurst parameter. In

67

 

l

 

: High BER
0.1 Moderate ' Interval

BER Interval-

...-1

Low BER interval

 

p p
o o
o: oo

Averaged BER
f;
A

.0
o
N

 

    

A- - .4“ ““u‘ -

2000 3000 4000 5000 ‘60l00
time (sec)

 

0 1000
Figure 4.5: BER process of trace MC-25 after filtering by 600 point averaging filter.

this work we use the Aggregate Variance, the R/ S method, the Periodogram method,
the Absolute Value method, the Abry-Veitch estimator and the Whittle estimator,
details for all of which can be looked up in [10]. We use the SELFIS tool deveIOped
by Karagiannis, Faloutsos and Riedi as part of their work [63]. Figures 4.6a and 4.6b
plot H B E R, the Hurst parameter of the BER process for the MC and ME trace sets
using all the above listed estimators. In addition, for each trace we plot the average
Hurst parameter over all estimators (thick line). The plots of the Hurst parameter
are accompanied by those of three other quantities, i.e. the average packet error rate
(PER), the average packet loss rate (PLR) and the average conditional bit error rate
(CBER). The average PER for a trace is the ratio of number of packets received
with failed CRC to total number of transmitted packets. The average PLR is the

ratio of the number of packets never received by the receiver to the total number

68

of packets transmitted. The average CBER is the expected value of the BER of all
received packets whose CRC failed. As all plots show, there are significant variations
in the estimates of the Hurst parameter computed by different methods. Estimates
of obtained by the absolute moments and aggregate variance methods consistently
provide the highest estimates, whereas the ones obtained by Abry-Veitch and Whittle
estimators are consistently the lowest. The ones provided by the Periodogram and
R/ S methods fall approximately in the middle of this range. This observed ordering

holds true for H B E R across MC and ME trace sets.

4.3.1 Observations For MC Trace Set

For the MC traces we observe that most estimates of H B E R rise in three distinct
places, peaking for channels 13, 17 and 25. Channels 13, 17 and 25 occupy the
centers of the spectrum bandwidth occupied by interfering channels 1, 6 and 11 of
IEEE 802.11b/g. The degree of interference experienced by channels is measured
by the average PER and average PLR. Thus H B E R bear some correlation with the
average PER and average PLR and, hence, the interference from nearby 802.11b/ g
W'LANS. The Abry-Veitch estimator is the only estimator that defies this obser-
vation and in fact exhibits inverse correlation with average PER and average PLR.
Estimates of H B E R for channels experiencing strong interference are high enough
to conclude LRD in the packet-level error process. But at the same time we observe
that for 802.15.4 channels that do not experience interference from 802.11b / g WLANs
estimates of H are close to or below the critical threshold of 0.5. It appears that in-
herently the packet-level error process in 802.15.4 channels is not LRD, except when
it is subjected to 802.11b/g interference. The explanation for this behavior comes
from Karagiannis, Faloutsos and Riedi [63] who concluded that periodic interference

in an otherwise memoryless channel can give the appearance of LRD (when LRD is

69

 

0.2" Q 0
A A A A Q
n A a u D a A
U D 8
ii a ° 8 8 5’ ° 7 8 a 9 i ? ° 2

 

 

12 14 16 18 20 22 24 26
MC-Channel #

(a) MC trace set.

 

+Aggregate Variance
+ R/S
+Periodogram
+Absoiute Moments
+Abry—Veitch Estimator
L +Whittle Estimator
‘ *Average H

c Average CBER
0 Average PER

A Average PLR

 

 

 

 

 

 

0.2T ‘ A °
'3 A ‘ 8 a a
0 E if D 4' L a R A A A A
2 6 10 12
ME—Trace #

(b) ME trace set.

Figure 4.6: Plots of estimates of the Hurst parameter obtained using various tech-
niques along with their average BERs, PERs and PLRs.

determined by estimating H). Moreover, LRD is also unlikely to exist in a process
if estimates of H B E R by various methods do not converge, which is surely the case

in our analysis. WLANs periodically transmit a beacon frame every lOOmsec that is

used to synchronize the network. It is the periodic interference from these beacons

70

frames that is giving the appearance of LRD in the packet-level error process.

4.3.2 Observations For ME Trace Set

Recall that all these traces in the ME set were collected in channel 26 to reduce
effects of interference. Most traces in this set experience lower average PER and
average PLRs. For many traces the average Hurst parameter often remains close to
0.5. Nevertheless, as in the case of the MC trace set, the estimates of H B E R still
do not clearly converge to one value, and when they do it is in a range very close to
0.5. The lack of consensus among estimates, and the close proximity to 0.5 otherwise

leads us to believe that the channels in the ME trace set are also not LRD.

4.4 Memory Length Measurement By Relative Mu-

tual Information

4.4.1 Shannon Information Measures

For random variables X and Y, with probability density functions (PDF) p X(:r) and

PYW) and joint PDF gym/(1r, y), the mutual information I(X; Y) is defined as,

 

(2:, y)
1(X;Y) = Z pXY(3’5> y) 10s2 pXY 1 . (4.6)
Var,y;p(:v,y)>0 (px($)pY(J)> ,

Mutual information [32] can also be understood more intuitively in terms of en-

tropy as,

1(X; Y) _—. H'(X) + H(Y) — H(X, Y). (4.7)

More generally, mutual information is defined between two sets of random vari-

71

ables. If random variables X and Y in equation 4.6 are replaced by sets of random
variables X1,X2, . . . ,Xn and Y1,Y2, . . . ,Ym, respectively, then pX(:z:), py(y) and
pxy(x,y) are replaced by respective joint PDFs pX1X2.HXn(:r1,:r2, . . . ,crn) and

leYg...Ym(l/1a92, . . . ,ym), and the joint PDF of all random variables

leXZ...XnY1Y2...Y7n(I1’ 11:2, ' ' ' ,In, yl’ 3127 ' ' ' 33/771)’

4.4.2 Description: Relative Mutual Information

We are essentially faced with the challenge of evaluating;

o How much information random variable Y can provide about another random

variable X;

0 While at the same time providing a measure of the remaining uncertainty about

X.

The Shannon mutual information I (X ; Y) achieves the first goal. However, mutual
information is not restricted to a fixed range. Thus, an evaluation of I (X ; Y) does
not give us a sense of how much information about X or Y remains unknown. We
use relative mutual information (RMI), denoted RM 1 (X ; Y), previously described by
[29] and adopted by us for the purpose of memory length measurement in [57]. RMI

is defined as;

I(X;Y)

(4.8)

 

Note that while I (X ; Y) is a symmetric measure, by definition RMI is non-
symmetric, i.e. RMI(X; Y) 7é RMI(Y; X) because H(X) 7E H(Y). Since 1(X; Y) S
min (H (X), H (Y)), the RMI’s value is limited to the interval [0, 1]. An RMI close to 1
implies Y contains most of the information contained in X, leaving little uncertainty,

while an RMI close to 0 implies the opposite.

72

 

l3(1m)

 

 

 

ARM|B(1,m) RM,
.0 .o .0

 

 

MB(5)

 

 

 

0.05 0.1 0.156 0.2 0.25 +Mc—15

Figure 4.7: For traces MC-ll, MC—12. MC-13, MC-14 and MC-lS each subfigure,
(from top to bottom): [Top] RM’ I 3(1, m) of BER process observed in MC traces for
lag m varying from 1 through 40. [Middle] ARMIB(1,m) of BER process for the

same channel traces. [Bottom] The memory length MB plotted as a function of 6,
the increments in RM I 3(1, m).

 

In the current context we replace. X by the BER process B (0), and replace Y by
the BER process of preceding packets 8(1), i.e. a one—right shifted version of 3(0).
Then RM I (3(0); 3(1)) measures the amount of information that a packet’s BER
,B[n] shares, on average, with the following packet’s BER [fin — 1]. A natural extension
of this measure would be to include more than just the immediately following packet.

but include any arbitrary number m such that,

 

73

 

9999
thoo

  
 
 

 

.0
OP—x
AU!

ARM|B(1,m) RMIB(1.m)
9

MB(5)

 

   

 

0.15 8 0.2 0.25

Figure 4.8: For traces MC—16, MC-17, MC—18, MC-19 and MC—20 each subfigure,
(from top to bottom): [Top] RMIB(1, m.) of BER process observed in MC traces for
lag m varying from 1 through 40. [Middle] ARMIB(1,m) of BER process for the
same channel traces. [Bottom] The memory length M B plotted as a function of 6,
the increments in RM I B(1,m).

Since we are assuming the BER process to be wide sense stationary, the RMI
becomes a function of m, the number of immediately following measurements. We
use the abbreviated notation RM I B(1, m) for the RMI in equation 4.9. Recall that
RM I B(1,m) is a function of the Shannon mutual information (and hence the joint
PDF) of BER processes B through 3(7”). Ideally, once m exceeds the channel’s
memory length M B, B become independent of B (m) and the Shannon mutual in-
formation equation 4.6 will drop to zero, producing a zero RMI in equation 4.8. But
as in the case of correlogram based analysis, the presence of slow fading complicates
interpretation of RMI. Therefore, for the duration of the slow fade a packet’s BER
will remain weakly correlated with its followers. Hence, after m > M B, this trans-
lates into a slowly increasing RM I B(1,m) for successive values of m until it finally
becomes 1. However, for the problem at hand we are not particularly interested in

slow fades but the memory length of fast fades. Therefore, we define channel memory

74

AA AA A A A A‘A‘AiiA'iA A A A AA"A’A‘LAA A
~——————— __—'j—__——__

  

 

io-o-J-o-o-if-c-o-O'O"

  

l L J

 

 

 

 

 

Figure 4.9: For traces MC-21, MC-22, MC-23, MC-24, MC-25 and MC-26 each sub-
figure, (from top to bottom): [Top] RMIB(1,m) of BER process observed in MC
traces for lag m varying from 1 through 40. [Middle] ARM I B(1, m) of BER process
for the same channel traces. [Bottom] The memory length MB plotted as a function
of 6, the increments in RAMBO, m).

length MB as the time it takes for the RMIB(1, m) function to rise to a level after
which it grows only slowly. Then channel memory M B is determined by computing
the RMI function for increasing values of m and setting A! B equal to the largest lag
m for which the amount of additional RMI provided by including the (m + 1) delayed

BER becomes smaller than 6, i.e.

618(6) 2 nmx (|RAIIB(1,m — 1) —— RJUIB(1,m)| 2 6) \7’m 6 [1,00].
(4.10)

Thus the memory length 1113(6) is a function of 6, the significant RMI increment
threshold. That leaves us with the choice an appropriate value of 6. RMI is under-

stood to be the fraction of Shannon information of one random variable contained

75

collectively in another set of random variables. Thus, in terms of RMI the memory
length of the BER process can then be understood as the lag m after which every
subsequent BER measurement will contribute less than 100 X ‘70 of new information
about B (O) The feature that makes RMI attractive for use as a tool for memory
measurement is not the fact that provides an unequivocal measurement of the BER
process’ memory length. Like the correlation coefficient before it, the RMI depends
on the subjective selection of a threshold that determines the cutoff between signif-
icance and insignificance. Rather, its strength lies in the fact that there is a clear
interpretation of the threshold (in this case 6) in information theoretic terms, e.g.
6 = 0.15 implies that on average, an MB delayed measurement contains at least 15%
information about the current measurement. Another advantage of using RMI over
the commonly used correlation coefficient is that the changing trends operating on
the timescales of slow fades that cause non-stationary behavior of the BER process

do not complicate the analysis.

4.4.3 Discussion

The plot at the top in figure 4.7 is the R11'IIB(1,m) of traces of IEEE 802.15.4 links
operating channels 11, 12, 13, 14 and 15 for a range of lag values m. The middle
figure plots the ARM I (1, m) functions of the same traces for the same range of lags.
The bottom plot in figure 4.7 shows the memory length as a function of increments
6. Figures 4.8 and 4.9 plot the same quantities for the remaining channels in the
MC trace set. Similarly, figures 4.10 and 4.11 plot RMIB(1,m), ARMIB(1,m)
and M 3(6) for the traces in the ME trace set. For our reading of the plots let
us take 6 = 0.15 ,i.e. last measurement within memory length contributes at least
15% information. Then the plots of M 8(6) for all traces adequately demonstrate

that all 802.15.4 traces exhibit a memory length within a narrow range of O to 10

76

 

 

 

ARMB(1,m) RMB(1,m)

 

 

 

 

 

AA AAAA AAA A.

5 10 15 204m 25 ' 30 35' ;.ME_1

 

 

 

 

 

 

 

 

 

 

 

A : —~—ME—2
:11, la, . ' I—-—ME-3
: . , +ME-4

0 +ME-5

0.05 0.1 0.15 5 0.2 0.25 _+_ MEJB

Figure 4.10: For traces ME-l, ME—2, .\lE~3, ME—4, ME—5 and ME—6 each subfigure,
(from top to bottom): [Top] RM I 3(1, m) of BER process observed in MC traces for
lag m varying from 1 through 40. [Middle] ARM I B(1, m) of BER process for the
same channel traces. [Bottom] The memory length MB plotted as a function of 6,
the increments in RMIB(1, m).

packets which, at the packet transmission rate of 10 packets per seconds used for
trace collection, corresponds to a time period of 0 to lsec. Thus, diversity in physical
environments and channel selection do not appear to have any significant bearing on

the duration of fast fades, as measured by RMI.

On a side note, there is a major practical challenge to the online and/ or real-
time computation of RMI. The computation of RM I B(1,m) is based on an m +
1 dimensional joint PDF of all processes 3(0),B(1),B(2),...,B(m). Populating
such a high dimensional PDF requires large data set even for moderate values of
m. Collecting data points to fill a modest 10 dimensional PDF takes significant time,
especially when considering that IEEE 802.15.4 is a low rate communication standard.
This makes this methodology especially unsuitable for applications in which this

computation is to be performed online. We repeatedly performed this computation

77

ARMB(1,m) RMB(1,m)

MBe)

ME-1 1
ME-

 

Figure 4.11: For traces ME—7, ME—8, MEI—9, ME-lO, ME—ll and ME—12 each subfigure,
(from top to bottom): [Top] RM I B(1, m) of BER process observed in MC traces for
lag m varying from 1 through 40. [Middle] ARMIBU, m) of BER process for the
same channel traces. [Bottom] The memory length MB plotted as a function of 6,
the increments in RM I B(1, m).

offline on different data sets, and found that even on a mid-level server class machine
the computation of the RMI function based on 50,000 data points for m ranging from

1 to 40 takes more than 30 minutes.

4.5 Conclusions

1. All IEEE 802.15.4 channels, regardless of channel selection or physical environ-

ment, exhibit a memory length of at most 2 bits and 2 symbols, respectively.

[\3

. Based on the correlation function and LRD based analysis we conclude that
various estimates of Hurst parameter may or may not detect packet level mem-
ory in 802.15.4 channels. The ’memory’, however, is not due to the channel’s

inherent properties at those frequencies, but due to interference from 802.11b/ g

78

traffic and beacon frames, i.e. if interference is periodic the channel appears to

have memory, if it is not periodic there is no memory.

. The Abry-Veitch and Whittle estimators’ consistent relative insensitivity to
changes in average PER, average PLR, average CBER and interference across
different traces leads us to conclude that they are better measures of the 802.15.4

channel’s inherent degree of LRD.

. The aggregate variance, R/ S, periodogram, and absolute moment estimators’
strong dependence on average PER and average PLR leads us to conclude that

these estimators are good detectors of (W’LAN) interference.

. The average CBER to which a packet is subjected by a channel is inversely
related to the average PER/ average PLR. Thus, it appears that interference

produces higher BERs in packets than do channel fading effects.

. We present use of RMI as a standardized version of the mutual information and
apply it to the BER process captured in bit traces. We observe that interference
free 802.15.4 channels are memoryless, while channels experiencing significant
interference from 802.11b / g networks sharing the 2.4GHz ISM band, a common
source of interference, have ”true” memory lengths varying in the narrow range

of 0 to lsec.

79

Chapter 5

A Statistical Measure Of Network
Lifetime For Wireless Sensor
Networks

80

5. 1 Introduction

One of the most fundamental challenges in wireless sensor networks (W SN ) is the
short and often limited supply of energy. Due to the disposable nature of WSN nodes
power sources are often non-replenishable or replenishable very slowly at best. In
either case this forces prudent use of battery power for all operations. Since WSNs
can be spread over large geographical areas multi—hop communication is employed
in transmitting sensor measurements to the data collection point, also known as the
base station. The problem is further compounded by the many-to-one traffic flow
pattern that is imposed by the data collection process. It produces a traffic hot-spot
or bottleneck around the base station and, depending on the positioning of nodes, in
other regions of the network as well. This phenomenon is called the reachback and
was investigated by Barros and Servetto in [9]. If the same sl'iortest-path-first (SPF)
routes [30] are maintained, as in the case of most present day mobile ad-hoc networks
(MANET) and W SN routing algorithms, nodes start running out of energy. Nodes
gradually start disappearing from the sensor network beginning with those handling
the highest traffic volume, the ones communicating directly with the base station.
Such nodes are referred to as critical nodes. Each node going offline will reduce
coverage provided by the WSN. Eventually all nodes in communication range of the
base station will run out of power and the base station will stand disconnected from
all surviving nodes, effectively dropping situational awareness at the base station to
zero. While the nature of the trafiic flow makes the degradation of system capabilities
over time inevitable, it is desirable to make it as graceful as possible. This leads us to
consider a solution that will redistribute the volume of traffic handled by critical nodes
more evenly. It is noted here that any deviation from routes selected using a routing
algorithm based on SPF means selecting a route that is suboptimal in the traditional,

greedy sense (of which there are many even for small sets of critical nodes).

81

The above discussion provides us with two objectives; 1) reducing the differences
between energy consumption rates of nodes, 2) but at the same time keeping the
average energy consumption rate low. This requires the joint minimization of both
objectives. Since these two objectives run counter to each other the selection of an
operating point is a trade-off between mean and variance of power consumption rates.

The remaining chapter is organized as follows. Section 5.2 reviews some recent
efforts that attempt to increase longevity of WSNS and positions our work. Section
5.3 describes our interpretation of network lifetime and how the advantages of its use
as an objective over previous definitions. Section 5.4 describes our network model and
introduces terminology. Section 5.5 formulates the problem as a quadratic program.
Section 5.6 describes some results for small scale examples and section 5.7 concludes

the chapter.

5.2 Previous Work

The volume of works spanning energy efficient routing protocols for WSN is extensive.
Early WSNs borrowed routing protocols from ad-hoc wireless networks and MAN ETs.
The routes selected by dynamic destination-sequenced distance-vector (DSDV) [99],
dynamic source routing (DSR) [61], ad—hoc on-demand distance vector (AODV) [100]
and directed diffusion [58] protocols are ” optimal” only in a greedy, SPF sense which
worked well enough for networks without power constraints. The performance of a
system using these protocols is as much subject to the reachback problem as one
making use of naive SPF routing.

In [25] Chang and Tassiulas formulate the lifetime problem as a minimax linear
program (LP) that seeks to maximize the minimum sensor lifetime. However, while
the approach is theoretically sound and provides a bound for any attempt at maxi-

mizing that particular notion of network lifetime, there are scalability problems which

82

are exacerbated by the very large number of optimization variables for which the LP
is solved. This was followed by several other LP formulations of the lifetime problem
[11],[107],[135], [78], all based on the same or similar meaning of network lifetime, of
varying degrees of usability. In [6] Back and de Veciana proposed a proactive multi-
path routing scheme by introducing joint minimization of the ”spreading factor” w,
and the probability of battery depletion of a sensor which is similar to Ilyas and
Radha’s parallel work in [53] that uses average and variance of power consumption
rates in sensors. However, Back and de Veciana’s mechanism used for path discovery
does not take into account several other available conununication links. While the
authors demonstrate the improvements offered by their energy balancing algorithm
in networks with any-to-any data flow the proposed solution does not seem to offer an
improven'ient when the traffic flow is many-to-one/ all-to—one. More recently Khanna,
Liu and Chen [65] took an evolutionary approach. However, this was marked by a
high complexity due to the inherent nature of Genetic algorithms and poses challenges

to scalability.

5.3 Novelty Of Approach

The bulk of previous work on the lifetime problem defines network lifetime as the time
until the first sensor runs out of power. The. rigidity of this definition is of advantage
because it provides a clear objective function for optimization. However, any set
of routes that deviate from greedy SPF routes produce an increase in the power
consumption rates of some nodes, decreasing their individual lifetimes. The prior LP
approaches that maximize the minimum sensor lifetime are no exception. However, by
solely focusing on one sensor’s lifetime (the minimum lifetime sensor), it ignores the
cost, the decrease in other sensors’ lifetimes at which this maximization is achieved.

This also implies a higher rate of failure of sensors as a network approaches the end of

83

its life, as defined under LP minimax problem formulations. This definition disregards
the inherent redundancy in WSNs and their ability to cope with a limited device
failure rate. In this chapter we use a notion of lifetime that takes these ”shortcomings”
into account. We propose the joint use of two statistics of P, the random variable

modeling the energy consumption rates of sensors in a W SN, namely;

1. E[P] : the mean of P.

2. Var[P] : the variance of P.

The problem then becomes a joint-minimization problem. This notion of lifetime
takes into account the lifetimes of the entire population of sensors making up the
network. Some previous solutions such as Singh, Woo and Raghavendra [113] describe
the independent minimization of only the variance of node power levels to extend the
lifetime of a network. l\«'Iinimization of E [P] alone is achieved by SPF routing protocols
based on energy as a cost metric. However, selecting routes based solely on the
minimization of E [P] will inevitably lead to the aforementioned reachback problem
where sensor nodes closer to the base station transmit packets at significantly higher
frequency compared to sensor nodes farther away. This problem can be formulated
as a budget constrained allocation problem: The minimization of Var[P] defined as
equation 5.1 is subject to the constraint equation 5.2 that E [P] be less than some
maximum budget value E[P]*, where P,- is the energy consumption rate of node

n,,‘v’ g i g N, and N is the number of nodes.

ZN (Pi. - E[P])2

 

min Var[P] = 2:1 . (5-1)
1’\/
Subject to,
N P
E[P] .. ZZZ-,1 7* g E[P]* (5 2)

Using the constraint formulation highlighted above, we can summarize our opti-
mization framework as identifying the set of routes throughout a given WSN such

that the following is satisfied: minE[P]<E[P]* {Var[P]}.

5.4 Network Model

To model devices we are adhering to the IEEE 802.15.4 low-rate wireless personal area
network (LR-WPAN) draft standard [5]. The standard defines two device classes,
reduced-function devices (RF Ds) and full-function devices (FF Ds). Since RFDs are
incapable of performing routing functions and since we are investigating a routing

solution we use the term node to refer to FFDs only.

A W'SN consists of N F F D + N R F D sensors and one base station. Since we
will only be dealing with F F Us in a routing solution we abbreviate N F F D by N.
FFD routers are numbered 1 through N and denoted by 721 through 17. N The base
station is assigned FFD ID 0 and denoted by ”0' F urtherrnore we are assuming the
nodes participating in the WSN to be capable of measuring received signal strength
indication (RSSI) and link quality indication (LQI) for received packets and adjusting
their transmission power as laid down in the standard. Nodes are capable of varying
transmission power at run—time on a packet-by—packet basis. Transmission power is

chosen as a function of the spatial separation between transmitter and receiver.

For the channel model we adhere to the model of the 802.15.4 Physical Channel
Modeling Subgroup [86]. We are modeling the conu‘nunication range of a sensor in
each direction by a Gaussian random variable Z with mean )1 Z and variance 022. As
a result, some links in the network are unidirectional. Each link from a node n.) to

another node n j is assigned a link cost Cij obtained by,

85

d“<n«.n->+z<u .0?)
Ci): 2' J B, Z Z 20. (5-3)

2

 

Here a is an exponential decay factor which varies for wireless communication
from 2 to 4, depending on the type of environment. An assignment of c, j denotes
the absence of a direct link from n, to 729-. Here d(a, b) is a function that returns the
Euclidean distance between the nodes provided in the argument and Bi denotes the
battery reserve of 71.). The cost Cij is the cost of transmitting the packet from 72.,- at a
power that ensures reception at nj with a minimum signal—to—noise ratio (SNR) [104]
of S I N Rmz’n with required certainty.

For the MAC model we refer to the 802.15.4 LR—VVPAN standard [5]. The stan-
dard is geared towards energy-conservation. Since the protocol supports a slotted
collision sense / multiple access (sCSMA) mode based on time division multiple access
(TDMA) we can make the reasonable assumption that there is little interference as

long as link utilization remains under the maximum possible data throughput rate.

5.5 Quadratic Program Formulation

In view of the new understanding of network lifetime and the associated objective
function we formulate the lifetime maximization program as a quadratic program
(QP). Let Q,- denote the data produced by n,- for transmission to no. This value
can depend on the spatial distribution of the entropy [32] of the underlying event
being sensed by the WSN. Let (1.1-j denote the flow from n.) to its neighbor nj with
which it communicates directly. Furthermore, qij Z 0. If C'ij denotes the cost of
communicating a unit of information directly from n,- to nj, then the cost incurred
by "2' in communicating qz’j to nj is cijqzjj. We are not assuming the use of any in-

network processing that might violate the conservation of flow in the network. This

86

1—

 

 

 

 

 

 

 

 

jS

\
/I\\
oi

Figure 5.1: The law of conservation of flow requires that the sum of incoming flows
qjfl- and data 6.2,; generated at node n.1- must equal the sum of all outgoing flows q'ik'

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

condition is illustrated in figure 5.1. Then, if Si denotes the set of nodes n, can relay
its flow to, the condition of flow conservation for any single node n,- can be expressed

as,

2765., 963,-, (5.4)

Note that the base station n0 has been deliberately excluded from the condition
of flow conservation in equation 5.4 since it is a consumer of flow. Since links between
nodes are not necessarily bi-directional, hence generally Cz’j # Cjz” If m,- denotes

the cardinality of 5,, then individual elements of 52' are referred to by 51(3) where

87

1 S j 3 mi. For the purpose of simplified notation elements within Si are assumed
to be sorted in ascending order of their node IDs. The total number of links in the
network is denoted by M = 22:17”? We formulate the problem as a quadratic

program in matrix form. Let (1 denote the flow vector that is to be optimized and Q

the flow generation vector as in.

T
Q = [(11,810) ' ' ' (11,Sl(7711)q2,5'2(1) ' ° ' q2,SQ(mg)qN,SN(1) ' ° ' qN,SN(mN)]
(5.5)
_ ... T (5 6)
Q. - [Q1522 QNl '
Let the separable cost matrix C be defined as,
- a T
01.510) 0 "' 0
01,310,”) 0 "' 0
C = (5.7)
O C2,Sg(mg) ” ' 0
O 0 U. CfV,SN(1)
_ 0 0 ... CN,5N(mN) -

 

 

C contains the same information as the adjacency matrix of the network, but is
organized in a sparser and separable fashion more suitable for later use. Since the bulk

of power consumed in sensors is due to data transmission we are associating a cost

88

only to the data transmission process. Power consumption for reception is usually
constant and Slgnificantly less than transm15s10n power. Since P, = ZjES, 0,311,]-

the objective function of minimizing the variance of power consumption in equation

5.1 can be expressed in the terms defined above as,
2
1 N 1 N
min 75 Z Z We — 7v- :1 Z 66qu
2:1 jeS, 2=1€,~S
2 2
N 1 N
z z + N 2: we
i=1 jES,‘ i=1j€3ic
2 N
W a] (119 :2: 02'jo
jeS, i=1jeS,

Note that we dropped the 7%, term. Equation 5.8 can be rewritten in matrix form

as.

T 1
min C C +—
(q) q N

 

[V
=n1ianCTCq + < N2 ) (lTCgXlTCg)

1 - 2N
=n1ianCTCq + (T) (“7971) (lTCg)

I

1 — 2N
= min QT (CTC + < V2 ) CTlNXNC) q.
j _

Here _1_ denotes a vector of ls. Since the second order coefficient matrix in equation

(5.9)

 

5.9 is symmetric this QP has a. solution, provided that the constraints are well defined.

The budget constraint in equation 5.2 can be expressed in matrix form as,

T
1 (.'

 

89

gtO (5.11)

Finally we need to define the constraint based on the conservation of flow. We

define a flow matrix F of size N x AI as below.

 

 

FT =
f1,Sl(1)(1a1) f1,31(1)(Na1)
f1,Sl(7I'll)(1"ml) f1331(1)(N, T711)
: 3 (5.12)
f2a52(m‘2) (1’ 212:1 771,-) H. f2,52(m2) (N’ 2:2le m,)
fjv,S]V(1) (1i 2:11:11 771., + 1) H . ffV,Sjv(l) (17V, 26:11 77% + 1)
fN,S./V(mN) (1’ 23:1 722,-) fNi'SmeN) (N’ Z:z'A—L—l mi) 1

Elements f

(1,Sb(c) (d, e) for which d = b are set to 1. All elements for which d 6 Sa

are set to —1. All remaining elements are set to 0. That allows us to express the

condition for the conservation of flow like in equation 5.13.

pg 2 Q (5.13)

Thus equations 59,510,511 and 5.13 constitute the QP formulation. Note that
the base station or data collection point no is exempted from the condition of conser-

vation of flow. The nature of this formulation is such that the solution g’“ provides

90

the optimal distribution of traffic with which outgoing links should be utilized.

Any routing strategy that seeks to redistribute the traffic load deviates from the
SPF routing strategy that provides minimum global energy consumption, thereby
raising E[P]. The increase in Chang and Tassiulas’ [25] defined network lifetime
comes at the cost of decreased individual lifetimes of some other nodes in the network.
However, the LP formulation is such that there is no control over the cost at which
this increase in lifetime is achieved. A benefit of the QP formulation over previous LP
formulations is that the objective function provides a global View of node consumption
rates.

Unfortunately, the complexity of solving this QP even for moderate values of M
and N is too high to be of interest for practical use. Nevertheless it provides us with

a bound on the best possible solution given a set of constraints.

5.6 Results

We applied the QP formulation to some example networks of varying, yet manageable
sizes of 10, 15 and 20 nodes. The nodes are randomly scattered in a square region
of 10 x 10 dimensions. All nodes are assumed to have equal initial battery reserves
B1(0) 2 82(0) = = BN(0) = 1 with maximum transmission range 7‘ == 10. The
Gaussian noise source producing irregular link costs is set to Z (10, 2). The decay
factor is taken a = 3, typical of omni-directional antennas in open spaces. The
spatial distribution of the sensed event’s entropy is assumed to be uniform. Hence,
all sensors are generating data at a uniform rate as well, i.e. Q1 = Q2 2 . . . = Q N'

To illustrate the QPs ability to offer a gradual tradeoffs of Var[P] for E[P]. Each
operating point in figures 5.2, 5.3 and 5.4 is obtained by successively relaxing the
constraint in equation 5.10 by incrementally raising E [P]* and solving each resulting

QP. As expected, each successive solution offers decreasing Va'r[P]. However, after a

91

Mean vs Variance of P

I

10,000
‘5' 40,000- . .
3 Au; ..3' ' 3:" ' .'

> 30,000 ”saw-":1"

20,000 .

 

 

1090930 100 150* 200 250
E[P]

Figure 5.2: Tradeoff of Var'[P] versus E[P] for a network with N = 10.

certain point Var[P] starts increasing again.

5.7 Conclusions

We propose a new definition of network lifetime consisting of E [P] and Va'r[P]. This
notion of network lifetime provides a more inclusive view of the power consumption
of sensors across the network. The objective function offers an alternative view of
network lifetime. We went. on to formulate the optimization problem for the new

objective function in the form of a QP and showed that a solution exists.

92

MeanvsVarianoeofP

E in

> " -- ' '
60,000 " -. I'- . - ' I
m,m P— . ... -' . l. :. l .

 

 

o 200 400 600

Figure 5.3: Tradeoff of Var[P] versus E[P] for a network with N = 15.

93

800

VARIP]

E t

MeanvsVarianoeofP

 

 

do so 160* 1.5.0 1210 150
E[P]

Figure 5.4: Tradeoff of Va7‘[P] versus E [P] for a network with N = 20.

94

Chapter 6

A Dynamic Programming
Approach to Maximizing Lifetime
of Sensor Networks

95

6.1 Introduction

Wireless sensor networks (WSN) are the enabling technology for applications rang-
ing from infrastructure protection and Operation, emergency and crisis intervention,
all the way to surveillance and environmental monitoring systems. One of the most
fundamental. constraints of WSN is the short and often limited supply of energy.
Difficulties accessing sensors post—deph)yment, hostile deployment environments, and
impracticality of performing maintenance operations on individual sensors requires
making sensors disposable. Power sources are often non-replenishable or replenish-
able very slowly at best. In either case this forces prudent use of battery power for
all operations. Since W SNs can be spread over large geographical areas, multi-hop
communication is employed in transmitting sensor measurements from sensor nodes
to the data collection point or base station. The problem is further compounded by
the many-to—one traffic flow behavior witnessed in information gathering for in-situ
and remote sensing applications. It produces a traffic hot-spot around the base sta-
tion and other regions of the network with traffic bottlenecks. Barros and Servetto [9]
called this phenomenon the reachback effect. Wan et al. [125] named it the funneling
effect. If the same shortest-path-first (SPF) routes [30] are maintained, as in the case
of most present day mobile ad-hoc networks (MAN ET) and WSN routing algorithms,
some nodes will run out of energy sooner than others. Nodes will gradually start dis-
appearing from the sensor network beginning with those handling the highest traffic
volume, i.e. the ones connnunicating directly with the base station. These nodes
are referred to as critical nodes. Eventually all nodes in communication range of the
base station will run out of power and the base station will stand disconnected from
all surviving nodes, effectively dropping available coverage to zero. It is desirable to
stave off this event for as long as possible. This leads us to consider a solution that

will redistribute the volume of traffic handled by critical nodes more evenly. It is

96

noted here that any deviation from routes selected using a routing algorithm based
on SPF means selecting a suboptimal route (of which there are many, even for small
sets of critical nodes). We discuss and compare several methods for obtaining or-
dered listings of suboptimal paths. A dynamic programming algorithm (DPA) then
selects an optimal set of routes. The link cost metrics are derived from physical layer
information, thus qualifying this route selection method as a cross-layer approach.
The above discussion provides us with the objective of reducing the differences
between energy consumption rates of nodes while keeping the average energy con-
sumption rate low. This requires the joint minimization of both objectives. Since
these two objectives have conflicting requirements the selection of an operating point
becomes a trade-off between mean and variance of power consumption rates. We
evaluate a number of route discovery algorithms that produce paths other than the
shortest paths between source and destination. This is followed by picking one path
for each source-destination pair in a way that energy consumption is spread out, yet,

kept low. The core contributions of this work are summarized below.

1. We propose mean and variance of the distribution of node power consumption

rates as alternative optimization objectives for network lifetime.

2. We provide a low-complexity, dynamic programming formulation of the opti-

mization problem rooted in operational rate-distortion theory.

3. We evaluate various algorithms for the discovery of paths deemed sub-optimal

in the SPF-sense as inputs to the dynamic programming algorithm.

The rest of the chapter is organized as follows. Section 6.2 reviews some recent
efforts that attempt to increase longevity of W SNs and positions our work. Section
6.3 describes our network model. Section 6.4 is the formal problem formulation.

Section 6.5 describes four route discovery algorithms. Section 6.6 explains in detail

97

the working of the DPA inspired by operational rate—distortion (RD) theory. Section
6.7 presents an in-depth analysis of the performance of DPA and the route discovery

algorithms used in conjunction with it. Finally, section 6.8 concludes this chapter.

6.2 Previous Work

The volume of works spanning energy efficient. routing protocols for WSN is extensive.
Early VVSNs borrowed routing protocols from ad-hoc wireless networks and MAN ETs.
The routes selected by dynamic destination-sequenced distance-vector (DSDV) [99],
dynamic source routing (DSR) [61], ad—hoc on-demand distance vector (AODV) [100]
and directed diffusion [58] protocols are ”optimal” only in a greedy, SPF sense which
worked well enough for networks without power constraints. These protocols are
prone to the reachback reflect. In [25] Chang and Tassiulas proposed an optimization
approach based on linear programming. However, their formulation maximizes the
minimum node lifetime because it defines the system lifetime as the time until the
first node runs out of power. More recently, Baek and de Veciana proposed in [6] a
proactive multi-path routing scheme which bears some similarity to our own inter-
pretation of the problem by introducing joint minimization of a ”spreading factor” to,
and the probability of battery depletion of a sensor which is very similar to Ilyas and
Radha’s parallel work in [53] that uses average and variance of power consumption
rates in sensors. However, Baek and de Veciana’s mechanism used for path discovery
does not take into account several other available communication links. While the
authors demonstrate the improvements offered by their energy balancing algorithm
in networks with any-to-any data flow the proposed solution does not seem to offer
an improvement when the traffic flow is many-to-one. More recently Khanna, Liu
and Chen [65] took an evolutionary approach. However, this was marked by a high

complexity due to the very nature of Genetic algorithms and poses challenges to seal-

98

ability. As we will show in the following sections the DPA proposed here resolves

traffic hotspots in a WSN irrespective of the nature of traffic flow.

6.3 Network Model

6.3.1 Device Model

To model devices we are adhering to the IEEE 802.15.4 low-rate wireless personal
area network (LR-WPAN) draft. standard [5]. The standard defines two device classes,
reduced-function devices (RFDs) and full—function devices (F F Ds). RF Ds feature only
a limited implementation of the features defined by the standard. They are capable
of associating and communicating with F FDs only and are incapable of performing
routing functions. FFDs on the other hand have a full implementation of the standard
and are capable of associating and communicating with both F FDs and RFDs. FFDs
are also capable of performing routing functions. Since the DPA is a routing algorithm
and only FF D3 are capable of performing routing functions all nodes in our network

model are assumed to be FFDs.

A WSN consists of N _FFD sensors capable of performing routing functions, ad-
ditional RFD sensors and one base station. FFD routers are numbered 1 through
N. The base station is assigned FFD ID 0. Furthermore we are assuming the nodes
participating in the W SN to be capable of measuring received signal strength indi-
cation (RSSI) and link quality indication (LQI) for received packets and adjusting
their transmission power as laid down in the standard. Moreover, nodes are capable
of varying transmission power at run-time on a packet-by-packet basis. Transmis-
sion power is chosen as a function of the spatial separation between transmitter and

receiver and the signal decay factor described next.

99

6.3.2 Link Model

We are employing and adhering to the channel model proposed by the 802.15.4 Phys-
ical Channel Modeling Subgroup in [86]. Early exploratory work by Ilyas and Radha
[53] assumed the simple disk model for a node’s communication range. A consequence
of this model was that it made all links bidirectional which is not necessarily true in
real wireless networks. We are modeling the communication range of a sensor in each
direction by a Gaussian ramqhnn variable 3 with mean #3 and variance 0%. Each
j is assigned a link cost Lisj' The link cost is

a function of the energy cost of receiving, processing and (re-)transmitting a packet

link from a node n,- to another node n

(reliably) and the remaining battery level at the transmitter. Although the total
energy consumed is the sum total expended in receiving, processing and transmitting
a packet, this total is dominated by the energy of the transmission step. The cost. of
receiving a packet depends on the platform used for a sensor and is approximately
constant for equally sized packets and the processing energy is negligible in compari-
son. The link cost Li, j in equation 6.1 is the cost to node "i of transmitting a packet
to node nj, that is dw- distance away. For an omni-directional antenna the decay
factor a is usually around 3 [86]. Li,j also takes into account the remaining battery
level in a node. The sum of reception, processing and transmission cost is divided by

the fraction of remaining battery level 8,.

(10 .
«lay-Z where did- 5 Ci
Liaj = l
00 where di‘j > c, (61)

100

Table 6.1: Symbols and notation.

 

 

 

 

 

Symbol Description
"0 Base station
nlj FFD sensor node with identifier 1
EM- Maximum communication range of n,- in the direction of
G (V, A) Graph consisting of vertex set V connected by directed edge
set A.
Ai,j Directed edge from n,- to nj.
T,- Rate of traffic generated at ”2"
Pi, j k-th best path from 71.2 to n].
E,- Energy consumption rate of 71.,- under the global traffic flow
under consideration.
,u E Sample mean of energy consumption rates E, for 1 S 2' S
N.
0%: Sample variance of energy consumption rates E-z' for 1 _<_
2 S N.
S P(i —-> j, G) Shortest path from n,- to n]- on graph G.
Lm- Link cost on ”i for link from ”2' to nj.
3.,- Fraction of total battery power remaining in "i-

 

Then the power consumption rate of a node n,- V 1 S i S N is denoted Ez- and is

defined in terms if link cost terms like in equation 6.2,

N N
E: Z Z L,“- for all 1 g is N
n=1 .
321k mm
Naefih

101

6.4 Problem Formulation

WSNs serve to provide situational awareness and collect data from the entire region
covered by them. An area is said to be covered as long as there is at least one sensor
taking measurements from it. Data reporting of an area can stop due to two reasons,
1) The sensor providing coverage runs out of power, and 2) the sensor is unable to
route its data to the base station due to a partition / fragmentation of the network.
When a node stops cormnunicating data to the base station there is a drop in coverage.
Therefore, the utility of a W SN is related to the time period for which a W SN can
maintain minimum coverage. The skewed distribution of energy consumption rates

in sensors requires a redistribution of traffic load that,
1. Produces a more evened out traffic load across sensors.

2. Yet, at the same time, guarantees low total energy consumption.

In mathematical terms, if E is a random variable modeling the energy consumption
rates of sensors in a WSN, we aim to minimize its sample variance 0% (defined in

equation 6.3) while keeping its sample mean “E low (defined in equation 6.4).

2 __ ZillEi ‘ HE)2

 

0'
E N (6.3)
v
M = ——Z"l=1 Ei.
E N (6.4)

This leads to a joint-minimization problem. l\~‘Iinimization of n. E alone is achieved by
SPF routing protocols based on energy as a cost metric. However, selecting routes
based solely on the minimization of n E will inevitably lead to a situation where sensor

nodes closer to the base station transmit packets at significantly higher frequency

102

compared to sensor nodes farther away. In a multi-hop WSN this implies significant
variations in individual nodes’ energy consumption rates. Due to the optimality
principle of subproblems, the SPF routes from sensors to the base station form a tree
rooted at the destination. Over time the region of failing nodes will expand from the
base station away. Once this region spreads to the maximum transmission range of
a sensor node in all directions around the base station it is cut off from the rest of
the network. The network will experience a partition [119] despite the fact that there
will be a large number of nodes with significant battery life left. This problem can
be formulated as a budget constrained allocation problem: The minimization of (7%:
in equation 6.5 is the objective function. Minimization of the objective is subject to

the inequality constraint that p E be less than some (user defined) maximum budget

value a E budget in equation 6.6.

 

- 2
mina
E (6.5)
subject to constraints,
N N L7,],
:1 Z N S ”E budget'
71:
. = 1
J _ (6.6)
. . 1
A27] 6 PTL,0

In [113] Singh, Woo and Raghavendra describe the independent minimization of only
the variance of node power levels to extend the lifetime of a network. We propose
joint minimization of [,L E and 0%. Using the constraint formulation highlighted above,
we can summarize our optimization framework as identifying the set of routes, one

for each source destination pair in a given W'SN such that 0% is minimized while

103

n E _<_ p E budget' This is analogous to minimizing a distortion measure given a rate
budget constraint under an RD framework. Consequently, our proposed dynamic
programming algorithm (DPA) is based on an operational RD—framework where we
strive to select the mean-variantte (MV) operating point that offers the lowest achiev—
able 0%) while maintaining it E S p E budget The solution takes the form of a set of
routes, one from each node to destination (base station). In terms of the optimiza-
tion problem formulated in equations 6.5 and 6.6, the solution is the set of values
k1, k2, k3, . . . , kN that select a corresponding set of paths 1311?), 192/6%, Pfifl’ . . . ,P:%
that minimize the variance of the power consumption under the simultaneous flow of
traffic from all nodes.

We have not, to this point, discussed the generation of sets of alternative routes
Pig) from every node 1 S 2' _<_ N to the base station. In the following section we

describe four different algorithms to obtain alternative paths between sources and

destination.

6.5 Route Discovery

Sub-sections 6.5.1 through 6.5.4 describe the four different route discovery algorithms
used for finding alternative paths from nodes to the base station. The four types
of routes are bottleneck edge disjoint (BED), bottleneck node disjoint (BND), edge
disjoint (ED) and node disjoint (ND). To explain and illustrate the differences between
the route discovery algorithms figure 6.1a through 6.1d show the different routes
determined from sensor node 7199 in a W SN consisting of a base station and N = 99
randomly placed FF D sensor nodes in a square shaped region of size 10 x 10 with
each nodes transmission range chosen from the Gaussian distribution N 01.30%)

(with mean #3 and variance 0.2:). For the following discussion we represent the

H

W SN by graph a C(V, A). Nodes are represented by the set of vertices V and the

104

Bottleneck Edge Disjoint Routes Bottleneck Node Disjoint Routes

 

 

   

1 O
(a) BED paths. (b) BND paths.

Edge Disjoint Routes Node Disjoint Routes

        
 
  

     

         

  
    

   

        

            
  

   

 

 

l , r: ’;_.’,4'// .\_
‘Ffi'f‘W/M’. (‘1‘?"7‘
"lint/affix“;--[lflgllfll" \
5 (WV/1V ‘ ‘9’;
‘(/’ Z ual/m (
//’l‘ ' fl:“"'//"""' 737” V<."'.r,’A‘-A
1727 "W’
0 5 10

(C) ED paths. (d) ND paths.

Figure 6.1: Paths from 7199 to no.

communication links between elements of V is denoted by the set of edges A. A
directed edge from vertex n," to nj is denoted by Ai,j- Furthermore, let F denote the
minimum spanning tree rooted at the base station on the connected component of G.
We also define the maximum power node function MaxPowNodefl") which returns the
node that experiences the highest power consumption when traffic flows to the base
Station along I‘. The predecessor function Pred(I‘, z") returns the parent node of n,-
in P. The indegree(nz~, F) function returns the in-degree of node n,- in graph I‘. The

Shortest path from vertex "2’ to vertex nj on a graph G is denoted by SP(i —) j, G).

105

6.5.1 Bottleneck Edge Disjoint Paths

We define the bottleneck edge to be the outgoing edge from node MaxPowNode(F).
This method repeatedly determines and removes the bottleneck edge, and rediscovers
the minimum spanning tree F and from it the shortest paths from each node to the

base station. Figure 6.1a depicts all BED paths from ”N to no.

 

Algorithm 1 Generate Bottleneck Edge Disjoint Paths
Require: n 2 0 V :2: 3f 0
Ensure: 5(2) 2 {},V1 3 2' S N
G := Topology of WSN with all available links
F := C
while indegree(n0, F) > 0 do
F := Minimum spanning tree on G rooted at no
for all "2' do
if SP(i —) O, F) 74 S(end) then
S(i) :2 {5(1), SP(2'. —> O,F)}
end if
end for
G I: C(V’ A — A(=.\-'IaxPowNode(F),Pred(MaxPowNode(F),F))
end while

 

 

6.5.2 Bottleneck Node Disjoint Paths

A bottleneck node is defined as the node returned by the hvt’laxPowNode(F) function.
This method repeatedly determines the bottleneck node under traffic flow along F,
removes all its incoming edges from the graph, and rediscovers the minimum spanning
tree F and from it the shortest paths from each node to the base station. Figure 6.1b

depicts all BND paths from 7199 to "0°

6.5.3 Edge Disjoint Paths

Edge disjoint paths between two vertices ”i and 'nj in a graph are paths that do not

have any edges in common between them. To determine an ordered set of edge disjoint

106

 

Algorithm 2 Generate Bottleneck Node Disjoint Paths
Require: n 2 0 V 3: ;£ 0
Ensure: 5(2)={},V1§2'S N
G := graph of WSN with all available links
while indegree(n0, F) > 0 do
F := Minimum spanning tree on G rooted at no
for all 222- do
if SP(2 -—) 0, F) # 5(67221] then
5(2) 2: {5(2) SP(2' —> 0,F)}
end if
end for
G := G(V — MaxPowNode(F), A)
end while

 

 

paths from a node ml: to the base station this algorithm repeatedly finds and saves
SP(2' a 0, G) and removes all edges it is composed of from G. The Edge() function
returns the edge set of the path provided in the argument. Figure 6.1c depicts all ED

paths from n N to 220.

 

Algorithm 3 Generate Edge Disjoint Paths
Require: n 2 0 V a: aé 0
Ensure: 5(2) = {},\7’1 _<_ 2' g N
G :2 graph of WSN with all available links
while indegree(nO, F) > 0 do
F 2: Minimum spanning tree on G rooted at 720
for all ”2' do
if SP(2' ——> 0, F) 75 5[end] then
5(2) := {5(2) SP(2 —> 0, F)}
end if
end for
TempM azN ode := MaxPowNode(F)
While Temp.M acrN ode 75 220 do

G == G(V/1 — ATempMacrNode,Pred(TempMaxNode,F)l
TempltfaxNode := Pred(TempMa:1:N0de, F)
end while
_end while

 

 

107

6.5.4 Node Disjoint Paths

Node disjoint paths between two vertices 22,- and nj in a graph are paths that do
not have any nodes in common between them. To determine an ordered set of node
disjoint paths from a node 22,; to the base station this algorithm repeatedly finds and
saves SP(2 ——> 0, G) and removes all nodes it is composed of from G. The Vertices
function returns the vertex set of intermediate odes between source and destination
nodes of the path provided in the argument. Figure 6.1d depicts all ND paths from
n N to no. It is worth mentioning here that the ND paths discovered by this algorithm
bear resemblance to the routes determined between source and destination nodes in

the multipath routing method described in Baek and de Veciana [6]

 

Algorithm 4 Generate Node Disjoint Paths
Require: n 2 0 V2: aé 0
Ensure: 5(2')={},\7’1_<_2'S N
G := graph of WSN with all available links
while indegree(n0, F) > 0 do
F 2: Minimum spanning tree on G rooted at 720
for all 22,- do
if SP(2 —-> 0, F) # 5[e'n.d] then
5(2) := {[5(2) SP(2 —-> 0,F)}
end if
end for
TempM axN ode := MaxPowNode(F)
while TempZW axN ode # 720 do
G := G(V — TempMaxNode, A)
TempM 0.ch ode := Pred(TempM azN ode, F )
end while
end while

 

 

6.6 Dynamic Programming

The algorithms in the preceding section described four different methods of generating

a list of paths in increasing order of path cost from every node to base station. In

108

 

 

P101 P201 F’so1 rn" PNO1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P1,o2 Paoz P392 1---- Pun2
[31,03 P2,03 P3,o3 ""- PN,O3
I l l I

. . . . [k1 k2 k3 kN]
l l |
4‘ k1 Paokz : J !
Pm i j k3 PN 0kN
i . Pao ’,
I l i J l
l l l l
[31,0r1 l:’2,or2 Paor3 F’N,orN

 

 

 

 

 

 

 

 

 

 

Figure 6.2: N lists of routes sorted in ascending order of path energies.

this section we describe the DPA that selects one path from each list while achieving
the optimization objective. Figure 6.2 illustrates this idea and depicts lists of paths
obtained by one of the route discovery algorithms. Each column corresponds to a
source node 222-, and every box in it represents one of the 7',- paths from source node
”2' to destination "0' The shaded boxes identified by path indices [151, kg, k3, . . . ,kN]
represents the solution vector to the optimization problem. The DPA described in
this section traverses the space of possible solution vectors and attempts to find one

that approximates the optimum solution. but at significantly lower complexity.

6.6.1 Theoretical Background

Similar to the operational rate-distortion [32] problem in source coding [94], this joint

optimization can be mapped into a Lagrange optimization framework [39]. This for-

mulation is feasible if there is an optimum tuple (ME, 02E] such that ,u E S l1 E budget

109

 

(i.e. an optimal/ minimal overall energy can be found such that it exactly equals the

budget constraint). In this case, one can view the problem as one that allocates the

total energy budget N x It E budget to all N nodes in the network such that the
2

variance 0 E is minimum. This is referred to as the budget constrained allocation

problem [94].

In brief, we can formulate the lifetime maximization problem by considering the
Lagrangian cost J (A) = 0%: + ApE, which depends on the Lagrange multiplier A 2 0.
Identifying the optimum value for the Lagrange multiplier is a crucial aspect of this
approach. In particular, the parameter A represents the slope of the curve in the RD
plane. However, the computational complexity of finding the true optimal solution
is staggeringly high. The MV plane consists of a E on the horizontal axis and 0%:
on the vertical axis. The MV region consists of all MV operating points achievable
by different coml.)inations of possible paths, one from each node, to the base station.
The desired optimal solution will lie on the hull of the MV region. This is very
similar to the problem of finding the optimal quantizer in an operational RD sense
from Information Theory [32]. We exploit some of the strategies used under RD
optimization to develop a DPA for identifying the optimum solution points in the

MV space.

6.6.2 Dynamic Programming Algorithm

In order to reduce the computational complexity of finding the hull of the MV region
we use a DPA similar to the one used in the determining the hull of the RD region
for optimal quantizer design. In the sensor network lifetime maximization problem
under consideration it is desirable to minimize 0%, the variance of node power con-

sumption rates, and “Ev the mean of node power consumption rates. The reduction

in computational complexity over an exhaustive search of all operating points comes

110

(0,4)
(0) 2(0) x1
[:Il’lE ao-E :]
- (0,4) 0_2(0, 4)
[:IuE 90- :]

3103)

(0,3) 02(03)
1(01) 0,2 2 0,2
[#E( >90; 1]

(0,1) 2(0,1) 02(1)
[#5 90E :[Z [#15090- :]

 
 
    

 

V»

”E

Figure 6.3: Selection of next optimal point in the MV-plane by DPA.

 

at the cost of DPA being only an operational optimization method approximating
the exact optimal solution. The DPA operates 011 N sorted lists, one for each node,
each containing [P 11,—0l — r, paths to base station ordered in ascending order of their
path costs. The algorithm described here is greedy in nature. It is possible that
this algorithm does not find the optimum operational [111;], 0%] point under certain
scenarios. Nevertheless, the algorithm does provide the optimum operational solution
under many practical scenarios. Furthermore, a set of RD variations of this algorithm

are rather popular due to their simplicity and low-complexity ([103],[109],[118]).

(0) 2 (0)

The starting point [12E ,JE ] of the DPA is the operating point that offers the
lowest possible ,12 E and is obtained by selecting all P310 where Vj,1 S j g N and

. . . . . 0 .
18 located closest to the 0% ax1s. The nnnnnum mean power consumption 11%) is

111

associated with a relatively high variance 0%(0). The DPA uses [11(3) , 033(0)] as a
(1) (1)]

starting point to search for the next achievable operating point [11E ,0’%—: closest

to the hull of the MV region that adheres to the budget constraint Ebudget' The

0) 2 (0)

shortest routes that correspond to [11%. ,0 E ] are the ones at the top of all columns

in figure 6.2. If pair [12%). 0%.(0)] satisfies the specified constraint 112.0) S ,uE budget

and [11(El),0%(1)] does not, [ughagjlm] is the optimum point and the algorithm

terminates. However, in general, the above condition is not met except in special
cases (such as where all nodes are at an equal distance and one hop away from the
base station), and the algorithm proceeds to the next step. Rather than considering all
possible route combinations, at any operating point Lug), 0%.(i)] during the (2+ 1)th

iteration, the algorithm introduces the smallest increase AE in energy to each route

- - . ,- -, (’1?) 2 (2) 2+1

by con81der1ng replacement of le,0 111 the set of routes used for [11E ,0 E ] by 133,0

in turn. In a network of N nodes this produces up to N new possible operating points
' 2' 1 . . .

for [#%+1)10%~( + ) ]. The exact number of alternative operating pomts depends on

the number of route lists that still have an alternate path left for consideration. Each

of these possible alternatives is obtained by replacing one of the routes that produced

[11%),0%(7)] by the next path and computing a new operating point. This way up to

N different alternatives lflg+1’j),a%(l+1’])] for 1 Sj S N are obtained. For each

of these N possible operating points we compute their slopes MM) with respect to

previous operating point Lag), diam] like in equation 6.7.

 

.. U2('i+1,j)_02(i)
190): E, . E
A?” — 11%) (6.7)

. , _ . 2 1,'
Equation 6.8 shows that out of these N alternative operating pomts “(EFF J),

112

2+1, ' . . .' ' -
(7%; '3) we select the one which offers the lowest gradient /\(l’]m'ln).

jmz’n = arg min(/\(i’j))
(6.8)

Equation 6.9 assigns the point obtained by changing node njn . 1’s path to be the
: 222

next achievable operating point of the MV plane and advances the the route pointer

in column jmz’n of the route list in figure 6.2 down by one position.

(2317- ) 2 (2327 v )
: [H mm ,0. 771.277. ]
E E (6.9)

This way the algorithm iteratively determines the next point on the MV region’s hull.

(0) 2 (0)

Figure 6.3 illustrates a simple example. [11E ,0 E ] is the starting point and there
are four possible points in the MV plane one of which has to be chosen as the next

operating point on the MV curve. In this example M021) < M02) < M023) < /\(024).

Therefore, the point [11(19’1),0%3(0’1)] corresponding to the smallest slope M021)

(1) 2(1)

be chosen for [11E , a E ]. 111 this manner a curve is obtained in the MV plane which

will

estimates the boundary of the MV region. Figure 6.4 depicts MV curves for a WSN
using different route discovery algorithms at our disposal. The differently sloped lines
(A’, A” and Am) represent different benefit/cost (variance reduction/mean increase)
ratios, that can be used to identify a suitable operating point on the hull of an MV
curve. The curves shown in this figure are truncated at the right end when the mean
value becomes greater than the mean at the initial starting point. If the DPA is
allowed to run to completion the number of points returned by it is approximately
50 to 100 times the ones plotted here. However, the region of the curve that is of

interest to us is the one between the starting point and the point offering I'ninimum

113

180

 

160 ' ‘ "
150 \

140- \
N L” Net

t>130» . '

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

120—
, —BED
11o— , - BND
. M ED
100 ._ END
908 9 ' '10 1'1 1'2 113 14
“E

Figure 6.4: Mean-Variance tradeoffs offered by BED, BN D, ED and ND paths.

variance.

6.6.3 Computational Complexity of Finding Optimal Solu-
tion

There are two sources of complexity in this problem; 1) The computation of achievable
operating points, 2) the traversal of the search space for the best achievable operating
point. The graph G(V, A) that represents the WSN consists of a vertex set V of
cardinality N + 1 and a directed edge set A of maximum cardinality N 2 + N (we
consider a worst case scenario in which the network topology forms a fully connected

graph). First, we derive an upper bound for the order of complexity of the size of the

114

search space for a brute force method. The total number of paths from a node ”2'
to base station n0 with q intermediate hops is (N—1)Pq' Then the total number of
possible paths from n, to no is the number of paths with 0, 1, 2, . . . , N —1 intermediate
nodes, i.e. 26V_1 (N—1)Pq = 281—1 Ali-ail! ! = O(N!). Then the search space
of solution vectors for routes from all N nodes to destination consists of 0(N 1N)
points. A linear search of this space leaves the complexity unaffected, i.e. 0(NlN).

Now we compute the complexity of the DP formulation proposed in this chapter.
The size of the search space for in DPA is I121]:- qu. The expression of T2 in terms of
N depends on the route discovery algorithm used. For BED the upper bound on the
number of paths 'ij generated for a node ”2' is (N + 1)N = 0(N2) bringing the total
size of the search space to N - (N + 1)N = 0(N3). The bound for BED can be used
as an upper bound on the number of paths per node generated by ED.

For BND the upper bound on the number of paths 2‘, generated for a node n,-
is N = 0(N) bringing the total size of the search space to N - N = 0(N2). The
bound for BN D can be used as an upper bound on the number of paths per node
generated by ND. Since the removal of each entry from the search space requires
the computation of N slopes, the complexity of the DPA for BED and ED is upper
bound by 0(N - N3) = 0(N4) and for BND and ND it is upper bound by N ' N2 =
0(N3). Regardless of whether BED / ED or BND / ND are used for route discovery,
their respective complexity terms 0(N4) and 0(N3), respectively, are still orders

of magnitude lower than O(N!N), the order of the exhaustive brute force search

method.

6.7 Performance Analysis

For the comparative evaluation of BED+DPA, BND+DPA, ED+DPA and ND+DPA

algorithms we generate 100 randomly generated networks deployed in a square shaped

115

 

region of size 10 x 10. In all networks the base station is located at coordinates (0,0)
to obtain large variations in path lengths. N = 99 nodes are randomly and uniformly
scattered in the plane. Transmission ranges E of sensors are Gaussian distributed
according to N (10, 2). The WSN is monitoring a process with an entropy uniform
across the space spanned by the network. This implies that the traffic generation
rates of all sensors are also equal, i.e. T1 = T2 = T3 = = T99 = T. Sensors
are equipped with omni-directional antennas with decay factor a = 3. Initial battery

reserves Bi for all sensors are assumed to be 1.

6.7 .1 Mean-Variance Trade-off

Figure 6.4 illustrates the variations in the estimate of the MV region boundary pro-
duced by the DPA when different route discovery algorithms are used for the same
network. In this particular example the BND paths seem to offer the greatest decrease
in 0%, ND the least and BND and ED fall in between the two extremes. However,
similar plots of other networks may reveal that this ordering does not hold true for
all cases.

We will attempt to determine if there is a route discovery algorithm which provides
consistent superior performance over the other algorithms. The four scatter plots in
figure (5.5a through figure 6.5d correspond to the four route discovery algorithms.
Each plot is obtained by applying the DPA to the list of routes produced by one of
the four route discovery algorithm. Each point in a plot corresponds to one of the
100 randomly generated networks. Its position on the horizontal axis indicates the
percent increase in ,a E (denoted A11 E) and its position on the vertical axis the percent
decrease in 0% (denoted [30%) offered by the minimum variance point (marked by
red cross hair) on the MV curve with respect to its starting point (marked by black

2

cross hair). The decrease in variance of power consumption rates A0 E are savings 5

116

that are achievable at a cost C that is the increase in mean power consumption rate
A11. E' In the remainder of this section we will use the terms savings and cost to refer
to the two quantities.

As such, it is desirable that most data points be concentrated in the upper left
corner of the scatter plots, indicating high savings 5 achievable at low cost G. Points
located close to the origin indicate little change from the starting point. Data points
located in the lower right corner indicate bad trade-offs, i.e. low savings in return for
high cost. The great overlap in the regions covered by the points indicates that there
is no clear winner among the route discovery algorithms.

To gain further insight we project data points in scatter plots in figures 6.5a
through figure 6.5d on the horizontal/C axis, and the vertical/5 axis. This process
is similar to finding the marginal distributions of a joint distribution of two random
variables. The four histograms in figure 6.6 are obtained by projection of data points
in figure 6.5a through 6.5d on the horizontal/C axis. Similarly, the histograms in fig-
ure 6.7 are obtained by projecting onto the vertical / 5 axis. In most cases the shapes
of the histograms in figure 6.6 and figure 6.7 all appear to indicate an underlying
gaussian distribution. We estimate the values of the mean and variance parameters
of gaussian distributions fitting the histogram using. We compute the maximum like-
lihood estimates (MLE) [122] of the parameters 12C, 0%, )1 5 and 0%. based on 100
data points for each histogram. The MLEs for BED, BND ED and ND are given in
table 6.2 and the resulting Gaussian distributions properly scaled for comparison are
plotted together with the histograms. Since we were previously unable to clearly iden-
tify one route discovery method as being superior to others in terms of performance
we now evaluate their performances in probabilistic terms. A clear winner will have
a distribution for C with low mean and low variance and a distribution for 5 with a
high mean and low variance. Based on MLE parameters in table 6.2 ND emerges as

the least attractive option because it offers the smallest average savings 115 = 24.37%

117

Table 6.2: MLE parameters of marginal histograms generated from applying DPA
with BED, BND, ED and ND route discovery algorithms to 100 randomly generated
network topologies.

 

10 cg. 13 02
BED 15.25 328.76 28.09 353.20
BND 20.94 289.93 36.22 303.03

ED 32.25 504.43 35.37 307.67

ND 29.06 390.07 24.37 250.96

 

 

 

 

 

 

 

 

 

at nearly the highest average cost “C = 29.06%. V’Ve eliminate it from further con-
sideration. Although with 35.37% ED offers on average one of the largest savings, it
does so at a cost that is even higher than ND, i.e. an average savings of 35.37% in
return for an average cost increase of 32.25%. In comparison, the BND+DPA option
offers an average savings of 36.22% in return for a 20.94% increase in cost, which is
significantly lower than from costs of ED and ND. Finally, BED offers average sav-
ings of 28.09% at an additional cost of 15.25%. In comparison with BND this means
BED offers lower savings for a lower cost. However, the variances of cost. and savings
distributions of BED in table 6.2 show that there is considerably greater variation
and uncertainty about BEDs costs and savings. relative to BND. The significant dif-
ferences in variance terms 0% and (7% can be attributed to the fact that the Gaussian
seems to be an ill—fit for BED’s histograms in figures 6.6 and 6.7 and approximates
an exponential more closely than a Gaussian. While multi-modal distributions could

offer more accurate fits for the data they will complicate performance comparison.

We explain the poor performance of paths generated by ED and ND route dis-
covery algorithms with respect to BED and BND by the fact that they exclude not
just one, but all elements (edges or nodes) of a discovered path from consideration in
subsequent path searches. While this cuts down the size of the search space it also

removes too many alternative paths from consideration.

118

Furthermore, the high concentration of values in the bins centered close to 0 in
ED’s histogram in figure 6.7 indicates a high rate of failure of the DPA, i.e. instances
where the DPA is not able to offer a significant trade-off. BED has the highest number
of failures followed closely by ED. A closer look at the scatter plots of BED and BND
in figures 6.5a and 6.5b respectively shows that BED offers better performance on
tight budget constraints on 12 E budget in the low range of savings 5whereas BND
performs better when the constraint on 11 E budget is more relaxed and higher higher
savings 5 are required. The higher rate failure rate of BED makes its use a more risky
option. Also, it should be noted that the scatter plots only plot trade—offs offered by
minimum variance point, i.e. when A = 0. As the curve plots in figure 6.4 there is a.
large set of points that offer intermediate trade-offs for when lambda is in the range
—-00 < A < 0. This is illustrated in figure 6.4 for three different values of A set to A],

A” and A”,

6.7 .2 Spatial Redistribution of Energy

That leaves open the question of how effectively this method evens out the spatial
distribution of energy consumption rates observed under SPF routing. The diffusion
plot in figure 6.8 depicts the spatial distribution of energy consumption under SPF
routing. This plot is obtained by averaging it over the same set of 100 networks
used in the previous section. As expected, as one moves from the upper right corner,
towards the lower left corner where the base station is located there is a gradual
increase in the power consumption rate in nodes. The lower left corner, the region
in immediate vicinity of the base station occupied by critical nodes, forms a traffic
hotspot. To illustrate the differences in the spatial distribution of energy consumption
rates produced by the BED, BN D, ED and ND route discovery algorithms and DPA

we first subtract the diffusion plot of the SPF routes in figure 6.8. The results are the

119

 

 

 

 

 

 

 

 

NDUJ 1 NbUJ 1
.9 . . '. _ .9 _
g #1.. a: .... I g I
v:- r .
°\° - , o\° ,
50 100 100
%inorinnE %inorinuE
(a) BED paths. (b) BND paths.
ED ND
NbUJ 1m, NbUJ 1m
.E ' I I ,E
.5 a) . ‘I" 5' . gm I II ...-I
'- II' .-~.I ' . 5,,- II '
\O l I ‘ . °\o fl... .
° (5'7 50 100 (57 100
% incr in 11E °/o Incr in 11E

(c) ED paths.

(d) ND paths.

Figure 6.5: Plots of percent decrease in variance against percent increase in mean.

differential spatial diffusion plots in figures 6.9 through 6.12. Like the plot in figure
6.8, these too are averaged over all 100 networks. All route discovery algorithms
produce a reduction in power consumption rates in the region occupied by a peak in
the SPF’S diffusion map in figure 6.8. This is shown by the blue negative region in the
same place, representing a reduction in node power consumption. We also observe an
increase in power consumption rates marked by a yellow-red-yellow band that envelOps
the blue region on its upper-right side. This signifies a partial shift of relay traffic

from critical nodes to their immediate neighbors that are farther away from the base

station. A closer look at figure 6.9 and 6.10 shows that the redistribution of traffic is

120

 

BED

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

30
20 _
10 m
G a;— I
o 20 4o BND 60 80 100
m 30"
'5 20—
f
g 10-
4—0
G —A—— I
(D 0 20 40 ED 60 so 100
C 30~
H—
O 20-
2 °W
Gl .———q .
o 20 40 ND 60 80 100
30
20
10‘ 7:]LW
0] I J l
o 20 60 80 100

40. '
% Incr In 11E

Figure 6.6: Marginal histograms of percent increase ”A1113 for the scatter plots in
figures 6.5a through 6.5d.

Similar under BED and BND, although BND spreads the energy out over a slightly
larger region and that the reduction in power consumption in BND+DPA is around
8000 units, while BED+DPA produces a maximum reduction of only 7000 units.
Rom our previous analysis we would expect the changes in the traffic distribution
between SPF and ED to be less stark which is confirmed by figure 6.11. Finally, ND

seems to be the worst performer because it produces the least change figure 6.12.

121

20 BED

i

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 ' 1
200 20 4o BND 60 80 100
(I)
x
L _—
g 10 , r.
04 J
(D 0 20 4o 60 so 100
C 20 ED
q—
0 F—
- 10
O __
Z 0 ,
0 20 40 ND 60 80 100
20
10 r"
0‘ . .
0 20 60 80 100

40 _ 2
% decr In O'E

Figure 6.7: Marginal histograms of percent decrease ”A02 for the scatter plots in
E

figures 6.5a through 6.5d.
6.8 Conclusions

In this chapter we presented an statistical interpretation of network lifetime that
takes into account a limited degree of node redundancy in WSNs. The interpretation
of network lifetime in terms of the mean and variance of network-wide node power
consumption rates provides us an optimization objective that does not narrowly focus
the on the lifetime of a single node as is the case of most prior work. We provide a

dynamic program formulation of the problem that seeks to optimize variance of power

122

10000

 

*isooo

--6000

[44000

 

 

 

0 2‘ l 4 6 8 10
X

Figure 6.8: Diffusion plot of energy consumption rates averaged over all 100 networks
under SPF routing.

consumption rates while constraining the average consumption rate. We develop the
DPA that chooses routes in a way that optimizes for our objective from sets of paths
that would be considered sub—optimal in the shortest path sense. Four variants are
developed based on the BED, BND, ED and ND route discovery algorithms. We
also observe that the routes discovered by the ND algorithm are very similar to
those proposed in previously proposed load balancing techniques such as Baek and
de Veciana's in [6]. Interestingly, under our understanding of network lifetime and a
many-to—one traffic flow ND is the worst performing of all route discovery algorithms.
A statistical performance comparison of these four techniques for the case when A = 0

shows that on average the BND and BED in conjunction with the DPA yield the best

123

 

   

0 2 “it 6 8
x

Figure 6.9: Differential diffusion plots of energy consumption rates averaged over 100
networks using BED paths and DPA.

 

 

 

0 24's 8
x

Figure 6.10: Differential diffusion plots of energy consumption rates averaged over
100 networks using BND paths and DPA.

performance. BND and BED yield reductions of up to 28% and 36% in variance
of power consumption rates at the cost of raising average node power consumption
by 15% and 21%, respectively. The computational complexity of variants of the
DPA vary from 0(N3) to 0(N4) which is significantly lower than the full search of

124

6 8

 

X

Figure 6.11: Differential diffusion plots of energy consumption rates averaged over
100 networks using ED paths and DPA.

1o.
$315000

710000

 

0 2' 4 6 ' 8

Figure 6.12: Differential diffusion plots of energy consumption rates averaged over
100 networks using ND paths and DPA.

the solution space which is of complexity 0(NIN). Analysis by means of diffusion
plots verifies that. DPA indeed reduces power consumption of sensors that experience
highest power consumption under shortest path routing algorithms. Diffusion plots
also show that the reduction power consumption is highest under BND, followed

125

closely by BED.

The resulting route selection method is one that is suitable for applications with
many-to—one traffic flows. Route discovery algorithms and DPA assume availability
of global network connectivity which is very likely going to be the base station. While
this may exclude its use in some applications we envision it having great application
in critical infrastructure protection / control/ monitoring, surveillance network and en-

vironmental / agricultural monitoring applications with infrequent topology changes.

126

-.. (K 1.]. ]

Chapter 7

Mean-Field Solution of
Small-World Wireless Sensor
Network Models With Range
Limited Shortcuts

127

 

7 .1 Introduction

The limited range of wireless channels naturally imposes geometric or Euclidean graph
topologies [98] on wireless networks [117]. Wireless sensor networks (WSN) [2] are a
class of mobile or stationary, multi-hop ad-hoc wireless networks of power constrained
nodes. WSNs are employed by sensing, detection and data gathering applications
which impose a 111any-to-one data flow. ()11 the spectrum of randomness of graphs
the endpoints are occupied by lattice graphs on one end (no randomness) and random
graphs on the other (complete randomness). In between these extremes lie small—world
networks [128]. Lattice graphs are characterized by strongly connected neighborhoods
that imply a high degree of connectivity and resilience, but rather large diameters.
Random graphs [13], on the other hand, are characterized by small diameters (that
imply an ability to retrieve and disseminate information quickly) but low connectivity
between neighborhood nodes. Note that the same topological properties that enable
the fast dissemination of information within a network also enable fast information
retrieval. Clearly, in the context of WSNS both strong connectivity at the local
level and small diameters are desirable, both properties of small-world networks. In
[128] and [127] Watts provided analytical models for 1-dimensional lattice graphs,
connected caveman graphs and Moore graphs. Several attempts have been made to
leverage the small-world network effect in WSNs ([111],[27],[125],[52]). However, to
the authors’ best knowledge there are no analytical models of 2—dimensional Euclidean
graphs with range limited shortcuts that are applicable to WSNS. This research
derives analytical models for the clustering coefficient and characteristic path length
of VVSNs whose topologies are augmented by a small number of shortcut links whose
range is limited by practical limitations. Results show that in spite of the fact that
shortcuts are range limited and significant differences in construction method the

phase difference between drop in clustering coefficient and characteristic path length

128

 

that is the hallmark of small-world networks still appears. Although shortcuts are
range-limited only about 0.005 — 0.05% of nodes had to be equipped with the ability
to communicate over long ranges.

The remaining chapter is organized as follows. Section 7.2 provides a brief back-
ground of small-world networks, their properties and their standing relative to Eu-
clidean and random graphs. Section 7.3 describes different system models for W SNs
with small-world topologies and provides a literature review of prior methods used
to this end. Section 7.4 uses mean field analysis to derive generalized expressions of
clustering coefficient and characteristic path length that can be used to model small
W SNs based on any of the system models in section 7.3. Section 7.5 evaluates the

model for different ranges of parameters. Section 8.8 concludes the chapter.

7.2 Background: Small-World Networks

Small world networks were first discovered by Milgram in social networks in [82].
Watts in [128] and [127] analyzed social networks which included a collaboration net-
work of actors and a collaboration network of mathematicians, to verify Milgram’s
idea of six degrees of separation. More recently Horvitz and Leskovec verified the
presence of six degrees of separation in social networks using a much larger data
set from Microsoft’s Windows Live lVIessenger instant messaging traffic [130]. Since
Milgram’s original experiment several other works have analyzed networks in natu-
ral and man-made systems to discover that their topologies are in fact small world
graphs. Latora and Marchiori [73] and Watts [127] analyzed the neural network of
the C.elegans worm. Montoya and Sole [87] studied food webs in nature. Moore
and Newman [88],[89] studied the transmission of infectious diseases in populations
with small world connectivity. Studies of transportation networks include Latora,

Marchiori [73],[74] on the Boston subway network and Sen et al. [110] on the Indian

129

railway network. Small world analysis of communication networks include Adamic’s
[1] on the World Wide Web. i Cancho and Sole [50] studied the lexical networks of
human languages and determined them to be small worlds. More relevant to our
work are studies on creating small world topologies in wireless networks by various
means. Wan et al. [125], Cavalcanti et al. [23] and Costa and Barros [31] showed
that selectively equipping a small fraction of nodes in a WSN with two radios (one
regular, short range IEEE 802.15.4 and one long range IEEE 802.11b) induces a small
world topology. Hubaux et al. [49] and Dimitar et al. [36] propose a small world
application layer for ad hoc networks similar to logical links in peer—to—peer networks.
Dixit, Yanmaz and Tonguz [37] analyze the topology of cellular wireless networks.
Helmy [46],[47] proposes mobility assisted wireless networks to create shortcuts that
mimic random links. Sharma, hilazumdar [111] make selected use of wired links to

create shortcuts.

Among prior models of small world networks are Watts [128] models for the clus-
tering coefficient and characteristic path length of 1-dimensional (circular) lattice
graph, connected caveman graphs and Moore graphs. In [91] Newman and Watts
found a mean-field solution of the small world network model for l-dimensional lat-
tices. Arnaral [3] studied the statistical properties of classes of small world networks
using a taxonomy based on the form of degree distribution of nodes. Some recent
advancements in W SNs have enabled the addition of a number of links with longer
communication range, called shortcuts or global scale links, to a network by various
means (at a cost). In this chapter we derive analytical expressions for two defin-
ing properties of small-world networks, the clustering coefficient C' and characteristic
path length L of geometric graphs in a 2-dimensional plane with a number of range
limited shortcut links. Let G (V, E) denote a graph consisting of a set of N vertices

V = {221, 212,213, . . . ,vN} and a set E of M edges consisting of all edges, where 6M

130

 

 

Figure 7.1: Illustrated examples for three different classes of graphs; a) Geometric
graph, b) Random graph, and 0) Small world graph.

denotes an undirected edge from ”2' to vj.

7 .2.1 Characteristic Path Length

Characteristic path length is defined as the average length of geodesic paths between
all pairs of nodes. For a graph G, the characteristic path length L(G) is defined as
the number of edges in the geodesic path between two vertices, averaged over all pairs

of vertices. If Le - is the number of edges on the geodesic path from v- to Us, then,
3:] Z J

L =Zly=12jv=1Ltj
NUV - 1) (7.1)

 

7 .2.2 Clustering Coefficient

The Clustering Coefficient is a measure of the cliquishness, the degree to which ver-
tices in a graph coalesce into tight groups. For a graph G, the clustering coefficient
C (G) is defined as follows. Suppose a vertex v has 100 number of neighbors, then the

maximum possible size of the set of undirected edges between 2) and all its neighbors

is k” ICU-H . Then CU is the fraction of these edges that actually exist in E. The

131

 

clustering coefficient of the graph is then defined as the average GU over all vertices

V, as in equation 7.2.

N ,
C : 27:1 Cl
N (7.2)

The above definitions of characteristic path length and clustering coefficient are

easily extended to directed and weighted graphs.

7 .2.3 Small World, Geometric and Random Graphs

We will be using the terms graph and network and the terms node and vertex inter-
change-ably. Geometric graphs have (see figure 7.1a), both large L and C by virtue of
their strong local connectivity. Random graphs (see figure 7.1b) are the other extreme
and are characterized by both small L and C. Small world networks (see figure 7.1c)
have small L but large C.

There are several construction methods for small world networks. Among the
simplest to understand is the 13-model described by Watts in [128]. A small world
network can be constructed from a lattice or geometric graph by rewiring one end of
every edge to a randomly selected node with probability 5’. The type of graph that is
constructed by the fi-model depends on the value of rewiring probability 5. If 6 = 0
the graph remains a lattice/ geometric graph, while 13 == 1 yields a random graph.
Watts and Strogatz [129], Newman and Watts [92], Pandit and Amritkar [96] and
Watts [128] repeatedly demonstrated ,3 =2 0.01 — 0.05 z (9 (7%,) as a typical range of
values for constructing small world networks, which means only a very small fraction
of links has to be rewired. Hence, on the scale of randomness where lattice / geometric

graphs occupy one extreme and random graphs the other, small world graphs fall in

between the two, but closer to the former than the latter.

132

 

 

 

In the context of W SNs the property of random graphs to propagate information
quickly means that nodes can relay important data back to the base station quickly.
The property of Euclidean graphs to have high G and strong local connectivity pro-
duces a stable network structure difficult to partition. It also enables the formation
of clusters which facilitates collaboration among groups of nodes. Thus, the W SN

application den'iands the best of Euclidean and random graphs.

7 .3 Small-World Topology Construction Methods
for Wireless Networks

There are several approaches to realize shortcuts in wireless networks with the intent

of creating a small world topology. We conducted a survey of practical techniques.

7 .3.1 Hybrid Sensor Network

Sharma, l\=’Iazumdar [111] and Chitradurga [27] proposed the judicious use of wired
links between a small subset of nodes in a W SN . The resultant network is not a
pure wireless network and is called a hybrid sensor network. The graph topology of
hybrid sensor networks uses range limited links for shortcuts instead of truly random
links. From a practical perspective the wired shortcuts limit the possible deployment

scenarios.

7.3.2 Multi-radio Network

The multi-radio node as a means to building small worlds was first proposed by Wan
in [125] with the objective of easing congestion in the primary network consisting of
ubiquitous low-rate links. The multi-radio node architecture has the disadvantage

that it requires a heterogeneous mix of motes. The links of the long range radio

133

interface serve as small world shortcut links. The implementation used in [125] uses
IEEE 802.15.4 [5] in conjunction with IEEE 802.11b [51] network interface. The
significantly higher power consumption of IEEE 802.11b is a disadvantage in power

constrained wireless SGDSOI‘ IIGUVOI‘kS.

7.3.3 Receiver Side Cooperation

The third solution is based on single-input multiple—output (SIMO) principles and
is dubbed by Ilyas and Radha as the ’Poor-Man’s-SIMO-System’ (PMSS) because it
uses receiver side diversity combination techniques in a network of commercial-off-the-
shelf components. Its principal strength over hybrid sensor networks and multi-radio
networks is that it circumvents the need for customized or reconfigured hardware.
Instead, it relies on receiver side cooperation in motes to form a SIMO system. This
way c00peration reduces losses and retransmissions while increasing throughput and
channel utilization. In [52] Ilyas and Radha describe the PMSS and how three well-
known diversity combination techniques are adopted for use in a network of commer-
cial IEEE 802.15.4 devices. The diversity combination techniques are derivatives of
diversity combining methods for analog signals presented by Brennan in [19]. Simply
put, the purposes of diversity combining are twofold; a) Select an error-free version of
a received transmission from among multiple received versions or, b) If the first goal
is not achievable, obtain another version of the transmission, with no errors or fewer

errors than any of the individual received versions.

In conclusion of this survey it should be understood that as diverse as these
different implementations of small-world networks in wireless networks may be, the

analytical model can be parameterized accordingly.

134

 

7 .4 Mean Field Analysis of Small-World Wireless

Networks

7 .4.1 Clustering Coefficient

Sensor nodes are distributed as a 2—dimensional Poisson point process with mean
,0 points (sensor nodes) per unit area. Let there be two nodes '01 and 212 capable
of communicating with nodes within range R1 and R2, respectively. For simplicity’s
sake we will take R1 = R2 = R. If the two nodes are separated by distance r < R then
communication coverage regions of nodes 211 and '02 will overlap, as depicted by the
shaded region in figure 7.2. Let A denote the size of the overlapping communication
coverage regions. Then A is computed by adding the areas of sectors X OX ’ and
X 0’ X ’ and subtracting the areas of triangles X OX I and X 0’ X ’ because they have
been added twice during the addition of both sectors. The resulting expression for A

is shown in equation 7.3.

()
A(6l, R) :2 g—NRZ — 2 - Rsind - Rcosél
7r

=26R2 — R2 sin 26 (7-3)

Here 0 is the central angle 1X 0’ X ' in figure 7.2. We can exm‘ess (I as a function of

r.
6(r, R) = arccos “—2
R (7.4)

135

 

Substituting the expression for 6 in equation 7.4 back into equation 7.3 give us A as

26f 2 /1_ 7"

A ,R =2R6 ,)R-— R —— —
[ 7.5
=2R2arccos(r2R)—-7R-1—— ( )

As we already described, for the network topology to be a small world some nodes

a function of r.

have to provide shortcuts or global scale links to other nodes. These nodes will be
referred to as shortcut nodes or global scale nodes. In literature, the graph formed by
global scale nodes is also referred to as a network’s substrate [128]. The remaining
nodes are called local scale nodes. The ratio of the maximum communication range
of global scale link to that of a local scale link is called the scaling factor 5. Now,
let p denote the average node density with which nodes occupy the covered region.
The cooperative communication model for small world WSNS in section 7.3.3 for the
implementation of shortcuts in a wireless sensor network requires multiple nodes to
function as a single receiver. Depending on how many nodes are collapsed into a single
node will reduce the effective node density ,0. Let nc denote the average number of
nodes that combine to function as a global scale node. This reduces the node density
in the equivalent small world graph (spread over area A) to p, which relates to {2

according to equation 7.6.

p, -p _ Ngloballnc — 1)
A (7.6)

 

Note that when shortcut links are implemented in multi-radio networks (subsection
7.3.1) and hybrid sensor networks (subsection 7.3.2) the node density in the small

world graph remains unchanged, i.e. nc— —— 1 which implies p - —p. Then A(r, R)p

136

 

 

 

Figure 7.2: Overlapping communication regions of two communicating sensor nodes
in a WSN.

denotes the number of edges 3. node (221) distance r away from a reference node (222)
has with the set of nodes that can also communicate with the reference node. Let
F(-v) denote the neighborhood of a vertex v E V, where V denotes the set of all
vertices in the graph G of the wireless sensor network’s topology. We treat the entire
region covered by sensors as a continuous, homogeneous sea of sensors populated with
a uniform density of p, nodes per unit area. Then the total number of edges d in the
neighborhood of vertex u can be computed by integrating A(r, R)p’ with respect to r
over the interval [0, R], the radius of the communication range; The number of links
in the neighborhood F(u) of a node v is denoted by d(F(u)). The exact derivation
of the expression for d(F(v)) in equation 7.7is provided as lemma 1 in this chapter’s

appendix.

are)» =7rp/2R3(2.9604R — V3)
(7.7)

137

Let D(F(u)) denote the number of edges in the completely connected graph of node
v and its neighborhood F(v).

D(F(v)) =7rp’R2 X 7rp’R2 = «262124
(7.8)

The clustering coefficient is defined as the expected value of the ratio of the number
of links in a nodes neighborhood to the number of links in a complete graph of its
neighborhood. This can also be defined in terms of previously computed values as;

_ d<r<v>>
C(Gl ‘EV [Danni

 

(79)

For a node that is part of the sea of sensors communicating over local scale links only

we define the clustering coefficient Clocal as,

 

 

_ dam) _ 2.960412 — \/a
local _D(F(U)) _ 7TH
~2.9604R — \/§ (7.10)
NR

 

Hence, Clocal grows O (712) with respect to R. For a node that is part of the
substrate communicating over local as well as global scale links we define the clustering

coefficient Cglobal as,

 

C dfrlvll + kglobal
l b l:
9 O a D(F(’U)) + kglobalOO/WRZ) 'l' (kglobal _ 1)! (7-11)

138

 

1
-, 9.‘ .
,- l. I
I, ..
‘r 1
.1». .11
r ,.,
s. 1
. [q

’V '1

‘ .
u... g
I

0 -..

1. '. « '
I. “F, ‘

,_ 1

-¢-.

 

.5
71.0]
'i‘
l“-A
-2 1
h i'.
‘1' [l‘
.l.
“';.I
l |
9
L.”
| .
\ «
h, “\

flu? ]

”:31ij

Note that in doing so we assumed that 6 Z 2. Here k l is the average degree of

globa

nodes in the global scale network or substrate with other nodes in the global scale

network. The function for C rows as O n 1 ). It must be noted here
global g (W

that in the computation of Clonal and Cglobal ignores border effects and assumes
nodes are spread on a torus. To obtain the clustering coefficient of the network G local

and Cglobal are combined in the ratios in which local and global scale nodes appear.

N
If we define a = . globarl then C is obtained by equation 7.12.

f global‘l'Alocal

 

C =12C' +(1— u)C
global local (712)

7 .4.2 Characteristic Path Length

The characteristic path length of a graph is defined as the average length of all geodesic
paths (measured in hops). In a graph G(V, E) the characteristic path length L is
defined as the average of geodesic path lengths [(211, uj) between all connected pairs
of vertices vi, uj E V and u,- # uj. Like the clustering coefficient, the characteristic
path length is also a function of the number of nodes with global scale links. Since the
global scale links are range-limited, and since global scale nodes have to be placed
within communication range of one other to be useful, the area covered by them
is also limited. The region of the WSN that is serviced by global scale nodes may
be fragmented into many different patches. Figure 7.3 shows a model of a WSN.
For analytical ease the shape of the region covered by all nodes of the network is
approximated by a circle of radius VA and center 0. The region covered by global
scale nodes is approximated by a circle of radius m < x/A also with center 0.
The difference between coverage areas of the entire WSN and only the global scale

nodes is a result of the constraint imposed by a) the limited range of shortcuts and b)

139

A

f
..l

the global vertex degree (the number of shortcut links incident on a substrate node).
Figure 7.3 also shows a node :1: distance removed from O. For the node in the figure
:1: > g and the sensor region can be divided into three separate regions, numbered
1 through 3 in the figure. These regions are defined relative to every node 2). Region 1
consists of all sensors that are reachable from 22 without having to traverse any global
scale links. In figure 7.3, for node v at distance :1: from center 0, region 1 corresponds
to the uncolored white region. Region 2 is the area of size Ag occupied by global scale
nodes and is colored solid gray. Region 3 is the region occupied exclusively by sensors
with local scale links but which is reachable (by a shortest path) only after traversing
region 2. In figure 7.3 region 3 is identified by shading and it’s area is denoted by
5 (:13) Assuming the deployment of sensors is sufficiently dense, we approximate the
geodesic path from one node to another by a straight line between them. To proceed
we define three central angles (1 = %ABOB’, [3 = .480] = ABIOJ’ and ’y = AJOJ’,
where 2(02 + 13) + 'y = 277. Since a and c): are functions of :r we will be denoting them

by 07(2) and 7(2).
A
9

x (7.13)

 

(1(a) = arccos

Ag
fl (7.14)

 

13 = arccos

';r=7r— (1:1:— ’3: 7r—(11t—
a() 2 2f) 21 2f (l 6) (7.15)

140

 

 

\

't'[_
., .

~.[‘4V.
w ‘4

 

 

 

 

 

 

 

 

 

\/A

 

 

 

 

 

X z

 

 

VA—i

Figure 7.3: Geometry of W SN deployment and regions within it with respect to an
individual sensor 22,-.

The area of the region in figure 7.3 labeled region 3 is (see lemma 2 in the appendix)

given in equation 7.16,

 

26+y
Ag

5(1)) :2 AAg sin/3 + £0? " 02(2) _ B) — 2 (716)

141

 

Characteristic path length is denoted by L. It is the average length (in hOps) of
all geodesic paths from every node u,- E V to every other node uj E V. Llocal is
the average number of hops contributed to geodesic paths by local scale links, while

L l is the average number of hops contributed to geodesic paths by global scale

globa

links. Then L can be expressed as equation 7.17.

L =L + L
local global (717)

Based on the 3 regions of the \NSN in figure 7.3, geodesic paths are classified into one

of four different types.

Type 1: Geodesic paths that originate from and terminate at nodes in region

1 and (outside the region covered by the substrate). Type 1 paths consist of

local scale links only.

Type 2: Geodesic paths that originate from a node in region 1 and terminate

at a node in region 2.

Type 3: Geodesic paths that originate from a node in region 1 and terminate

at a node in region 3. Such paths pass through region 2.

Type 4: Geodesic paths that originate from and terminate at nodes in region

2.

We compute Llocal (L ) in equation 7.18 (equation 7.19) as the sum of Llocalffl

(L

global

globalalli the contributions of local (global) scale links to type 2 paths, weighted

142

 

 

i“ ll
F3: Ll ..

 

'1'“.
\ .V

by the normalized frequency of occurrence of type 2' paths.

4
1 .
Llocal zfi Z Llocalm
i=1 (7.18)

4
1 .
Lglobal 26—}? Z Lglobalm
' 2:1 (7.19)

(1)

[0601(1) denote the average

For a node v in region 1 at a: > Ag distance from 0, let l
spatial distance on type 1 paths traversed over local scale links. Then for a particular
node the size of region1 is A — Ag — 5 (.13) Since the deployment of sensors is in 2

dimensions we approximate the size of the region traversed by these paths by the

 

square root of its area, \/ A — Ag — 5 (2:) Although region 1 is irregularly shaped
we approximate the spatial distance by one half of it, assuming the sensor is located

(1)

approximately in the center of region 1. Then [localml is given by equation 7.20.

 

<1) 1 .
local( ) 2\/ g ( ) (7.20)

A node v at distance a > , /Ag from O has pllocalffl paths to other nodes in region

1. To obtain Llocalu) we integrate this term over the entire region of deployment

of the W SN, i.e. all circumferences of radii ranging from , /Ag through VA. This is

143

 

shown in equation 7.21.

L (1) =—1——— fl 277.1:- (A — A — 5(1)) 1(1) ($)d:r
local (p/A)2 mp p 9 local (721)

Since the paths from u to other sensors in region 1 do not traverse region 2 no global
scale links are used. Hence L globalU)’ the component of global scale links in paths

to sensors in region 1 is 0, as in equation 7.22.

L (1) = 0
global (7.22)

(2)

For a node v in region 1 at I > Ag distance from 0, let l l
oca

[(2) denote the average

spatial distance on type 2 paths traversed over local scale links. It is approximated

by the mean of the shortest (:r —— Ag) and longest distances (from node v to point B)

from v to a node in region 2, i.e. %(:r — fl + :rsin (1(2)). To include paths from

nodes in region 2 back to node 22 we multiply this term by 2. This is approximated
(2)

by equation 7.23. We roughly approximate lgloballx) by the square root of the area

of region 2 in equation 7.25.

,(2)

local<$l =11: — ”Ag + assin (1(2) (7 23)

Llocal(2) (equation 7.18) is the component of Llocal contributed by node u’s type 2
paths, i.e. to other nodes in region 2. A node v at distance :L' > ,/Ag from 0 has
p”Ag paths to other nodes in region 2. To obtain Llocal(2) we integrate this term

over all circumferences of radii ranging from «Ag through \/A. This is shown in

144

equation 7.24.

(NA)? local

. 1 VII 2
LZOCGZQ) =—/ . 102711" ' PHAgl( ) (.L')(l;r
V A9 (7.24)

The length of global scale paths from v to sensors in region 2 is approximated like in
equation 7.25.
1(2) (I) = Ag

global (7.25)

Equation 7.26 weighs this by the number of paths from all 2: outside of region 2 to all
sensors inside region 2 (and vice versa) relative to the number of paths between all

pairs of nodes, i.e. (p’A)2.

Lglobalgll =(/—)-——/14)2/\/_A_g/)27r.”rp Am/A gda:

:ppllnAg 32/ (A— Ag) (7.26)
(P’A)2

 

(3)

For a node v in region 1 at :1: > A9 distance from 0, let [locallxl denote the average
spatial distance on type 3 paths traversed over local scale links. It is approximated

by equation 7.27.

 

1‘3) (1:17) ()+’/‘72_7

local local

(.1: + V_— (/A 9+ 2:1: sin 0( (727)

145

A node v at distance :1: > ,/Ag from O has p5 (:13) paths to other nodes in region 3.
To obtain Llocall3) we integrate this term over all circumferences of radii ranging

from ,/ Ag through VA. This is shown in equation 7.28.

. (3)
p27r1: - p5(;r)l , (:1:)d.1:
local (7.28)

I
L . . 3 :—
1()C(ll( ) (/)/A)2 / /—‘49

For Lglobalw) in equation 7.29 we approximate the length of global scale links by
the diameter of region 2, 2,/A , and weigh it by the total number of paths from all

u to nodes in region 3, {)5d(r).

1 VA
L 3 277.1: )5(.1)2g(l.1:
globall l: (p A—T—fg / r—4 P I 1W (7.29)

(4)

Final[fifor a node 22 in region 2 at distance x < Ag from 0, let [local denote the
average spatial distance on type 4 paths traversed over local scale links. We assume
that on average, a node ’U in region 2 communicating with another node in region 2
will route (over local links) to a nearby node with global scale links. Once traffic has
I""‘l‘lu‘d the vicinity of the target destination node traffic is routed over local scale
links to the destination. Let hmag; denote the maximum distance of a node from its

Closest global scale node, as expressed in equation 7.30.

l A
h h=1 ‘ global (7.30)

146

Assuming that equidistant nodes form concentric bands around global scale nodes
the probability mass function p H(h) of the hop distance H is modeled by equation

7.31.
«(1.12)? — «((11 —1)R)2 2h — 1
pH(h) = 2 = 2 '
7r(hmaiL‘R) hmaa: (7.31)

 

 

Then the 1(4)

local is approximated by the expected value of H as in equation 7.32.

hmaa: 2h _ 1

(4)
l . =5 H lhl = h
local ’12::1 hgzax (7.32)

 

A node v at distance :1: < ,/Ag from 0, within region 2, has pi/Ag paths to other
nodes in region 2. To obtain Llocal<4> we integrate this term over all nodes in region

2. This is shown in equation 7.33.

L1 . 1(4) =_—2(p”A9)21(4)

06“ (1M)2 local (7.33)
L (4) —2(p”A9)2 , /A

910’” (1)/A)? 9 (7.34)

NOW We can back substitute equations 7.21, 7.24, 7.28 and 7.33 into equation 7.18
\0 Obtain Llocal' Similarly, equations 7.22, 7.26, 7.29 and 7.34 are put back into

equation 7.19 to obtain L Equations 7.18 and 7.19 are substituted into 7.17

global“

to obtain the characteristic path length L.

147

Clustering coefficient C vs. u Characteristic path length L vs. n

 

   
 
  
   

 

 

 

 

 

 

 

 

 

 

 

0.09- , 120- '
IIIIIIIIIIIIIIIIIIIIIIIIIIIII +R = 3
0.088""""'"-""‘--‘--'--'- +R =4
_.-.-.-.-._.______, 100— +R=5
0.086~ . :§;:
0'084— I I —+———R:9
"""""‘-'?-----'-------j . _+R= 10
0 0.082- 4' .4 60 .
—R = 3 ‘
0.083'"R=4 ------.-..
..,.R:5 40'
'''''' R=6
0.078~
—R= 7
--- = 20' l n V
0076:," 2 =3 - . _____ ’ _____
O 074 lllllll R =10\ G ":-:-: -: i 247-: :.-;-. , :: : ; L;
' 0 0.5 1 0 0.001 0.002
H H

Figure 7.4: Plots of clustering coefficient C [left] and characteristic path length L,
lfight] as functions of u for different values of R. Network parameters that remain
fixed are A = 10000, kglobal = 4, p = 10, f = 3 and 11c = 1.

7-5 Observations

111 this section We explore the behavior of the analytical models of clustering coefficient
C and characteristic path length L as we vary the length of local scale links R, the

average degree of shortcut nodes in the substrate k l and the scaling factor 5.

globa
For the following evaluations we assumed a network of 100,000 nodes in an area of
size A = 10, 000 and 110 = 1. Unless stated otherwise, the default parameters of the
network are R = 3, g = 3 and kglobal = 4. Figure 7.4 plots the clustering

CDeficient and characteristic path length of networks as functions of 11, the fraction

Of nodes that are shortcut nodes, for different values of R. The clustering coefficient

148

 

 
 
  
 
 
  
  

 

 

 

 

 

 

 

 

 

 

 

 

 

Clustering coefficient C vs. u Characteristic path length L vs. a
0_O3924..., ' ‘ 350 - - " ‘
—B—§ : 3
_ 300 4 —.e.—§ = 4
0.039 : +§ = 5
0.0388— ' 1 +5, = 7
—n—§ : 8
O H —+—§ : 9
0.0386——§=3 +§=10
. . .g = 4
a = 5
0.0384 IIIIIII é ___ 6 -
_g = 7
0.0382—---§=8 _
.... E“ = 9
0 ““““ i = 10 ; . _ .
'0380 0.5 1 00 0.005 0.01 0.015
11 11

Figure 7.5: Plots clustering coefficient C [left] and characteristic path length L, [right]
88 a. function of 11 for different ratios of global scale link to local scale link communi-
cation range é. Network parameters remain fixed at. A = 10000, kglobal = 4, p = 10,
"c = 1 and R = 4.

0 appears as a linearly decreasing function of )1 with identical slopes. The clustering
(30(‘fficient has higher values for longer range R of local scale links. The right pane of
figure 7.4 plots the Characteristic path length for various R. Higher values of R imply
that destinations can be reached in fewer hops, thus reducing the characteristic path
length_ This is born out by the curves of L. Moreover, a side—by—side comparison
Clearly illustrates the phase difference in the reduction of of C and L. This means the
derived analytical model indeed captures the small world effect in the network. Unlike
in Watts’ fl—model [128] of constructing small world networks, C lacks a sharp drop

OE as l4 ~+ 1. The reason for this is the range limitation on shortcuts which imposes a

149

Clustering coefficient C vs. u Characteristic path length L vs. ll

 

 

 

 

 

 

 

 

 

 

 

 

 

+kglobal = 2
+k =
0.08 global
4 4 . , 4 . __ kglobal = 4
0.078 .. . .' . r 4 +kglobal = 5
_. I .. d .. d P i +kglobal =
(1076" kglobal -
O —+—k =
0.074 ---- global
“rte-oval '
0.072 '—
0.07r - .
O 068 """" global = O . _
. 0 0.5 1 0 0.001 0.002

n 11

Figure 7.6: Plots clustering coefficient C [left] and characteristic path length L, [right]
as a function of a for different values of kglobal' Network parameters remain fixed at
A = 10000,p=10,§=3, R=3andnc=3.

degree of localization in a shortcut link’s reach which leads to clustering. In addition,
all construction methods of small world WSN in section 7.2 add shortcuts on top of
the existing geometric graph instead of rewiring existing links as the ,B-model does
Where each global scale link is added at the cost of removing a local scale link.
Figure 7.5 plots clustering coefficient C and characteristic path length L as func-
tions of l1 for different values of global scale link scaling factors 6. Recall that in
deriVirig the analytical model for the clustering coefficient we made the assumption
that 5 2 2, i.e. shortcut links have a communication range of at leats twice that

Of local scale links. Note that in figure 7.5 all plotted function of C overlap, i.e. as

long as as the assumption 5 2 2 holds, the clustering coefficient as a function of p is

150

independent of the global scaling factor g. The small world network effect is clearly
visible by the large gap between clustering coefficient and characteristic path length.
AS expected, the drop in characteristic path length L drops off earlier as 5 increases,

i.e. fewer shortcuts are necessary to achieve the same reduction in L as shortcuts get

longer.

For the networks in figure 7.6 we varied the average number of shortcuts kglobal
incident on global scale node. Larger values of kglobal imply a slower expansion of
the area within a WSN that is serviced by shortcut nodes (region 2 in figure 7.3).

Figure 7.6 shows that as kglobal increases, the rate at which the clustering coefficient
drops increases. We also observe that the number of global scale nodes needed to
achieve a significant reduction in L also increases as evidenced by higher value of a
to achieve the same characteristic path length L for higher kglobal' That is because
at higher kglobal the range limited nature of shortcuts forces more global scale nodes
to be deployed in closer proximity.

However, we observe consistently that the small world effect holds in 2-dimensional
Spatial graphs with range limited shortcuts such as those formed by WSNS with some
shortcuts. What is also interesting is the fact that for a sufficiently dense WSN the

number of shortcuts that is needed to achieve a significant reduction in L is very

small.

7-6 Conclusions

We derived analytical models for both clustering coefficient and characteristic path
length of a 2-dimensional spatial graph with range limited shortcuts that models the

tOpological constraints to which W SNs are subject.

1' The model is sufficiently general to accommodate any small world network

COnstruction methods in wireless networks previously proposed for W SNs and

151

is, to the authors’ best knowledge, the first analytical model for networks with

these constraints.

2- VVe observed that for sufficiently dense networks characteristic path length can
be reduced significantly by replacing a 11 z (9(0005 — 0.05) fraction of the local
scale nodes by global scale nodes providing shortcuts in the network. The order
of ,u, the fraction of nodes that are designated shortcut nodes, is about the

same as the value of b, the rewiring probability, in Watts’ small world network

construction method.

3. Whichever small world network construction method is applied carries with it
a cost. The model lends itself for the task of designing WSNs, e.g. determining
the number of shortcut nodes required to achieve a certain characteristic path

length.

APPENDIX

Lemma 1. We derive the expression for d(f (11)), the number of links between the
36’: of nodes consisting ofv and all its neighbors T(u). The region covered. by F('U) is
approximated by a circular region of radius R centered at v. Consider another node
”I at a distance r from node 1) shown in figure 7. 7. Then the number of links from v,
to other nodes in Nb) is A(r)p’. The narrow ring of width dr is inhabited by 27rrp’
Other nodes at the same distance r from u. The number of links from all nodes at

distance 7~ from v is 24.,.p/A(.,.)p’4 Then, to find d((F(’U)), the total number of links

between nodes '1) and [(11), we integrate 27r7~p’A(r)p’ with respect to r over [0, R], the

152

ra di'us of communication of v.

R / /
d(F(i')) 2/0 /) 27rrA(r)/) dr

R R
:/O p’227rr 2R2 arccos _QLR — _2_r 1 — 2172—?) dr

R . R (7.A-1)
=47rR2p/2 / r arccos —r—dr — Zp/Z/ rQV 4R2 — r2a'r
0 2R 2 0

=47rR2p’2 . A(R) — gp’2.3(1{)

W’here A(R) and BU?) are evaluated in equations 7.A-2 and 7.A-3.

. , 1 l .2
All?) = arcsin (57-1?) R2 — §r 1— 47—52-51?

+1 2 < " > R
—" arcc °‘ —
27 Ob 2R 0
1 l
= arcsin (g) R2 -— jig—R + 5R2 arccos (5)
=13? _ —\/—§R
3 4

2
1, ,/ . . . .
313 4R2 _722F1(15’—05,25,4—7l€2)

 

7.2
1 __
4R?

 

153

 

(7.A-‘2)

(7.A-3)

2171 (a; b, c, 2) denotes the hypergeometric function {equation 7.A-4) and is evaluated

in equation 7. .4-5,

 

. . .., __ Fe) ”5,404,044
2F1(a,b,c,4) —F(b)P(C- b)‘/0 (l —t2)a (it (7.A-4)

 

 

 

1 —r5 _t2
= — 0.1875 / t (1 ) dt (7'A'5l
0 .

Substituting the value of the hypergeometric function from 7.A-5 into equation 7.A-3

P’I‘Od‘uces equation 7.A-6.

BU?) =§R4 «2 F1(1.5; —-O.5; 2.5; 0.25) — 0 = 06142-134
(7.A-6)

Substituting A(R) and BU?) from equations 7.A-2 and 7.A-6 back. into equation 7.A-1

gives equation 7.A-7.

d(l‘(v)) 22.9604np/2R4 — flap/2R3 (7 A 7)

LeInma 2. We compute the area of region 3 in figure 7.3 for a node a? distance from
O denoted as S(at), where :1: > ,/A9. In order to proceed we compute the area of
triangle BOJ denoted by A(:r). For :2: > ,/Ag, A(r) can be expressed in previously

defined terms as,

154

 

 

Figure 7.7: Graphical representation of integration term of d(F('v)).

A: A ‘fls' i3
l7 ... (7.A—s)

Similarli, the area of sector JOJ’ with central angle 7' and radius #7 is denoted by

,\
A(CL’). For :1: > ‘ Wig, A(at) can be (;'.'17pr(_ess(_zd in previously defined terms as,

 

lilac) =A/éir)7r\/ZZ = g(n —- oz(a') — [3)
7T (7.A-9)

The area of sector 308’ with central angle 2/3 + 7 and radius ,/A9 is denoted by
A(l‘). The, for :1: > M] , like) can be expressed as,

2 awn/(as) 2 2s+'y(:r-)
A(x):—27r—7Tl/Ag = —-—2——Ag

(7.A-10)

155

We now use Mr), like) and [KL/t) to obtain the area of region 3 denoted by 5(a).

3(1?) =2A(;L‘) + Ru) — Km
. A 2.13+“/(~T)
: ./ _ _ ,.._ __ 7.A-11
2,/AAgana+2(n an) a) 2 Ag ( )

156

 

 

Chapter 8

Enabling Cooperative
Communication and Diversity
Combination in IEEE 802.15.4
Wireless Networks Using
Off-the-shelf Sensor Motes

157

 

8 . 1 Introduction

Channel fades and interference effects limit the throughput, useful communication
range and (in case of battery powered devices) lifetime of nodes. In this chapter we
describe the generalized ‘Poor Man’s SIMO System’ (gPMSS), a readily deployable
lovv—cost, low-power, protocol centric approach that enables cooperative communi-
cation in IEEE 802.15.4 [5] wireless sensor networks (WSN). We demonstrate that
gPlVISS reduces the fraction of packets that are received with bit errors or not re-
CE‘iVEEd at all by an order of magnitude, thus reducing the number of retransmissions.
It Illakes the use of long range links that are unfeasible due to high packet loss and
retransmission rates for sible again. We also show that even in instances where gPMSS
is not able to correct all errors from a packet it still succeeds in reducing the number
of bit errors. At the receiver side gPMSS uses diversity combining methods adapted
fI‘Onl their analog domain counterparts of the same name [19] for digital signals. W hat.
nl'cxkes the application of SIMO diversity combining principles novel from traditional
“'38 is that they are applied to the demodulated version of received packets, after
p1lysical layer processing. W‘ den'ionstrate the efficacy of gPMSS by applying it to
bit error traces collected from IEEE 802.15.4 channels that allow detailed analysis and
precise reproduction of results. we also demonstrate gPMSS’ effectiveness under real-
WOrld conditions by implementation on Crossbow’s Imote2 .NET Micro Framework
SEinsor platform [35].
Enabling the use of long range links (that would otherwise not be used) makes

gPMSS a viable protocol due to the benefits and utility of such links by several

applications in wireless sensor networks.

Network Lifetime Extension: Funnel-ing is the effect of network traffic from multi-
ple sources flowing to a small number of sink nodes [125]. This traffic surge produces

COngestion in the region around the sink nodes / base station, forcing nodes near sink

158

s sis-3E!-

 

‘FT-m

 

nodes to relay more traffic than other nodes and consume power at correspondingly
higher rates. Since nodes in W SNs have only limited power resources this means that
the sink node’s neighbors will run out of power sooner, leaving the sink node discon-
nected from the rest of the network. Load balancing techniques like [53] attempt to
distribute the burden of relaying traffic to increase the lifetime of sensor networks.
Em ploying gPMSS in such a scenario will grow the set of neighbor nodes of the sink

IlOde and allow load balancing among more nodes.

 

Small—world Networks: Several attempts have been made at building small-world
network [128] topologies in wireless networks to simplify resource discovery and re-
ducin g average path length to facilitate data dissemination. Proposed architectures
required hardware modifications such as adding a secondary radio frequency interface
(l1 24],[125]) or building hybrid networks by augmenting wireless networks with wired
ShOI‘tcuts ([111],[27]). Since gPMSS is a protocol centric approach it does not require
any hardware modifications which adds to its appeal as a lox-x-r—commexity and low-cost
SOlution.
Network Connectivity: Long range links can be used to add links between two
COIliponents of a network that are only sparsely connected with one another.
gPMSS adopts well—understood diversity combining methods for analog signals
and applies them to digital signals (packets). Specifically, gPMSS implements selec-
tion diversity, equal gain diversity combining and maximal—ratio gain diversity com-
bining. The latter relies on a model of the instantaneous bit error rate (BER) driven
by channel state information (CSI) [55], i.e. received signal strength indication (RSSI)
and link quality indication (LQI). We provide proof of concept by applying gPMSS
tO channel traces and demonstrate one order of magnitude reduction in packet losses.
Applying gPMSS to traces allows more detailed analysis and reproducibility that is
rlot possible in a live setup, i.e. the event when receivers are not able to reconstruct

an error-free version of the transmission. We show that even then we are able to

159

 

 

 

. Low packet loss ’ .
communication range,

FigLIre 8.1: Application of generalized gPMSS in a wireless sensor network with mesh
toDOlogy. Path from transmitter T to receiver R1 marks the multihop path that would
be taken in a network without gPMSS. Dashed line links between T and receivers R1,
R2 and R3 denote the longer range but high loss links that are used under Generalized

gPMSS.

Siglnificantly reduce the average BER of incorrigible packets. Finally, we implement
gPMSS on Imote2 sensor motes [35] using C# and demonstrate a clear reduction
in packet losses. Experimental results from IEEE 802.15.4 links indicate that using
diversity combining raises packet reception rate (PRR) by up to an additional 130%
Over those in a single receiver.

Our contributions are threefold;

1. gPMSS is a protocol centric, cross-layer approach which means it can be used in
presently deployed wireless sensor networks by making software changes only.
It does not require any modifications to hardware but. runs on networks of

commercial off-the-shelf (COTS) single antenna sensor motes.

2. gPMSS is non-intrusive in the sense that it does not require changes to the

pre—existing IEEE 802.15.4 standard.

160

 

3 - gPMSS is able to reduce power consumed at the transmitter per packet delivered
by up to 68%. This represents a significant increase in the lifetime of sensor

node.

Figure 8.1 illustrates the difference between routes traversed by a packet sent by
transmitter T to a distant node R1 when gPMSS is used (dotted arrows represent
long range links, solid lines represent links between R1, R2, R3 that form a fully
connected graph), and the multi-hop path from node T to R1 when it is not used
( SOlid arrows).

The remainder of this chapter is organized as follows. Section 8.2 reviews some
related works. Section 8.3 describes the three diversity combining techniques for
paCket recovery. Section 8.4 describes the gPMSS that enables cooperation between
ITlllltiple receivers. Section 8.5 describes the trace collection setup and demonstrates
a proof of concept of gPMSS in a manner that can be reproduced. Section 8.6
deSCribes the gPMSS implementation on Imote2 and its results. Section 8.7 discusses
0111‘ results in terms of PRR, retransmission attempts and energy consumption per

13 a-Cket. Section 8.8 concludes this chapter.

8 ~ 2 Related Work

T119 concept of spatial receiver diversity is not new and has been studied extensively
in the analog signal domain. Chakraborty et al. proposed the extended automatic
reliDeat-request (ARQ) scheme [24] that recombines spatially diverse versions of a re-

CeiVed packet to detect bit errors and an exhaustive search to correct them if their
n\Unber is less than a threshold value. Extended ARQ has a lot in common with
the version of gPMSS that uses equal gain combining and is agnostic of what MAC
Standard is used, but the results provided in [24] are based on theoretical analysis

Only. Miu et al. [83] proposed a system that used transmitter diversity to increase

161

 

‘C

packet reception rate in IEEE 802.11 [51] networks with multiple access points (AP)
as senders. The scheme roughly corresponds to gPMSS with selection diversity, with-
out diversity combination for error correction. Miu et al. generalized this approach
i n [85] for applications beyond streaming video. In [84] Min extended the idea further
to reduce packet losses on the uplink (mobile device to AP). However, this required
n‘lodifications to IEEE 802.11b AP hardware or deployment of more APs, and uses
EL dedicated frame combiner connected to all APs through a wired network. It used
the equal gain method for detecting bit errors and, like extended ARQ, relied on
all exhaustive search of the correct bit values. Cheng and Valenti [26], [123] ex-
tGlided the idea for improving throughput on uplinks in IEEE 802.11a networks by
l-lsirlg maximal ratio combining based on CSI measurements. However, like Miu’s
85" Stem it still required a dedicated combiner connected to all APs. In [60] Ji et al.
Dr C>I)osed an approach for improving the throughput of downlinks by scheduling trans-
nl iSSions to multiple receivers in IEEE 802.11a/ b networks based on explicit feedback
from receivers while maintaining fairness. In [7] Bahl made the case for multi-radio
tr‘Efl—rlsmeivers, but as figure 4 in his paper showed, collaboration between network in-
terfaces is possible only when they are all located on the same device. More recently,
VVOO described SOFT [134] which also exploited receiver diversity for the uplink in
IEEE 802.11 networks similar to Miu’s in [84], but with diversity combining being
I)erformed using maximal ratio combining. Therefore, it too requires a centralized
combiner on the wired network that all APs are connected to. To summarize, the
g1DIVISS system presented is distinct from all these prior works on cooperative com-
r111lliication and diversity combining in wireless networks because it is (1) designed
for IEEE 802.15.4 networks, (2) is purely implemented in software and COTS motes
‘Vithout modifications to mote hardware, (3) is tested on bit error traces collected

from real IEEE 802.15.4 channels, (4) as well as actual implementation on motes.

162

 

‘Ii! MPH Int-“r

8 - 3 SIMO Diversity Combining Techniques

T118 solution that is described in this section is dubbed the generalized ’Poor-Man’s-
SIB/I O-System’ because it uses receiver side diversity combining techniques and is
1 ) uilt using commercial-off-tlie-shelf components, without customized or reconfigured
llardware. Receiver diversity improves link quality of wireless channels with high
losses. This way we reduce losses and retransmissions and increase throughput and
(:11 annel utilization. This subsection describes linear diversity combining techniques.
All these techniques are derivatives of the techniques by the same names presented by
Brennan in [19]. Brennan describes scanning diversity, selection diversity, equal gain
(iiVersity and maximal-ratio diversity combining. Although the methods described
})3’ Brennan were meant for analog signals, we have suitably modified and adapted
tllern for use with demodulated, digital signals. we have included the last three,
Selection, equal gain and maximal-ratio diversity combining. Readers should know
t" 118-t even when the diversity combining method used is either equal gain or maximal-
r'd'toio combining, selection diversity is used whenever at least one receiver possesses
an error-free version of a transmissions. Equal gain or maximal ratio combining are

0 . . .
111)? used when none of the gPMSS receivers was able to receive error—free (1.e. the

Si . . . . . . .
t' 11 ation described in figure 8.5c). The purposes of diverSIty combining are twofold.

1 . Select an error-free version of a received transmission from among all received

versions.

2. If the first goal is not achievable, obtain another version of the transmission,

with fewer errors than any of the individual received versions.

163

8 - 3 -1 Selection Diversity

Selection diversity is the simplest diversity combining technique. Figure 8.2 is an
equivalent system diagram of the selection diversity process. The basic idea in is to
select from all received packets the one that is expected to have the fewest errors.
T11 is is advantageous when it is used in conjunction with forward error correction
( 2F EC) because fewer bit errors are easier to correct than more bit errors. When all
r€2C‘eived versions have errors, the best selection diversity can hope to achieve is pick

tl‘le version with the fewest bit errors. We define the BER of the nth packet in a

Sequence as,

# of error bits in nth

 

a la
BER=6[n] = "BC“ 10 .

# of bits in nth recvd pkt (81)

Tlllls the underlying random process producing the sequence of BER observations
[3 [71] is called the BER process and is denoted by B. The term BER is not used in its
St 1‘51 Ct traditional sense where it denotes the long term average probability of bit errors,
811 Ch as in a binary symmetric channel (BSC). instead the BER is computed over each
re‘Qeived packet. Unfortunately, under ordinary circumstances the BER process is not
directly observable. A packet’s failure to pass the cyclic redundancy check (CRC)
test only tells us that the number of bits with errors is non-zero (B > 0), but it
(10 es not give any information about the number of errors. Therefore, we must rely
on estimates of the BER. The performance of selection diversity will be determined
by the accuracy of the model used to predict the BER of packets that fail the CRC
test. We have used Ilyas and Radha’s [55] CSI measurement-based model of the BER
process on IEEE 802.15.4 links. For each received packet the model relies on two CSI

parameters, i.e. LQI, and RSSI. Measurement of both RSSI and LQI is mandated by

164

ill .'

tlle IEEE 802.15.4 LR—WPAN standard for every received packet. The RSSI random
pro cess is denoted by P, and RSSI measured by a receiver R for the nth packet in a

sequence is denoted by pR[n].

VVe used the MICA2 [34] to demonstrate proof-of-concept and the Imote2 [35] to
(ienjonstrate the functioning gPMSS protocol implementation, both of which use the
Chipcon CC2420 radio transceiver [120]. Technically, the CC2420 does not measure
t1'le LQI directly. Instead, it measures the correlation C between the first 8 received
SDv’ITIbols (of the PHY header) and the corresponding 8 known symbols (preamble).
IEEE 802.15.4 uses lG-ary ()ffset-Quadrature Phase Shift Keying modulation which
el”Icodes 4 bits in one symbol. The first 8 symbols, 4 bytes, of the PHY header

C()1'r1prise of the Preamble sequence consisting of 32 binary zeros. The LQI is then

C1EEfined as,

LQI 2: (C — Cl) - 02. (8 2)

In the Chipcon CC2420 c1 and (:2 are functions of the packet error rate (PER)
I11‘38.sured over an extended period of time and are determined experimentally. c1
and c2 scale the 7 bit value of the correlation to the range of an 8 bit number. Since
eq‘u ation 8.2 is merely a shifting and scaling of the measured C we take Cl = O and
C2 2 1. The LQI random process is denoted by A, and LQI measured by a receiver

R for the nth

packet in a sequence is denoted by ARM].

Coming back to our description of the CSI-driven BER model of [55], each pair of
LQI and RSSI inputs produces a probability density function (PDF) of the BER of
packets received with those particular CSI measurements. To be useful in the current
C30ntext, the output of the CSI-driven BER model has to be mapped to a single value.
VVe use flcho to denote the X th percentile of the BER process’ PDF (550% is 8’3

mean). The instances of the BER model return BER estimates denoted as MR1),

165

5 ( R2) and BUB). The output selector in figure 8.2 receives as input the estimated
BERs MR1), MR2) and BUB). Based on these estimates it selects the receiver with
t.11e lowest BER estimate as the least error-prone one and accepts its received copy

e1.s the best one and outputs it as D(Sel), i.e.

D661) 2 D<Rr) : r = argininMRi).
z (8.3)

8 - 3.2 Equal Gain Diversity

Tlle equal gain diversity combining method described here is depicted by an equivalent
SE’Stem diagram in figure 8.3. Recall that like D(T), the three received copies Bun),
B ( R2) and 5(R3) are vectors of binary numbers (representing bits) obtained after
del‘l‘lodulation of the received carrier signal. Essentially, equal gain diversity com-
billing uses received data [DU—{1), 15022) and 5(R3) to vote on the value of each
Ollt«Ibut bit. In the example in figure 8.3 performs vector addition of 15031), 15(R2)
and 5(R3), stores the sums in integers and then adjusts the gain by dividing by the
IlLlIIlber of receivers N, where N = 3 in this example. The result will be an array of
rEltional numbers in the range [0, 1]. These numbers are thresholded such that values
less than 0.5 are. remapped to binary zeros, and values greater than (or equal to)
O ‘ '5 to binary ones. The output of the thresholder is D(EG). If S () is a function

EEIDI‘esentmg the operation of the binary deC1sion thresholder, then for an N-receiver

graph/188 cluster the equal gain diversity combining process can be represented as;

[V
. 1 Z i

166

Equal gain diversity combining has two advantages over the preceding selection di-

versity combining.
1. It has lower complexity because it does not rely on a BER model.

2. It offers the possibility to recover from all bit errors introduced by the wireless

channel, even when all received versions contain errors.

8 - 3.3 Maximal Ratio Diversity

The maximal ratio diversity combining method described here is depicted by an
e Cl Llivalent system diagram in figure 8.4. It combines elements from selection and equal
gain diversity combining. Maximal ratio combining can be described as equal gain
diversity but with weighted addition. 5(R1) , 5(R2) and 5023) are each multiplied
b},

Vveights ml, 102 and "(1)3 computed as,

w,- = l — ,Bin) V1 3 i g N
2 (8.5)

a‘11C1 added. The sum is then re-normalized by dividing by the number of receivers
[\r (in this case N = 3) and thresholded which returns the output DOV-IR) of the

1'11 aximal ratio combining process;

7

D(MR) = 5 (wt , Dean)

1
N i—l (8.6)

In the following subsection we proceed to describe the gPMSS protocol that enables

Cooperation between receivers.

167

 

 

 

 

 

.. \t ...D(R1)
\\\ R1 NR1) BER Wm)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

\\ (X \\ \ PG“) ;Model
i i l ~D<R2>
j "'1! A A(RZ) (R2) F A [3(8)
_r // fl / . R2 (R 2) > BER B
/ / / p > Model
./ ///\[ ~D(R3) 0
(R3) utput
R3 A ‘ BER BR” > Selector

 

 

 

 

 

 

 

PR3) > Model

 

 

Figure 8.2: Illustration of logical functioning of selection diversity.

 

~D(R1)

 

 

 

 

(JO

 

 

R2

 

R1
1 .,
~ (R2)

,c ... D<EG>

 

 

 

 

 

 

 

 

 

 

 

 

~D(R3)

 

R3

 

 

 

Figure 8.3: Illustration of logical functioning of equal gain diversity.
8 .4 gPMSS Protocol

This section describes the operation of the gPMSS protocol. Assume a W SN consist-
ing of a large number of single-antenna COTS receivers communicating over multiple

hops with the base station collecting data. According to some topology construction

168

 

 

 

T ~D<Ru
\\ R. w

\ \ \\ p(R1) :Model

T) ( )xt ~D(R2)r
T ./ / / , R2 MR2) BER

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

// / p(R2) :Model D(MR)
/T ~D<R3>
(R3)
R3 A = BER
9(R3) : Model

 

 

 

 

 

Figure 8.4: Illustration of logical functioning of maximal-ratio gain diversity.

algorithm, a node R1 is chosen as an upstream end-point of a link. To use R1 as
part, of a set of multiple receivers we propose the gPMSS protocol that defines the
message exchange between cooperating receiver nodes to handle transmissions that
at e received with errors or not received at all. The following subsection provides a

br ief overview of gPMSS protocol message exchanges for four important operations.

For
T

illustrative purposes we assume a scenario in which there is a distant transmitter
and a receiver R] with two neighbor nodes R2 and R3 that are located close enough

to communicate with R1 with few losses.

8 «4.1 gPMSS Cluster Creation

Tlle ’Poor Man’s SIMO System’ described by Ilyas, Kim and Radha in [52] differs
from gPMSS in the way clusters of receivers are formed. Cluster creation in PMSS was
e3-(plicit, and required the operation by which nodes R2 and R3 come to be associated
as cooperating receiver nodes of a node R1. In gPMSS nodes take advantage of CSI
0f overheard messages. Figures 8.9 and 8.10 density functions of LQI and RSSI of

packets originating from R1, R2 and R3. Nodes in a network with the gPMSS protocol

169

_ A"
j ..""":’====1:::::_- _~_-:_-:_-:_._.::::: _________ / ....................
...... R2) ‘

D(
ACK ACK

  

 

 

 

 

(a) Reception of an error free packet by a (b) gPMSS message exchanges when parent re-
gPMSS cluster. ceiver R1 receives message with errors but child
R2 receives error-free.

 

Diversity
combining

 

 

 

 

 

(c) gPMSS message exchanges for recovery of data when neither parent R1 nor
children R2 and R3 receive error-free.

Figure 8.5: gPMSS protocol operations.

will maintain such histograms for all neighbors from which they overhear traffic. A
high mean, median or mode of LQI and RSSI density functions is indicative of a link
with high PRR. In this way, once a node determines it enjoys good link conditions
with a neighbor it will act as a member of that neighbor’s cluster of receive nodes.

However, unlike in PMSS this association will not be made explicit by an exchange of

170

 

 

 

(YI

messages. Instead, the node will make its contribution by participating in selection
diversity and diversity combining functions of the neighbor node. For the following
discussion we will assume that this way two nodes R2 and R3 placed close to R1 make
the assessment that they enjoy a reliable wireless channel with R1 and volunteer to

assist it as cooperating receivers.

8.4.2 Error-free Reception by at Least One Recipient

This section describes the exchange of messages under the gPMSS protocol that
occurs when at least any one of the receiving nodes receives a transmitted packet
without errors. Figure 8.5a depicts the simplest case. The solid lines represent the
transmission and reception of a message between source and destination node. The
dotted lines represent communication that occurs implicitly as a result of a receiver
operating in promiscuous mode, (deliberately) eavesdropping on messages exchanged
between other nodes (marked by solid lines). Here T sends a data message D(T)
to R1 that is overheard by R2 and R3. R1 will promptly responds to T with an
acknowledgement (ACK). R2 and R3 overhear the ACK and recognize that the packet

was successfully received by R1 and no further action is required.

Figure 8.51) depicts the case where R1 is not the final destination. In addition,
T)

let us also assume that R1 receives the transmission D( with errors (marked by a

zigzagged arrow), whereas R2 and R3 receive the same error-free. What happens is

that all receivers that receive D(T)

error-free choose a random wait-time t1 from an
exponential PDF limited to the range [0, T1]. Let t1(R2) and t1(R3) denote R2 and
R3’s random wait-times, respectively. Let t1(R2) < t1(R3), then R2 will transmit
ACK back to T before R3. R3 will overhear R2’s ACK and cancel transmission of
its own ACK. At any time, if an ACK packet is lost and not received by T, T will

T)(

retransmit D( although it may already have been received and ACK’ed). This

171

way the power consumed in nodes forming the gPMSS cluster to relay packets will

be more evenly distributed.

8.4.3 Erroneous Reception by All Recipients

This section describes the exchange of messages under the gPMSS protocol that
occurs when all nodes that form a gPMSS cluster receive a transmission with errors.

T)

Figure 8.5c depicts this entire transaction. Here T sends a data message D( to
R1 that is overheard by R2 and R3. Since all receivers R1, R2 and R3 receive
with errors none of them is able to respond to T with an ACK within time T1.
Let Run), Run) and 15023) denote the different versions of D(T) as they are
received by R1, R2 and R3, respectively. Thus, there is no error—free copy of the
transmitted message at any receiver. Nodes R1, R2 and R3 all wait for one another
to respond to T with an ACK. When none of the receivers R1, R2 and R3 overhear
an ACK going back to T within the T1 time of receiving, they infer that none of them

(T)

received D error-free. Instead of requesting a retransmission from T, R1 collects
the error-prone versions of the D from cooperating receivers, acknowledging each
one immediately as it receives them. R2 will transmit 15(R2), A(RZ), MR2), which
denotes the concatenation of EU”), the LQI A(RQ) and RSSI {)(RQ) with which it
was received from T, to R1 between [T1,T1 + T2] after it received 15022). Similarly,
R3 will transmit [Di-R3), A(R3),p(R3) between [T1, T1 + T2] after it received 15023).
Once R1 has received {15(R2), A(Rg), p(R2)} and {E(R3), A(R3), p(R3)} it executes
one of the diversity combining algorithms described in the preceding section in an
attempt to recover D(T). If the CRC computed from the recovered packet matches
the appended CRC the attempt is successful. On the receiver side T waits for an

ACK, any ACK from any of the receivers R1, R2 or R3, for a timeout period of TT

until it attempts retransmission of DiT). Note that TT > T1 + T2.

172

/

Host PC

 

 

MICAz Mote

 

Ethernet
Gateway

 

Receiver 1

 

MICAz Mote

 

Ethernet
Gateway

 

Receiver 2

 

MICAz Mote

I
Channel 2x

 

 
   
  
 

IEEE 802.15.4
Channel 26
(2.480 GHz)

 

 

MICAz Mote

Transmitter

Ethernet
Gateway
Receiver 3

 

 

 

Figure 8.6: Equipment setup for trace collection.
8.5 Trace Based Proof of Concept

In this section we provide proof of concept of gPMSS by testing its performance on
bit error traces. We collected error traces of a few million packets in a way that
provides, to the authors’ best knowledge, the BER a packet is subjected to and the

LQI and RSSI with which it is received.

8.5.1 Experimental Setup

The trace—collection setup is depicted in figure 8.6 and consists of a Crossbow MPR2400
MICAz mote [34] transmitter and another three MICA2 motes mounted on Crossbow
MIB600 Ethernet gateways [33] as receivers. The three receivers R1, R2 and R3 are
connected to a host personal computer (PC) running three instances of Xlisten (a
data logging application), one for each receiver. This way a data collection session
produces three traces. All traces were collected while operating in channel 26 in

the 2.4800Hz band. The reason for choosing channel 26 was that it is least prone

173

 

 

Octets:

 

 

 

 

 

 

 

 

 

 

 

 

1 2 1 2 2 1 1 29

Frame Sq Best Best Data /
Le” Control No PAN ID Addr TV” (3'9 Payload FCS
2 1 1 4 4 4 4 4 4 1
0x8 Src Dst SeqNo SeqNo SeqNo SeqNo SeqNo SeqNo 0x
401 Adr Adr (1) (2) (3) (4) (5) (6) 00

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 8.7: CC2420 MAC frame format used for experiments.

to interference from any 802.11b / g frequency channels. Our own experience shows
that selecting channel 26 does not completely eliminate interference from co-located

802.11b/g VVLANs, but reduces it significantly.

Packet Payload

TinyOS [75} is one of the most widely used Open source operating system in WSN
devices. T inyOS vl.1 allows various packet formats to be transmitted. We suitably
modified code to enable the standard 802.15.4 frame format which TinyOS v1.1 la-
bels CC2420 Frame Format (after the Chipcon CC2420 chipset [120] used in MICAz
devices). Strictly speaking, the term packet refers to the protocol data unit (PDU)
exchanged between network layers of the transmitter and receiver while the term
frame is used for PDU’S exchanged between MAC layers. However, since our analysis
is restricted to the MAC layer there is little cause for confusion and we use these
terms interchangeably to refer to MAC layer PDUS. The exact MAC frame format
used is shown in figure 8.7. The size of the frame is 41 bytes and comprises of a 1

byte length field, 2 byte frame control field (FCF), 1 byte sequence number, 2 byte

174

destination PAN ID, 2 byte destination address, 1 byte type field, 1 byte group field,
29 bytes of data followed by a 2 byte frame check sequence (FCS) containing a CRC.
The contents of the payload field are of our own choosing and consist of 3 unused
bytes, the source address, the destination address and 6 cepies of a 32 bit sequence
number. The sequence number in the payload is used to keep track of lost packets. If
the sequence number between two consecutively received packets skips one or more
numbers that is indicative of a packet loss. The sequence number field alone proves
too small for this task in the face of long fades. Note that transmitted packets differ
only in the 1 byte sequence number in the header and the six 32 bit sequence num-
bers in the payload, and the CRC. For a particular trace all remaining bits remain
unchanged. However, since the wireless channel will introduce bit errors the copies of
the sequence number used to track packet losses in the received packet may differ. For
this purpose we use a majority vote of the received sequence numbers to reconstruct

the transmitted sequence number and from it the entire packet.

Trace Generation

Bit-level error traces can be generated by comparing a transmitted packet with its
received version. A simple bit-wise XOR operation of the transmitted and received
packets yields a. bit pattern in which a zero (’0’) signifies a bit that is received without
error while a one (’1’) represents an inverted bit. We observe that in some cases the
length of the received packet is shorter than the transmitted packets. This constitutes
a partial loss and we use the term partially lost packets to refer to such packets.
Partially erased packets are logged when bits in the MAC header’s length field are
inverted and the receiver stops listening to the wireless channel prematurely. It has
also been observed that if bits in the length field are inverted in such a way that the

length of the incoming packet appears longer than actual the length of the logged

packet still equals that of the transmission. Although the length field in the received

175

packet may falsely indicate a longer packet, the absence of a carrier signal allows the

receiver to detect the end of transmission.

8.5.2 Channel State Information

Each received packet’s logged entry is accompanied with three pieces of packet level
CSI parameters. The first is the F CS status of the packet modeled by random variable
(I) with the nth packet’s FCS status is represented by q§[rl.]. Ordinarily receivers only
distinguish between two states, i.e. F CS Pass (denoted o = 0) if the CRC value in
the FCS field matches the CRC of the received packet, and FCS Fail (denoted (lb aé 0)
if it does not. Since we have knowledge of packet erasures and size of transmitted
packets we extend the definition of F CS status to accommodate the reason for failure.
We restrict the definition of FCS Fall BE (denoted o = 1) to mean that the size of a
received packet matches the size of the transmitted packet and the CRC failure is due
to bit errors (BE). Furthermore we classify a packet as being F CS Fail PL (denoted
o = 2) and F CS Fail CL (denoted cf) 2 3), where PL and CL are abbreviations for
partial loss and complete loss respectively. Packets that are partially lost cannot pass
the CRC test and are marked FCS Fail PL. Packets that are not received at all, i.e.
when the decoded sequence number at receiver skips, are marked FCS Fail CL.
Among other CSI there are RSSI and LQI which we described in earlier sections.
Completely lost packets, with db = 3, are assigned p = —128, /\ = 0, and £3 = 1.
Thus each received packet is characterized by its FCS Status, LQI, RSSI and BER

processes.

8.5.3 Implementation Results

Due to shortage of space and due to the consistent similarities in results, we restrict

our discussion to a subset of traces. This particular data. set was collected in an

176

 

 

+R2
. +R3

 

 

 

 

 

 

0.02 0.04 0.06 .08 0.1 0.12 0.14
B R-|3

Figure 8.8: PDF of BER experienced by receivers R1, R2 and R3 (pB(,3 = 0) is
cropped out for better view of non-zero range.

office environment over a period of 24 hours consisting of approximately 800,000
transmitted packets. The link between transmitter and receiver was non-line—of—sight,
with a wall, a door and several furniture items in the direct line between them. The
receivers were separated by a distance of 0.25m. The gPMSS cluster consisted of
three receivers, also Crossbow MICAz motes mounted on MIB600 Ethernet gateways.
Figure 8.8 is a cropped portion of the PDF of BERs observed in packets at gPMSS
receivers R1, R2 and R3 that excludes ,8 = O for enhanced visibility. Figure 8.9
depicts the PDF of the LQI of all received packets at R1, R2 and R3. Figure 8.10
depicts the PDF of their RSSI. These three figures clearly show that. all three receivers

experience different channel conditions.

177

 

 

 

 

 

 

 

 

   

40 W 60 80 100
LQI - 7t
Figure 8.9: PDF of LQI experienced by receivers R1, R2 and R3.

PER and PLR Analysis

We define two quantities based on the FCS status, the packet error rate (PER) and
the packet loss rate (PLR);

# of reed packets "with. d) = 1,2

PER =
# of transn'zxitted packets ’ (8.7)

 

PLR _ # 0f rcvd packets with q) = 3
— # of transmitted packets ’ (8.8)

 

The packet reception rate (PRR) as PRR 2 1 — (PER + PLR). In figure 8.11 the
first three entries on the horizontal axis plot the PER, PLR and the sum of the two,

PER+PLR, for R1, R2 and R3. For individual receivers PER+PLR happens to be

178

 

 

 

.0
0)

 

 

.0
<11

 

.0
i?

 

.0
N

.0
—\

 

 

 

 

-95 _QRSSI - p(d§5m) '80

Figure 8.10: PDF of RSSI experienced by receivers R1, R2 and R3.

approximately 7%, 17% and 12%. These figures are followed by plots of these same
quantities for the three diversity combining techniques. The simplest technique, se—
lection diversity, appears to track the PER+PLR of the best performing receiver, in
this case R1. Equal gain and maximal ratio diversity combining both perform better
than any individual receiver and selection diversity. This was to be expected. Recall
that selection diversity merely tries to pick out the least corrupted version among a
set, whereas equal gain and maximal ratio actually attempt to correct errors in re-

ceived messages by un-weighted and weighted voting, respectively. This is adequately

reflected in the plot of PER, PLR and PER+PLRS. Both are able to reduce the PER

179

7

0.2

 

 

 

 

->-PER
—o—PLR .
g 7 +PER+PLR 4
0151 . . ..... . .. . ...
0.1-

 

 

 

911 R2 R3 Select Equ Gain Max-ratio
Receiver '

Figure 8.11: PER, PLR and PER+PLR experienced by receivers R1, R2 and R3
without gPMSS diversity combining and with selection, equal gain, and maximal
ratio diversity combining.

BER Analysis

In figure 8.12 we plot the histogram (not PDF) of packets with non-zero BER as
experienced by individual receivers R1, R2 and R3 without any diversity combining,
as well as with different diversity combining methods. Again, the trends exhibited by
diversity combining methods are the same across all traces. Figure 8.12 shows that
the histogram of the selection diversity combining closely matches that of the best
receiving individual receiver, i.e. R1. The close match of the histogram of selection
diversity with that of R1 shows it manages to bring a gPMSS’ BER performance
up to that of the best receiving node. Thus, the BER model that is at the heart

of this diversity combining technique delivers good performance. The result of equal

180

 

60,000 l r l l l r m
; .R1
? .R2

 

 

 

 

 

T .SlMO-Selection Div.

.5 .SIMO-Equal Gain Div. _
% 40’000 :4 F’SlMQ-Maximal Ratio Div.
0. 30.000 1
l..—
o
4* 20,000 + ______ -

10,000

 
 

 

 

 

0.02 0.04 0.06 0.08 0. 0.12 0.14
BER

Figure 8.12: Histogram of BERs observed by receivers R1, R2 and R3 without gPMSS
diversity combining and with selection, equal gain, and maximal ratio diversity com—
bining.

gain and maximal gain diversity combining are even better. For every BER bin in
the histogram, both equal gain and maximal—ratio combining are able to reduce the
number of corrupt packets. Both are very close in their performance, but equal gain
is consistently beating maximal-ratio combining across all BER bins in figure 8.12,

and is also able to maintain this performance across different. trace sets.

8.6 gPMSS Protocol Implementation

This section describes our implementation of the gPMSS protocol for motes and ana-
lyzes its performance. For the mote platform, we selected the Crossbow’s Imote2 with
the pre-installed .NET Micro Framework edition [35]. Using this edition of the Imote2

enabled us to implement gPMSS in the C# programming language which simplified

181

and accelerated development. At this point we would like to clarify that although
the Imote2 used for the actual implementation in this section is different from the
MICA2 we used for trace collection in section 8.5, both use the same Chipcon CC2420
radio transceiver [120] which makes them equivalent for the purpose at hand. As the
description of the gPMSS protocol above showed, in a situation when a transmission
is received correctly by at least one recipient, gPMSS implements selection diversity
described in subsection 8.3.1. But when a transmission is received with errors by all
receivers, gPMSS either implements the functionality of an equal gain diversity or

maximal ratio diversity combiner. We have implemented both in C# for Imote2.

The maximal ratio diversity combiner depends on the CSI-driven BER model by
Ilyas and Radha [55]. Since the BER model takes as input an LQI, RSSI pair A, p
we still need to map it. to a probability value. In the first instance we find the 90th
percentile value of the BER’S predicted PDF, i.e. the BER for which the value of the
cumulative distribution function (GDP) is 0.9. In the second instance we map PDFs
of the BER to their corresponding 50th percentile. We analyze the performance of
the gPMSS protocol in a setting with one transmitter and N = 3 receivers. The
receivers run a complete implementation of the gPMSS protocol described in section
8.4. For the experiment the timeout constants were set to T1 = 10 sec, T2 = 12 sec
and TT 2 30 see. We deliberately chose large values for T1, T2 and TT to avoid
synchronization issues and justify them by the low—rate nature of target applications
for IEEE 802.15.4. For the time being we have not attempted to optimize them to

maximize throughput while still avoiding synchronization problems. The experiment

was conducted at a residence with moderate W i-F i network interference.

182

8.7 Results and Analysis

This section analyzes and compares PRR, energy per packet and effect of retransmis-

sion limits on packet delivery rate with and without gPMSS.

8.7 .1 Packet Reception Rate

We denote the total number of transmissions made from transmitter T by CT, and the
number of retransmissions among them by CR. Similarly, the number of transmitted
packets that are received at R1, R2 and R3 without errors are denoted by C1, C2
and C3, respectively. Finally, CS denotes the number of packets for which diversity
combining was attempted and succeeded, and CF the number of packets for which it
failed. All these values are tabulated in table 8.1. Each row in the table corresponds to
a trial experiment using a variant of gPMSS specified in the first column. The results
presented here are for three variants, i) Maximal-ratio combining using 690% for the
BER point estimate, ii) Maximal-ratio combining using the 650% for the BER point
estimate, and iii) equal gain combining. To make sense of the packet counts in table
8.1 and quantitatively assess the benefits of using only selection diversity, and using
selection diversity in conjunction with maximal-ratio/ equal gain combining we look
at PRRs, denoted by 8. Columns (1), (2) and (3) in table 8.2 contain the PRRs of the
baseline configuration in which receivers R1, R2 and R3 do not cooperate. Column
(4) contains the PRR when gPMSS is used with the diversity combination method
in column (0). Some of the packets received using gPMSS will have been received
as a result of selection diversity, while others will have been recovered as a result of
diversity combining. The following columns separate the gain in PRR over that in
the baseline configuration by providing the additive increase in PRRS of individual
receivers. Columns (5), (6) and (7) are additive contributions of selection diversity

in 9913111sz to the PRRs of individual receivers. Thus, AOSD,Rli AQSD,R2 and

183

Table 8.1: Packet counts.

 

 

 

 

 

 

 

 

 

 

 

 

Diversity. combining CT CR C1 C2 C3 C3 CF
Max-Ratio 590% 3170 855 937 893 957 597 0
Max-Ratio 050% 4167 1039 1254 1322 1198 739 0
Equal Gain 3683 879 819 844 825 497 0

 

 

A6 S D R3 are the increments in the PRR with respect to their respective baseline
performances 6R1, 6R2 and 6R3 in non—cooperating mode. Finally, column (8) is
the additive contribution of diversity combining ABDC to the PRR 691321155 of the
system with gPMSS. Thus, since the PRR gains in columns (5), (6), (7) and (8) are

all additive the relationship between the terms in table 8.2 is,

9913.11.93 = 9m + A95pm + Moe
= 9R2 + A950,122 + MDC

= 6R3 + A930,123 + MUC-

8.7 .2 Energy Per Packet

In this section we compute separately the energy expended by the transmitter T as
well as the receiver cluster R1, R2 and R3 per error free packet communicated to
any one receiver. Let E D AT denote the energy spent to transmit a data packet,
E AC K the energy spent to transmit an ACK packet and E AC K < E D AT' Then
the energy ET spent by the transmitter T during the course of an experiment is
CT x E D AT' The energy spent by receivers R1, R2 and R3 in acknowledging these
are E131: C1 X EACK’ ER2 = C2 x EACK and ER3 = C3 x EACK' Note

that although energy is consumed by motes in tasks other than radio transmissions,

184

Table 8.2: PRR of individual nodes without gPMSS, PRR with gPMSS protocol,
PRR gain for individual receivers R1, R2 and R3 due to selection diversity, and the

PRR gain due to diversity combining.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(0) (1) (2) (3) (4) (5) (6) (7) (8)
6R1 932 933 69PMSS 4950121 A9513M 49,919,123 Moe
Expl: 0.29 0.28 0.30 0.73 0.25 0.26 0.24 0.19
Max-
Ratio
590%
Exp2: 0.30 0.31 0.29 0.75 0.27 0.26 0.28 0.18
Max-
Ratio
550%
Exp3: 0.22 0.23 0.22 0.76 0.40 0.39 0.40 0.13
Equal
Gain
0.0020 —
ArszogPMSS
Rx: NogPMSS
VTXI Select Div
A A} Rx: Select Div
“4 Brx:gpl\/lss
a szgPMSS
at 0.0010 v
Q
\
*—
s V V :2
[a B
o _

 

 

Exp1zMax—Ratio (90%)

Exp’2:Max-.Ratio (510%)
Experlment #

Exp3quu Gain

Figure 8.13: The energy in nJ consumed by transmitter and receivers per successfully
delivered packet.

185

 

Table 8.3: Energy consumed by transmissions at transmitter and receiver side per
error-free received packet. Columns (1) and (2) correspond to the baseline scenario
using only retransmissions. Columns (3) and (4) correspond to the case when only
selection diversity is used by receivers. Columns (5) and (6) corresponds to the case
where a full implementation of gPMSS is used that employs diversity combining (equal
gain or maximal-ratio) in addition with selection diversity.

 

 

 

 

 

 

 

 

 

 

(0) (1) (2) (3) (4) (5) (6)

DlV PT PR PT PR PT PR

comb

Expl 3'383EDAT EACK 1.845EDAT EACK 1'369EDAT 1'516EACK
Max— +

Ratio 0.516EDAT
390%

Exp2 3'323EDAT EACK 1.744EDAT EACK 1.332EDAT 1'473EACK
Max- +

Ratio 0.473EDAT
350%

Exp3 4.497EDAT EACK 1'596EDAT EACK 1.314EDAT 1'355EACK
Equal +

Gain 0.355EDAT

 

 

 

the power consumed by computations is orders of magnitude less. Since the gPMSS

protocol has computational complexity of 0(N). We compute the energy per packet

consumed at the transmitter PT and the sum of energy consumed by all receivers

together PR as,

ET

 

PT = # of packets 7‘

ER

ecvd wo errors ’

 

P = .
R # of packets recvd wo errors

(8.10)

(8.11)

Thus, PT and PR are energy consumption rates of transmitter and receivers obtained

by normalizing by number of successfully delivered packets. Table 8.3 lists PT, the per

186

 

decodable packet energy at the transmitter, and PR, the per decodable packet energy
at all receivers combined for all three experiments (listed in column (0)). Columns
(1) and (2) in table 8.3 correspond to the baseline case when gPMSS is not used and
packets received by R1 are retransmitted. Columns (3) and (4) correspond to the case
when only selection diversity is used by cooperating receivers. Columns (5) and (6)
corresponds to the case where a full implementation of gPMSS is used that employs
diversity combination (equal gain or maximal-ratio) in addition to selection diversity.
To keep the relationship general the tabulated values are in terms of E AC K and
EDAT-

The Intel Imote2 consumes 79272.] / b when in transmit / receive mode when it op-
erates at 13.MH2 [35]. Based on this figure EDAT for a 41 byte frame is 260/1J and
E AC K for a 5 byte acknowledgement frame is 32/lJ. Figure 8.13 plots PT and PR
(in Joules) expended in experiments 1, 2 and 3 when when maximal-ratio combining
with 590%, maximal-ratio combining with 050% and equal gain combining. As in
table 8.3 we also evaluate energy for the cases if no gPMSS and if only selection di-
versity were used. Black lines correspond to energy consumption of the transmitter,
while blue lines correspond to energy consumption of the receivers. Clearly, the trans—
mitter power consumption rate PT and receiver power consumption rates do not vary
significantly across experiments and gPMSS variants. However, there is significant
variation in PT and PR when gPMSS is not used versus selection diversity versus
gPMSS. For all three experiments PT is highest when gPMSS is not used while the
corresponding receiver power consumption rate PR is lowest. Opting to use selection
diversity alone significantly reduces PT for maximal ratio gain variants (Exp 1 and 2)
by about 42% and about 64% for equal gain variant (Exp 3). PR remains unchanged.
Note from the previous section that this is accompanied by a 25% (for Exp 1 and
2) and 40% (for Exp 3) increase in PRR. Thus selection diversity is able to provide

significant power savings while increasing PRR at the same time. V’V hen gPMSS is

187

 

20 ............ ............ : ............ h
[ +Exp1: w/o gPMSS §

+ Exp1: Max—Ratio [390% 5 :l

15_ + Exp2- W/O gPMSS .......... ........... 1’
+ Epo: Max-Ratio B i
Exp3: w/o gPMSS ;

E 10 Exp3: Equal Gain """""" " '

50%

      
 

 

 

 

 

 

0 20 40 60 80 100
9 (043)

Figure 8.14: Maximum number of transmission attempts m versus delivery guarantee

g(%)-

employed PT is reduced by about 58% (for Exp 1 and 2) and 68% (for Exp 3) over the
baseline configuration not using gPMSS. However, this is accompanied by an increase
of approximately the same amount of energy on the receiver side. Thus, it appears

that gPMSS shifts some of the power consumption from the transmitter side to the

receiver side.

8.7.3 Packet Transmission Attempts

The number of times the IEEE 802.15.4 MAC will retry transmitting a packet is
controlled by the marMarFrameRetries attribute whose default value is set to 3 but

can be varied from 0 — 7 (refer to IEEE 802.15.4 standard [5]). This limit on the

188

number of transmission attempts m for a packet limits the maximum PRR that can
be guaranteed to 9. Conversely, we may ask what is maximum number of transmission
attempts m that the MAC must be allowed in order to ensure that at least 9% of
packets are received without errors? Figure 8.14 plots m against 9 for all three
experiments. Clearly, to achieve any delivery guarantee 9%, fewer transmissions are
required with gPMSS, regardless of whether maximal-ratio or equal gain diversity
combination is used, compared to the case where gPMSS is not enabled. For example,
figure 8.14 shows that to achieve a 95% delivery guarantee we have to allow 9, 9, 13
transmission attempts for the channel conditions observed in experiments 1, 2 and 3.
Using gPMSS, however, the maximum number of transmission attempts required to
achieve the same delivery guarantee 9 = 95% are 3, 3, and 3, respectively. Clearly,
the values of 777. required to achieve 9 = 95% without gPMSS exceeds IEEE 802.15.4’s
lap-abilities. From the plot in figure 8.14 we see that at IEEE 802.15.4’s default value
of m = 4 the maximum achievable delivery guarantee for the three experiments lies

in the range 65 — 75%.

8.8 Conclusions

We presented the gPMSS, a protocol-centric approach to enable receiver cooperation
and diversity combining without requiring any changes to mote hardware or the IEEE
802.15.4 LR—WPAN standard. We described three principal mechanisms enabled by
gPMSS, namely selection diversity, equal gain and maximal-ratio gain diversity corn-
bining. We provide proof—of-concept and demonstrate gPMSS’ efficacy by applying
these diversity combining techniques on bit error traces collected from a network
of IEEE 802.15.4 motes. we demonstrate gPMSS by implementing it on the Intel
Imote2 sensor mote running the .NET i\-"licro framework. We analyze the performance

of gPMSS in terms of PRR, retransmission attempts and power consumption per de—

189

livered packet. We saw that gPMSS raises the PRR from 22 — 30% to 73 — 76%, a
relative increase of 150 — 245%. Since gPMSS is a protocol-based solution it implies
a messaging overhead. we observe that power consumption by the transmitter per
correctly delivered packet. is reduced up to 68%. We evaluated the effect of retry limit
imposed by the IEEE 802.15.4 standard of the on the packet delivery rate that can
be achieved. At the default retry limit of 3, (m =2 4) gPMSS can achieve delivery
rates of greater than 99%, against only 65 -— 75% when gPMSS is not used. Thus
we demonstrate that gPMSS is capable of raising PRR, making use of highly lossy
links feasible, thus reducing the number of required retransmission attempts and re-
ducing the energy consumption rate of the transmitter per packet delivered. gPMSS
has direct application in the design of small-world topologies in wireless networks
to reduce the characteristic path length and diameter of networks which facilitates
service discovery and the routing of high priority data in a network. This has the
advantage of not needing any additional hardware([124] and [125]), or adding wired
conuections([27] and [111]). The extension of the effective communication range also
has applications in extending the lifetime of nodes surrounding the base station in
wireless sensor networks subject to the funneling effect. The larger communication
range allows more nodes to communicate with the base station directly and reduces
the traffic load from nodes positioned closer to the base station. More generally,
gPMSS can be used to connect weakly connected components of a network by adding

more links between nodes farther apart.

190

 

Chapter 9

Principal Component Centrality as
a Measure of Node Centrality in
Communication Networks

191

9. 1 Introduction

Centrality [14], [15], [17], [42], [106] is a measure to assess the criticality of a node’s
position. Node centrality as a measure of a node’s importance by virtue of its central
location has been in common use by social scientists in the study of social networks
for decades. Over the years several different meanings of centrality have emerged.
Naturally, the idea of ranking nodes for their ability to spread or detect (positive or

negative) influence is of significant interest to social network analysis.

Among many centrality measures, eigenvalue centrality (EVC) is arguably the
most successful tool for detecting the most influential node(s) within a social graph.
Thus, EVC has been a highly popular centrality measure in the social sciences ([45],
[126], [14], [41], [38], [40], [127], [121], [16], [15]) (it is often referred to simply as
centrality). As we demonstrate later in this chapter, one key shortcoming of EVC is
its focus on (virtually) a single influential set of nodes that tend to cluster within a
single neighborhood. In other words, EVC has the tendency of identifying a set of
influential nodes that are all within the same region of a graph. This shortcoming
may not represent a major issue for many social science problems and Internet appli-
cations, such as PageRank, where EVC has been used extensively [72]. Meanwhile,
when dealing with massive networks/ graphs, it is hardly the case that there is a sin-
gle neighborhood of influential nodes; rather, there are usually multiple infiucultial

neighborhoods most of which are not detected or identified by EV C.

In order to identify influential neighborhoods, there is a need to associate such
neighborhoods with some form of an objective measure of centrality that can be
evaluated and searched for. To that end, one can think of a centrality plane that is
overlaid over the underlying graph under consideration. This centrality plane may
contain multiple centrality score maxima, each of which is centered on an influential

neighborhood.

192

Local Centrality
(«~— Maximas

 

Centrality

 

 

Figure 9.1: This figure shows a graph on the lower plane, overlayed with another
plane of the interpolated surface plot of node centrality scores. The centrality planes
typically exhibit a number of peaks or local maxima.

Nodes that have centrality score higher than other nodes are located under a
centrality peak and are more central than any of their neighbors. We use the term
hubs to refer to nodes forming centrality maxima. Figure 9.1 illustrates this concept.
Thus, these hubs form the kernel of influential neighborhoods in networks. Hence, our
focus in this research is on identifying influential neighborhoods rather than influential
nodes. We will show that EVC has a tendency to be too narrowly focused on a
dominating neighborhood. To this end, we introduce a new measure of centrality
that we call principal component centrality (PCC) that gradually widens the focus of
EVC in a controlled manner. More importantly, PCC provides a general framework
for transforming graphs into a spectral space analogous to popular signal transforms
that operate on random signals.

In this chapter, we give a brief review of common centrality measures accom-

193

panied by a critique of their application to wireless network topologies. we then
introduce PCC, a node centrality measure that is inspired by the Karhunen Loeve
transform (KLT) and principal component analysis (PCA). In essence, PCC is a gen—
eral transform of graphs that can provide vital insight into the centrality and related
characteristics of such graphs. Similar to the KLT of a signal, the proposed PCC of
a graph gives a form of cou‘lrmct representation that identifies influential nodes and
more importantly influential neighborhoods. Hence, PCC provides an elegant graph
transform framework that outperforms EVC. In particular, early in this chapter, we
demonstrate EVCs shortcoming by using both EVC and PCC to compute node cen-
tralities in a network small enough to allow meaningful illustration. This is followed
by a thorough description of PCC, and its utility in transforming massive real-world
networks / graphs. we also develop the equivalence of an inverse PCC transform that
attempts to reconstruct a reprlrseutation of the original graph from its influential
neighborhoods.

The rest of this chapter is organized as follows. Section 9.2 gives a background
review of existing centrality measures for graphs, highlights problems in EVC and
motivates our development of a new node centrality. Section 9.3 defines the PCC
measure of centrality. Section 9.4 describes in detail the advantages, mathematical
interpretation, visualization and the effect of varying number of features of PCC.

Section 9.5 concludes the chapter.

9.2 Background

Let A denote the adjacency matrix of a graph G (V, E) consisting of the set of nodes
V =_— (01,122,223, . . . ,vN} of size N and set of undirected edges E. When a link is
present between two nodes u- and vj both A27 j and A j,’i are set equal to 1 and set to

l

0 otherwise. Let F (0,) denote the neighborhood of vi, the set of nodes vi is connected

194

to directly.

9.2.1 Degree Centrality

The degree centrality of a node in a graph is a measure of the relative importance to
the graph’s connectivity. The degree centrality of a node is defined as the number of
edges incident on it. Nodes with more incident edges have higher degree centrality
than nodes with fewer incident edges. If di denotes the degree of node u,- then its

degree centrality is computed by:

 

N “‘1 (9.1)

Degree centrality is a measure of a node’s rate of dissemination (of an infection) in
the immediate short term. It has the advantage that its computation does not require

nodes to exchange information. However, it has two significant disadvantages;

1. Without an exchange of centrality information with other nodes, it is not pos-
sible to interpret and evaluate an individual node’s centrality relative to that

of others.

2. Degree centrality does not take into account the centrality of its neighbors.

9.2.2 Closeness Centrality

The closeness centrality of a node is defined as the mean length of geodesic paths
to all other nodes. Intuitively, nodes occupying a more central location within the
graph are expected to have shorter paths. Closeness centrality is a measure of the
rate at which a node can spread an infection to all reachable nodes. Closeness is

a suitable measure of centrality when the flow of commodity in the network follows

195

geodesic paths. Closeness centrality is a good measure of the average detection time

in a network with flows of non-replicating commodity following geodesic paths.

9.2.3 Betweenness Centrality

The betweenness centrality of a. node is defined as the fraction of geodesic paths
(shortest paths) out of all geodesic paths between all pairs of nodes passing through
that node. Thus, nodes located on more geodesic paths have a higher betweenness
centrality than nodes located on fewer geodesic paths. Intuitively, since the subprob-
leln optimality principal holds for the shortest path problem, a node’s location on
a geodesic path implies close proximity to all other nodes on that path. A node’s
betweenness can be interpreted as a measure of disruption caused when the node is
removed from the network. Like closeness, betweenness too assumes that the flow of
commodity is along geodesics. Betweenness centrality is a good measure of the av-
erage probability of detection of flows in a network with non-replicating commodity

following geodesic paths.

9.2.4 Eigenvector Centrality

Eigenvector centrality (EVC) is a relative score recursively defined as a function of the
number and strength of connections to its neighbors and as well as those neighbors’

centralities. Let 17(1) be the EVC score of a node vi. Then,

:I:(i)=;’\- Z we)
J'EFW)

1 N
j=1

196

 

 

 

 

 

 

Figure 9.2: A spatial graph of 200 nodes. Node colors are indicative of the range in
which their EVC falls.

Here A is a constant. Equation 9.2 can be rewritten in vector form equation 9.3

where x = {x(1),:1:(2), z(3), . . . ,$(N)}’ is the vector of EVC scores of all nodes.

x = §Ax
Ax = Ax (9-3)

This is the well known eigenvector equation where this centrality takes its name
from. A is an eigenvalue and x is the corresponding eigenvector of matrix A. Obvi-
ously several eigenvalue/eigenvector pairs exist for an adjacency matrix A. The EVC

of nodes are defined on the basis of the Perron eigenvalue A A (the Perron eigenvalue

197

is the largest of all eigenvalues of A and is also called the principal eigenvalue). If A is
any other eigenvalue of A then AA > |A|. The eigenvector x = {20(1), 33(2), . . . ,11:(N)}’
corresponding to the Perron eigenvalue is the Perron eigenvector or principal eigen-
vector. Thus the EVC of a node 1), is the corresponding element 2(1) of the Perron
eigenvector x. Note that when the adjacency matrix A is symmetric all elements of
the principal eigenvector x are positive. As mentioned above, EVC is widely used in
the social sciences ( [82], [127], [14], [42], [40], [41], [128], [126], [16], [15]) and is often

referred to simply as centrality.

EVC does not suffer from the same problems as degree, closeness and betweenness
centralities. In computing a node’s EVC it takes into consideration its neighbors’s
EVC scores. Because of its recursive definition, EVC is suited to measure nodes’
power to influence other nodes in the network both directly and indirectly through
its neighbors. Connections to neighbors that are in turn well connected themselves
are rated higher than connections to neighbors that are weakly connected. Like close-
ness and betweenness, the EVC of a node provides a network-wide perspective. At
the same time it can take advantage of distributed methods of computing eigenvec-
tors / eigenvalues of a matrix but does not have to bear the overhead of excess network
traffic. Sankaralingam [108], Kohlschiitter [69] and Canright, EngQi-Monsen and Je—
lasity [20], Bischof [12], Bai [8] and Tisseur [121] proposed some parallel algorithms

for computing eigenvectors and eigenvalues of adjacency matrices.

9.2.5 The Need for a New Centrality Measure

In the preceding sections we highlighted some of the key characteristics of the most
common measures of centrality. Our discussion left us with only one viable measure of
centrality that takes into consideration the centrality scores of a node’s neighbors and

which provides a network-wide perspective, i.e. EVC. EVC has been used extensively

198

to great effect in the study and analysis of a wide variety of networks that are shown
to exhibit small-world and scale-free properties. In [21] Canright and Engo-Monsen
correlated EVC with the instantaneous rate of spread of contagion on a Gnutella
network peer-to-peer graph, a social network of students in Oslo, a collaboration
graph of researchers at Telenor R&D and a snapshot of a collaboration graph of the
Santa Fe Institute. In [90] Newman analyzed the use of EVC in a lexical network
of co-occuring words in Reuters newswire stories. In [22] Carreras et al. used EVC
to study the spread of epidemics in mobile networks. They used three sets of traces
collected by Intel Cambridge, a trace of the public transportation network from the
DieselNet project at the University of h-lassachusetts at Amherst and mobility and
interaction traces from MIT’s Reality Mining project.

Now consider the graph in figure 9.2. It consists of 200 nodes and is typical of
networks of stationary WSN as well as mobile \VSNS such as the cellphone based
Nokia SensorPlanet project ([95], [62]). Its nodes are assigned one of six colors from
the adjacent color palette. Each of the six colors represents one of six bins of a
histogram spanning, in uniform step sizes, the range from the smallest to the largest
EVCs. As the legend accompanying figure 9.2 shows, blue represents the lowest EVCs

and red the highest. We make the following observations:

1. EVCs are tightly clustered around a very small region with respect to the total
size of the network and drops off sharply as one moves away from the node of

peak EV C.

2. EV C is unable to provide much centrality information for the vast majority of

nodes in the network.

3. The position of the peak EVC node appears somewhat ‘arbitrary’ because a

visual inspection shows that almost equally significant clusters of nodes can be

199

 

  
  

 

 

 

 

 
 

 

 
  
 

 

 

 

 

 

 

a 50 z .............................. _Adjacency Matrix
C [:J Laplacian Matrix 3
a) 40 _ ---------------------- Averaged — Adjacency Matrix 3'
8. 3 - - - Averaged — Laplacian Matrix :
a) 20 .................. ‘ ......... ‘ ........ ‘. ........ ‘ ..........
L _' '
LI. : E
0 .
-1O -5 0 5 10 15 20 25 30
A.
l
i_ 1 .......................................................
‘_
z a
N . . ‘. I
\ 0.5 ............. , ‘. i P ...... — Adjacency Matrix
___ ‘ I l I Laplacian Matrix S
fi 3 Averaged - Adjacency Matrix 3
‘- : - - - Averaged — Laplacian Matrix
0- _u 0 i l 1 i
N 0 50 100 150 200

P

Figure 9.3: [Top] Histogram of eigenvalues of adjacency matrix and Laplacian matrix
A of network in figure 9.2; [Bottom] Cumulative sum of the sequence of eigenvalues
of adjacency matrix and Laplacian matrix of network in figure 9.2 when sorted in
descending order of magnitudes. In both figures the lines plotted in red color are
averages of 50 networks generated randomly with the same parameters.

visually spotted in other locations in the graph. Counter to intuition, the high

EVC cluster is connected to the rest of the network by a single link.

9.3 Principal Component Centrality

The EVC of a node is recursively defined as a measure of centrality that is proportional
to the number of neighbors of a node and their respective EVCs. As we saw in section
9.2.4, the mathematical expression for the vector of node EVCs is equivalent to the

principal eigenvector. Our motivation for PCC as a new measure of node centrality

200

 

 

 

Figure 9.4: Reconstructed topologies of the graph from figure 9.2 using only the first
1, 2, 3, 5, 10, 15, 50 and all 200 eigenvectors.

may be understood by looking at EVC through the lens of the KLT. When the KLT
is derived from an N x N covariance matrix of N random variables, the principal
eigenvector is the most dominant feature vector, i.e. the direction in N -dimensional
hyperspace along which the spread of data points is maximized. Similarly, the second
eigenvector (corresponding to the second largest eigenvalue) is representative of the
second most significant feature of the data set. It may also be thought of as the
most significant feature after the data points are collapsed along the direction of the
principal eigenvector. When the covariance matrix is computed empirically from a
set of data points, the eigendecomposition is the well known PCA [38]. Since we
are operating on the adjacency matrix derived from graph data we call the node
centrality proposed in this research PCC. In a covariance matrix, a non-zero entry
with a ’large’ magnitude at positions (2',j) and (3,1) is representative of a strong

relationship between the i-th and j-th random variables. A non—zero entry in the

201

 

adjacency matrix representing a link from one node to another is, in a broad sense, also
an indication of a ’relationship’ between the two nodes. Based on this understanding

we draw an analogy between graph adjacency matrix and covariance matrix.

In the preceding section we described various centrality measures from litera-
ture. Among them, EVC is the node centrality most often used in the study of
social networks and other networks with small-world properties. While EVC as-
signs centrality to nodes according to the strength of the most dominant feature of
the data set, PCC takes into consideration additional, subsequent features. We de-
fine the PCC of a node in a graph as the Euclidean distance/t2 norm of a node
from the origin in the P-dimensional eigenspace formed by the P most significant
eigenvectors. For a graph consisting of a single connected component, the N eigen-
values MN 2 [Agl _>_ _>_ |AN| = 0 correspond to the normalized eigenvectors
x1,x2, . . . ,xN. The eigenvector/ eigenvalue pairs are indexed in order of descending
magnitude of eigenvalues. When P = 1, PCC equals a scaled version of EVC. Unlike
other measures of centrality, the parameter P in PCC can be used as a tuning pa-
rameter to adjust the number of eigenvectors included in the PCC. The question of
selection of an appropriate value of P will be addressed in subsequent subsection 9.4.4.
Let X denote the N x N matrix of concatenated eigenvectors X = [x1x2 . ..xN]
and let A = [A1A2 . . . A N], be the vector of eigenvalues. Furthermore, if P < N and
if matrix X has dimensions N x N, then X Nx p will denote the submatrix of X
consisting of the first N rows and first P columns. Then PCC can be expressed in

matrix form as:

 

cp = \/((AXNXP) 9 (*A‘XNxP))11”><1 (9.4)

The ‘G’ operator is the Hadamard (or entrywise product or Schur product) op-

202

erator. Equation 9.4 can also be written in terms of the eigenvalue and eigenvector

matrices A and X, of the adjacency matrix A:

 

CP 2 l/(XNxP @XNxP) (APX1GAPXIA (9-5)

It is important to note a major difference between a traditional ”signal transform”
under KLT as compared with the proposed PCC ”graph transform”. First, recall that,
under KLT, a transform matrix T is derived from a covariance matrix C; and then
the eigenvector-based transform T is applied on any realization of the random signal
that has covariance C. Meanwhile, under the proposed PCC, the adjacency matrix A
plays a dual role: at one hand, it plays the role of the covariance matrix of the KLT;
and on the other hand, one can think of A as being the ”signal” that is represented
compactly by the PCC vector C p. Effectively, the adjacency matrix A represents
the graph (i.e., ”signal”) that we are interested in analyzing; and at the same time
A is used to derive the eigendecomposition; and hence, we have the dual role for A.
Later, we will develop the equivalence of an inverse PCC, and we will see this dual

role of the adjacency matrix A again.

9.4 Evaluation

9.4.1 Interpretation of Eigenvalues

The definition of FCC is based on the graph adjacency matrix A. For a matrix A
of size N x N its eigenvectors x,- for 1 S i g N are interpreted as N-dimensional
features (feature vectors) of the set of N —dimensional data points represented by their
covariance (adjacency) matrix A. The magnitude of an eigenvalue corresponding to

an eigenvector provides a measure of the importance and prominence of the feature

203

represented by it. The eigenvalue A, is the power of the corresponding feature x,- in

A.

An alternative representation of a graph’s topology is the graph Laplacian matrix
which is frequently used in spectral graph theory [28]. The graph Laplacian can be
obtained from the adjacency matrix by setting the diagonal entries of the adjacency
matrix to AM 2 — Z§v=1n7éj Am, i.e. a diagonal entry in a Laplacian matrix is
the negative of the sum of all off-diagonal entries in the same row in the adjacency
matrix. This definition applies equally to weighted and unweighted graphs. The graph
Laplacian is always positive-semidefinite which means all of its eigenvalues are non-
negative with, at least one eigenvalue equal to 0. The adjacency matrix, however, does
not guarantee positive semidefiniteness and typically has several negative eigenvalues.
This is the reason the ordering of features is based on magnitudes of eigenvalues. The
bar chart at the top of figure 9.3 plots histograms of eigenvalues for both adjacency
and Laplacian matrices of the network in figure 9.2. But why then, did we not use
the Laplacian matrix in the first place? The reason is that the eigendecomposition
of the adjacency matrix yields greater energy compaction than that of the Laplacian.
The middle plot in figure 9.3 shows the normalized, cumulative function of the sorted
sequence of eigenvalue powers. The line for the eigenvalue derived from the adjacency
matrix rises faster than that of the Laplacian matrix. The adjacency matrix’ curve
indicates that 25%, 50% and 75% of total power is captured by the first 15 (7.5%),
44 (22%) and 89 (44.5%) features, respectively. In contrast, the Laplacian matrix’
eigendecomposition shows that the same power levels are contained in its first 26
(13%), 61 (30.5%) and 103 (51.5%) features, respectively. Thus eigendecomposition
of the adjacency matrix of graphs offers more energy compaction, i.e. a set of features
of the adjacency matrix captures more energy than the same number of features of

the corresponding Laplacian matrix.

204

C15 - 3D Spectral Drawing

 

0 high
I

0 low

 

 

 

 

    

-3
2

x,<i> 01 ° x2e

Figure 9.5: Spectral drawing of graph in three dimensions using entries of x1, x2,
and X3 for the three coordinate axes. Nodes are colored according to their C15 PCC.

9.4.2 Interpretation of Eigenvectors

EVC interprets the elements of the Perron-eigenvector XI of adjacency matrix A as
measures of corresponding nodes’ centralities in the network topology (see section
9.2.4). Research on scale-free network topologies has demonstrated EVC’s useful—
ness. However, when applied to large spatial graphs of uniformly, randomly deployed
nodes such as the one in figure 9.2, EVC fails to assign significant scores to a large
fraction of nodes. For a broader understanding that encompasses all eigenvectors we
revert to the interpretation of eigenvectors as features. One way of understanding
PCC is in terms of PCA [38], where PCC takes part of its name from. PCA finds
the eigenvectors x1,x2, X3, . . . ,x N and eigenvalues of G’s adjacency matrix A. Ev-
ery eigenvector represents a feature of the adjacency matrix. To understand how

these feature vectors are to be interpreted in graphical terms, refer to equation 9.6

205

 

which uses eigenvectors and eigenvalues to reconstruct an approximation A P of the
adjacency matrix A. Reconstruction can be performed to varying degrees of accu-
racy depending on P, the number of features/ eigenvectors used. If we set P = N
in equation 9.6 (all eigenvectors/ eigenvalues are used), the adjacency matrix can be
reconstructed without losses (see He [45]). Here, A denotes the diagonal matrix of
eigenvalues sorted in descending order of magnitude on the diagonal (from upper left

corner to lower right corner).

~ A T
AP = XNXPAPXNXNXN

A1 0 0
A 0 A2 0
where A =
0 0 - - - A N

 

 

To illustrate, consider the unweighted, undirected graph G' (V, E) shown in figure
9.2 with adjacency matrix A. A’s entries are either 0 or 1. However, this is not
necessarily true for A p, the version of the matrix reconstructed using the P most
significant eigenvectors. The entries in A P will very likely contain a lot of fractions.
Therefore, before viewing the recovered topology in the reconstructed adjacency ma-
trix A P its entries have to be thresholded. Prior to plotting the topology, we rounded
values less than 0.5 down to 0 and round values larger than or equal to 0.5 up to 1.
Figure 9.4 plots the adjacency matrix reconstructed from the most significant 1, 2,
3, 5, 10, 15, 50 and all 200 feature vectors. The plot for A1 shows that the recov-
ered topology information is highly localized to the vicinity of nodes with the highest
EVC. The plot using A2 adds another highly connected but still very localized cluster

to the network. Adding more feature vectors extends the set of connected nodes in

206

 

 

   

 

 

 

200 150
150
g $100
3100 g
9 9 50
O 0 2 4 0 0 2 4
0,0) 020)

(a) (b)

Figure 9.6: PCC of nodes in network of figure 9.2 when computed using first (a) 1
and (b) 2 eigenvectors. The histograms accompanying each graph plot show the dis-
tribution of PCC of their nodes. The lineplot in the histogram represents the average
PCC histograms of 50 randomly generated networks with the same parameters as the
network in figure 9.2.

various parts of the network. As more eigenvectors are added to the computation of
PCC it has the effect of increasing the resolution of centrality scores in nodes lying

in less well connected regions of the network.

207

 

 

 

 

 

 

100
a a
s s
a s 50
e 2
LL LL.
2 _ 4 O0 2 .
03o 050)

(a) (b)

Figure 9.7: PCC of nodes in network of figure 9.2 when computed using first (a) 3
and (b) 5 eigenvectors. The histograms accompanying each graph plot show the dis-
tribution of PCC of their nodes. The lineplot in the histogram represents the average
PCC histograms of 50 randomly generated networks with the same parameters as the

network in figure 9.2.

9.4.3 Graphical Interpretation of PCC

In this section we evaluate the usefulness of the PCC scores assigned to nodes of
a network. Recall that a node’s PCC is its 62 norm in P—dimensional eigenspace.

Perceptional limitations restrict us from redrawing the graph in any eigenspace with

208

 

 

 

 

 

   

30‘
“Z" «2‘ 20 x
h |_ 10»
“- 10» LL
0 2 n 0 2 -
C100) C150)

(a) (b)

Figure 9.8: PCC of nodes in network of figure 9.2 when computed using first (a) 10
and (b) 15 eigenvectors. The histograms accompanying each graph plot Show the dis—
tribution of PCC of their nodes. The lineplot in the histogram represents the average
PCC histograms of 50 randomly generated networks with the same parameters as the
network in figure 9.2.

more than 3 dimensions. Figure 9.5 is a drawing of the graph in figure 9.2 in the 3-
dimensional eigenspace formed by the 3 most significant eigenvectors of the adjacency
matrix A. Nodes are colored according to their C15 PCC scores, derived from the 15

most significant eigenvectors, divided into 6 equally sized intervals between the lowest

209

 

 

 

A
O
A
0

Frequency
Frequency

N
C?
N
9

 

 

   

2 .
0500)
(a) (b)

Figure 9.9: PCC of nodes in network of figure 9.2 when computed using first (a) 50
and (b) all 200 eigenvectors. The histograms accompanying each graph plot show the
distribution of PCC of their nodes.

and highest PCC score. Based on the interpretation of PCC we expect nodes with
higher (red) PCC scores to be located farther away from the origin at (0,0,0) than
nodes with lower (blue) PCC scores. From figure 9.5 we can see that this is clearly
the case. For clarification, the cluster of low-PCC nodes around the origin (0,0,0) is

marked with a red, dashed oval.

210

9.4.4 Effect of Number of Features on PCC

In this section we study the effect varying the number of eigenvectors P has on PCC.
For an illustrated example we revert to the randomly generated network topology of
200 nodes in figure 9.2. We compute PCC while varying P from 1 through 2, 3, 5,
10, 15, 50 and 200. Figures 9.6a, 9.6b, 9.7a, 9.7b, 9.8a, 9.8b, 9.9a and 9.9b re-plot
the network with nodes colored to indicate their PCC scores. The bin size for all
histograms is set to 0.25. Recall that since PCC score at P = 1 are a scaled versions
of EVC, the figure 9.6a. represents the baseline case of EVC. In figure 9.6a, EVC
identifies a small cluster in the upper right corner as the nodes most central to the
network. Note that ironically this cluster is separable from the larger graph by the
removal of merely one link! On the other hand, clusters of nodes in the larger, better
connected part of the graph are assigned EVC on the low end of the scale. As P is
increased from figure 9.6b through 9.9b, more clusters of high PCC nodes pop up.
As expected, the accompanying histograms below each graph plot show that this has

the effect of increasing the variance of FCC scores.

Adding successively more features/eigenvectors will have the obvious effect of
increasing the sum total of node PCC scores, i.e. llxNCm > llxNCn when
m > n. However, it is unclear how much PCC’S scores change as P is varied from
1 through N. In [20] Canright et al. use the phase difference between eigenvectors
computed in successive iterations as a stepping criteria for their fully distributed
method for computing the principal eigenvector. we use the phase angle between
PCC vectors and EVC to study the effect of adding more features. We compute the

phase angle 95(71) of a PCC vector using it features with the EVC vector as,

 

C C
«6(P) = arccos ( P - E ) .
ICPI ICEI

211

 

 

 

O 50 100 150 200
P - # of eigenvectors

Figure 9.10: Plot of phase angles g6 (in radians) of PCC vectors with the EVC vector
for the graph in figures 9.6, 9.7, 9.8 and 9.9.

Here, ‘9 denotes the inner product operator. The relationship of the phase angle
with the number of features used in PCC for the network under consideration is
plotted in figure 9.10. Initially, the function of phase angle (25 rises sharply and then
levels off almost completely at 22 features. This means that, in this example, the
relative PCCS of nodes cease to change with the addition of more features beyond
the first 22 features. The phase angle plot may be used for determining how many

features are sufficient for the computation of FCC of a network.

9.5 Conclusions

We reviewed previously defined measures of centrality and pointed out their short-
comings in general and EVC in particular. We introduced PCC, a new measure of
node centrality. PCC is based on PCA and the KLT which takes the View of treating a
graphs adjacency matrix as a covariance matrix. PCC interprets a node’s centrality as
its 62 norm from the origin in the eigenspace formed by the P most significant feature

vectors (eigenvectors) of the adjacency matrix. Unlike EVC, PCC allows the addition

212

of more features for the computation of node centralities. We explore two criteria for
the selection of the number of features to use for PCC; a) The relative contribution
of each feature’s power (eigenvalue) to the total power of adjacency matrix and b)
Incremental changes in the phase angle of the PCC with P features and the EVC
as P is increased. We also provide a visual interpretation of significant eigenvectors
of an adjacency matrix. The use of the adjacency matrix is compared with that of
the Laplacian and it is shown that eigendecomposition of the adjacency matrix yields
significantly higher degree of energy compaction than does the Laplacian at the same
number of features. We also investigated the effect of adding successive eigenvectors
and the information they contain by looking at reconstructions of the original graph’s
topology using a subset of features.

In the future we intend to extend the definition of PCC so it can be applied to both
directed and undirected graphs. Furthermore, we propose to formulate a distributed
method for computing PCC along the lines of Canright’s method [20] for computing

EVC in peer-to-peer systems.

213

Chapter 10

Conclusions

214

The following three sections list conclusions we learnt from the results of our

research.

10. 1 Channel Modeling

Our analysis and modeling of errors observed in IEEE 802.15.4 traces in chapters 3

and 4 leads us to make the following conclusions.

1. LQI and RSSI exhibit moderate negative correlation with the BER process (and

strong positive correlation with each other).

2. LQI and RSSI can be used to reduce the variance of BER’s estimated PDF of
packets failing the CRC.

3. The CSI driven BER model remains valid across a variety of physical environ-

ments.

4. All IEEE 802.15.4 channels, regardless of channel selection or physical environ-

ment, exhibit a memory length of at most 2 bits and 2 symbols, respectively.

5. Based on the correlation function and analysis for LRD we conclude that various
estimates of Hurst parameter may or may not detect packet level memory in
802.15.4 channels. The memory, however, is not due to the channel’s inherent
properties at those frequencies, but due to interference from IEEE 802.11b/ g
traffic and beacon frames, i.e. if interference is periodic the channel appears to

have memory, if it is not periodic there is no memory.

6. The Abry-Veitch and Whittle estimators’ consistent relative insensitivity to
changes in average PER, average PLR, average CBER and interference across
different traces leads us to conclude that they are better measures of the IEEE

802.15.4 channel’s inherent degree of LRD.

215

7. The aggregate variance, R/ S, periodogram, and absolute moment estimators’
strong dependence on average PER and average PLR leads us to conclude that

these estimators are good detectors of in-band (WLAN) interference.

8. The average CBER to which a packet is subjected by a channel is inversely
related to the average PER/average PLR. Thus, it appears that interference

produces higher BERs in packets than do channel fades.

9. We introduced RMI as a standardized version of Shannon mutual information
and apply it to the BER process captured in bit traces. We observe that interfer—
ence free IEEE 802.15.4 channels are memoryless, while channels experiencing
significant interference from IEEE 802.11b / g networks sharing the 2.4GHz ISM
band, a common source of interference, have true memory lengths varying in

the narrow range of 0 to 2sec.

10.2 Network Lifetime

Our analysis and modeling of the network lifetime problem in wireless sensor networks

(W SN) in chapters 5 and 6 leads us to make the following conclusions.

1. We propose a new definition of network lifetime consisting of the tuple of mean
and variance of node power consumption rates in a WSN. This interpretation
of network lifetime is more inclusive and considers the power consumption of

sensors across the network.

2. We formulated the optimization problem for the new objective function in the
form of a budget constrained QP and showed that a solution exists. The solution

of the QP, however, has a high complexity.

216

3. As an alternative to the QP, we developed a greedy dynamic program formu-
lation that chooses routes in a way that optimizes for our objective from sets
of paths that would be considered sub-optimal in the shortest path sense. Four
variants are developed based on the BED, BND, ED and ND algorithms for the

discovery of alternative routes.

4. We also observe that the routes generated by the ND algorithm are very similar

to those proposed in previously proposed load balancing techniques such as

 

Back and de Veciana’s in [6]. Under a many-to—one traffic flow the same ND
paths, when used in conjunction with DPA) yield the worst performance out of

the four route discovery algorithms.

5. A statistical performance comparison of these four route discovery algorithms
for networks of 100 nodes and A = 0 shows that on average the BND and BED
in conjunction with the DPA yield the best performance. BND and BED yield
reductions of up to 28% and 36% in variance of power consumption rates at the

cost of raising average node power consumption by 15% and 21%, respectively.

6. The computational complexity of variants of the DPA vary from 0(NB) to
0(N4) which is significantly lower than the full search of the solution space
which is of complexity 0(NlN). However, for randomly generated networks of
100 nodes we consistently observed that the time to run the DPA on a PC is of

the order of a few seconds.

7. Analysis by means of diffusion plots verified that DPA reduced power consump—
tion of sensors that experience highest power consumption under shortest path
routing algorithms. Diffusion plots also Show that the reduction power con-

sumption is highest under BND, followed closely by BED.

8. The resulting route selection method is one that is suitable for applications

217

 

with rnany-to—one traffic flows. Route discovery algorithms and DPA assume
availability of global network topology information which is usually available at
the base station. While this makes the DPA a centralized solution we envision
it finding applications in critical infrastructure protection/control/monitoring,
surveillance, and environmental/ agricultural monitoring applications with in—

frequent topology changes.

10.3 WSN Topology

Our analysis small-world properties of networks with range limited shortcuts, the
application of cooperative communication and diversity combining concepts and cen-
trality measures for W SNS in chapters 7, 8 and 9 leads us to make the following

conclusions.

1. From the analytical model of characteristic path length and clustering coefficient
we observed that for sufficiently dense networks characteristic path length can
be reduced significantly by replacing a ,u z O(0.005 — 0.05) fraction of the local

scale nodes by global scale nodes providing shortcuts in the network.

2. The order of [,L, the fraction of nodes that are designated shortcut nodes, is
about the same as the value of 13, the rewiring probability, in \Natts’ small-

world network construction method.

3. The model lends itself for the task of designing WSNS, e. g. determining the num-

ber of shortcut nodes required to achieve a certain characteristic path length.

4. We demonstrate gPMSS by application to IEEE 802.15.4 SIMO channel traces
and implementation on the Crossbow Imote2 sensor mote. We analyzed the

performance of gPMSS in terms of PRR, retransmission attempts and power

218

10.

consumption per delivered packet. We saw that for a setup with 3 receivers
gPMSS raises the PRR from 22 — 30% to 73 — 76%, a relative increase of
150 — 245%.

. we observe gPMSS reduces transmission power per correctly delivered packet

by up to 68%.

. We evaluated the effect of retry limit imposed by the IEEE 802.15.4 standard of

the on the packet delivery rate that can be achieved. At the default retry limit
of 3, (m = 4) gPMSS can achieve delivery rates of greater than 99%, against
only 65 — 75% when gPMSS is not used.

By making use of lossy links feasible, gPMSS can be used as a mechanism for

adding shortcut links to enable small—world network topologies in WSNS.

Shortcuts implemented by gPMSS will reduce characteristic path length and
diameter of networks which facilitates service discovery and the routing of high
priority data in a network. It has the advantage of not requiring any hardware

modifications ([124] and [125]), or adding wired conuections([27] and [111]).

gPMSS used at the base station can increase communication range and the
number of sensors at 1 hop distance, thereby increasing the number of critical
sensors. This extends the lifetime of nodes surrounding the base station in
W'SNs subject to the funneling effect. The larger communication range allows
more nodes to communicate with the base station directly and reduces the traffic
load from nodes positioned closer to the base station. More generally, gPMSS
can be used to connect weakly connected components of a network by adding

more links between nodes farther apart.

We conducted a review of pre-existing measures of node centrality and their

shortcomings with regard to W SNs. We introduced PCC, a new measure of

219

node centrality. A node’s PCC can be interpreted as its 62 norm from the origin
in the eigenspace formed by the P most significant feature vectors (eigenvectors)

of the adjacency matrix.

11. While we see PCC as having wider applications, an immediate application of di-
rect relevance is its use in identifying nodes for the placement of shared network

resources, e.g. endpoints of shortcut links created through gPMSS.

12. To select an apprOpriate number of features for the computation of PCC we

explored two methods.

0 The relative contribution of each additional feature’s power (eigenvalue) to
the total power of adjacency matrix. The use of the adjacency matrix is
compared with that of the Laplacian. We concluded that eigendecompo-
sition of the adjacency matrix yields significantly higher degree of energy
compaction than the Laplacian at the same number of features. However,
the cumulative scree plot of neither matrix yields a clear cutoff region for

the selection of number of features P to use for PCC.

o Incremental changes in the phase angle of C P (PCC with P features) and
C1 (EVC) as P is increased. We concluded that PCC reaches steady state
\v'alues with the use of only 5 to 10% most significant features out of all
available feature vectors. Selecting number of features P using the phase

angle function d)(P) yields a very clear cutoff region.

220

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