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LLBRARY t This is to certify that the
MlCh‘gan Sta 6 dissertation entitled

University

Speed Disturbance, Absorption and Traffic Stability

presented by
Ruihua Tao

has been accepted towards fulfillment
of the requirements for the

Doctoral degree in Civil Engineering

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4/ Isl/03

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6/01 cJClRC/DateDue.p65-p.15

SPEED DISTURBANCE, ABSORPTION AND TRAFFIC STABILITY
By

Ruihua Tao

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Civil and Environmental Engineering

2003

ABSTRACT
SPEED DISTURBANCE, ABSORPTION AND TRAFFIC STABILITY
By

Ruihua Tao

Speed disturbance is a natural phenomenon of human-controlled vehicles. When
vehicles follow one another the response action of a following vehicle to a speed
disturbance from the lead vehicle can vary. The following vehicle may respond to a
speed disturbance by reducing its speed immediately if following very close, or may
respond to it at a later when the spacing between them is reduced to a certain level, or
may not respond to it by its speed reduction but just reduce the spacing between the two
vehicles. In the cases when the following vehicle does not respond to a speed
disturbance immediately, a speed disturbance can be absorbed (completely or partially)
through the reduction of spacing (i.e., distance headway). Limited research has been
found on the issue of absorption of a speed disturbance and the effects on traffic
stability as a result of speed disturbance absorption.

The work explores driver behavior in a car following mode in response to a
speed disturbance fiom the lead vehicle, and develops a concept of Expected State-
Control Action Chains. Afterwards, the study identifies three scenarios of car-following
behavior in response to a decrease in the speed of the vehicle immediately downstream.
The impact of the responding behavior of the following vehicle on dynamic spacing
between the lead-following vehicle pair was analyzed and minimum dynamic spacing

for each scenario were obtained. The research conducted as part of this study first

demonstrates the existence of an upper boundary on the magnitude of a speed
disturbance that can be absorbed by a single spacing, or by multiple spacings if the
speed disturbance is propagated backward through a line of following vehicles.
Mathematical models were developed to enable the calculation of the upper limit of
speed disturbance absorption. The conditions for local and asymptotic stability were
also determined. Results from two simulation experiments confirm the boundary
conditions proposed by the mathematical models.

The work presented here sheds some light on the effect that a single speed
disturbance of a lead vehicle has on the spacing and speed of the following vehicle in a
car-following mode at the microscopic level. This establishes the foundation for further
research on multiple speed disturbance absorption and its impact on traffic stability at

the macroscopic analysis level.

ACKNOWLEDEGEMENTS

It has been a time to express one’s feelings when finishing the final page of
his/her dissertation. It has been a place to leave his/her words for people to remember. It
is a right time and a right place for me to write down something for lasting memories.

Thinking about the years at Michigan State University, I am grateful to all of
you, the faculty members of the transportation group, for giving me opportunities to
study in its excellent program, which I am always proud of, and to learn fiom you,
whom I always admire to. I am profoundly grateful to Dr. Virginia P. Sisiopiku, my
advisor, for her always supports, encouragements and priceless advices; to Dr. William
C. Taylor for his guidance, suggestions and assistance whenever I needed; and to Dr.
Richard W. Lyles for value of the knowledge he taught and the example he set in his
teaching and researching that I am going to follow.

The years during which I studied were the years that my family worked together
to support each other. I am grateful to my husband for his always supports in my career
improvement; and to my daughter for her understanding, at her young age, of my
absence sometimes from my duty to take care of her.

Looking back the path I walked over, I saw the footprints left behind. Each of
them filled with love, supports and efforts of my parents. And each of them costs part
spans of their lives.

A sincere appreciation is to extend to Mr. Thomas Hicks, Mr. Erik Tabacek and
Mr. Robert Herstein. Without their supports and understanding, the dissertation could

not be finished in time. Also a special thanks is to send to Mr. Edward Countryman for

iv

his reading the early part manuscript of the dissertation, and to Ms. Terri Tabash for her
warmly friendship.

Receiving so much loves, supports, understandings, advices, guidance, helps,
and knowledge, I have been doing my best to contribute myself to science and
engineering in hope to make the world more beautiful through a powerful stream
combining each tiny effort from us. As a reward to what I received, I would dedicate

this academic work to those whom I love and admire.

TABLE OF CONTENTS

LIST OF TABLES .......................................................................................................... viii
LIST OF FIGURES ............................................................................................................. x
CHAPTER 1
INTRODUCTION .............................................................................................................. 1
1.1 Background ................................................................................................................... l
1.2 Literature Review ......................................................................................................... 2
1.3 Objective ....................................................................................................................... 9
1.4 Contributions and Contents ........................................................................................ 10
References
CHAPTER 2
HUMAN BEHAVIOR IN SIN GLE-LANE CAR FOLLOWING ................................... 14
2.1 Perceptive Variables and Expected States in Single-Lane Car Following ................. 15
2.2 Expected State-Control Action Chains in Single-Lane Car Following Process .......... 17
2.3 Normative Behavior Hypothesis ................................................................................ 21
2.4 Deviation of Assessment and Control Skill ................................................................ 22
2.5 Summary ..................................................................................................................... 28
References
CHAPTER 3
SINGLE SPEED DISTURBANCE AND DYNAMIC SPACING ................................... 32
3.1 Dynamic Spacing with Acceptable-Acceptable Expected States ................................ 34
3.2 Dynamic Spacing with Acceptable-Unacceptable-Acceptable

Expected States ............................................................................................................ 35
3.3 Dynamic Spacing with Unacceptable Expected States

at Target-Risk Level ................................................................................................... 42
3.4 Summary ...................................................................................................................... 43
References
CHAPTER 4
SPEED DISTURBANCE AND STABILITY .................................................................. 46
4.1 Value of Upper Limit of a Speed Disturbance that a Single Spacing

Can Completely Absorb .............................................................................................. 47
4.2 Value of Upper Limit of a Speed Disturbance that a Single Spacing

Can Partially Absorb ................................................................................................... 48
4.3 Local Stability ............................................................................................................. 57
4.4 Asymptotic Stability for a Single Speed Disturbance ................................................. 57
4.5 Summary ..................................................................................................................... 59
References

vi

CHAPTER 5

MODEL VERIFICATION ................................................................................................ 61
5.1 Description of the Experiments ................................................................................... 61
5.2 Data .............................................................................................................................. 63
5.2.1 Experiment One .............................................................................................. 63
5.2.2 Experiment Two ............................................................................................. 65
5.3 Case Study .................................................................................................................. 68
5.3.1 Cases for Scenario 1 - Acceptable-Acceptable Expected State ....................... 68
5.3.2 Cases for Scenario 2 - Acceptable~Unacceptable-Acceptable
Expected State ................................................................................................. 71
5.3.3 Cases for Scenario 3 -Unacceptable Expected State at Target-Risk Level ..... 73
5.4 Summary ...................................................................................................................... 76
CHAPTER 6
FINDINGS AND RECOMMENDATIONS ..................................................................... 94
6.1 Findings ................................................................................................................. 94
6.2 Recommendations for Further Research ............................................................... 95
APPENDICES ................................................................................................................... 97

vii

Table 2-1

Table 2-2

Table 2-3

Table 3-1

Table 5-1

Table 5-2

Table 5-3

Table 5-4

Table 5-5

Table 5-6

Table 5-7

Table 5-8

Table 5-9

Table 5-10

Table 5-11

LIST OF TABLES

Basic Expected State-Control Action Chains and Three
Scenarios .................................................................................................... 21

Driver’s Braking Behaviors under Three Situations ................................. 26

Earliest Perception-Reaction Time as a Function of Spacing
and Deceleration Rates ............................................................................. 27

Formulas to Calculate Minimum Spacing for the Scenarios
Defined in Chapter 2 ................................................................................. 44

Percentage of Driver Type in Two Experiments ....................................... 62

Driver Type and Minimum Time Headway of Each Vehicle in
Experiment One ........................................................................................ 65

Driver Type, Approximately Constant Speed, and Minimum Time
Headway of Vehicles 69, 70, 77, 81, 90, 93, 94, 96, and 97 during the
Simulation+ ............................................................................................... 66

Driver Types and Minimum Time Headways of

Vehicles 71 through 76 .............................................................................. 66
Driver Types and Minimum Time Headways of Vehicles 78 through 80
and Vehicles 82 through 89 ....................................................................... 67
Driver Types and Minimum Time Headways of

Vehicles 91, 92, and 95 ............................................................................. 67
Cases in which speed Disturbances Were Absorbed Completely or
Partially .................................................................................................... 68
Case Stud for Scenario 1-Acceptable-Acceptable Expected State ............ 70

Case Study for Scenario 2-Acceptable-Unacceptable-Acceptable
Expected State ........................................................................................... 72

Values of the Develop, Absorbed, Propagated, and Calculated Speed
Disturbance ................................................................................................ 73

Case Study for Scenario 3-Unacceptable Expected State at Target-Risk
Level .......................................................................................................... 75

viii

Table 6-1 Summary of Findings ................................................................................ 95

ix

Figure 1-1

Figure 2-1

Figure 2-2

Figure 2-3

Figure 2-4

Figure 2-5

Figure 2-6

Figure 2-7

Figure 3-1
with

Figure 3-2

Figure 4-1
Figure 4-2

Figure 4-3

Figure 5-1

LIST OF FIGURES

Inter-Vehicle Spacings of a Platoon of Vehicles versus Time
for Chandler’s Model .................................................................................. 4

Hierarchical Driver Modeling Framework ................................................ 14

Detailed Generalization of Car Following from Conventional
Control Theory .......................................................................................... 15

Interactive Perception and Reaction Process of Single-Lane
Car Following ............................................................................................ 16

Speed Diagram of Lead Vehicle and Its Two Phases of Speed
Disturbance ................................................................................................ 18

Interactive Speeds of Following Vehicle with the States of Perceptual
Variables Acceptable from t0 through T. ................................................... 20

Interactive Speeds of Following Vehicle with the States of Perceptual
Variables Acceptable at t0 and Becoming Unacceptable at Sometime

Between t0 and t0 + T. ................................................................................ 20

Interactive Speeds of Following Vehicle with the States of Perceptual
Variables Unacceptable at t0 ................................................................... 21

Possible Control-Action Strategies that Following Vehicle Can Take

Expected States of Acceptable-Unacceptable-Acceptable ........................ 36
Possible Control-Action Strategies that Following Vehicle Can Take

with Expected States of Unacceptable at Target-Risk Level .................... 42
Absorption of Speed Disturbance under Given 56;“ ................................. 52
Control-Action Strategy of Following Vehicle with 56;” = if; .................. 55

Illustration of Car Following of the Vehicle Following the Following
Vehicle ....................................................................................................... 58

Speed Interaction between Vehicles in Experiment One .......................... 77

Figure 5-2 Speed Interaction between Vehicles in Experiment Two .......................... 81

Images in this dissertation are presented in color

xi

CHAPTER 1 INTRODUCTION

1.1 Background

It has long been recognized that the breakdown of traffic flow is a direct
consequence of high traffic volume. In the Highway Capacity Manual (HCM, 2000),
capacity is defined as “the maximum sustainable flow rate at which vehicles or persons
reasonably can be expected to traverse a point or uniform segment of a lane or roadway
during a specified time period under given roadway, geometric, traffic, environmental,
and control conditions.” The definition of capacity implies that a breakdown occurs
immediately after the capacity has been reached. However, numerous field observations
confirm that breakdown does not always occur at the highest flow rate observed, nor
does a speed drop always correspond to the highest observed flow. Another
phenomenon observed in the field is that speed sometimes does not drop, even when
flow is at a high rate. These phenomenon supports the hypothesis that breakdown is a
probabilistic event (Elefteriadou 1995). If this is accepted, there is a need to determine
what random factors can be used to describe the observed traffic behavior.

A study by Hall et a1. (1993) examined the three types of speed drops suggested
in previous studies of Athol, Banks and Koshi (Athol 1965, Banks 1989, Koshi 1983),
and concluded that the speed drops seem to be related to the data collection locations or
to environmental conditions. The data in Hall’s study were collected at different
locations in the vicinity of congestion, and support his conclusions. His study provides
an explanation of different types of breakdown. However, his study does not explain the

phenomenon that speed sometimes does not drop, even when the flow rate is quite high.

Therefore, there must exist some other factors that affect speed drop under such
conditions.

Many previous spot speed studies suggested that speed drop is not a continuous
process. However, individual vehicular speed change is a continuous process both on
the dimensions of time and distance. When a vehicle follows another vehicle closely,
the individual vehicular speeds affect each other. A speed drop of a vehicle in a traffic
stream can be propagated upstream, if vehicles in a traffic stream are close enough to
one another. On the other hand, if traffic is not heavy, the following vehicles do not
necessarily respond to temporary speed changes of leading vehicles, and small speed
disturbance can simply be absorbed by the traffic stream. Lorenz and Elefteriadou
provide the evidence of this phenomenon through their field observations (Lorenz and
Elefteriadou 2000). The phenomenon and the evidence lead us to consider speed
disturbance as a possible random factor leading to a breakdown. In this respect, two
questions are raised: a) Does there exist an upper limit on the magnitude of a speed
disturbance that can be absorbed by traffic at a certain flow rate without causing a
breakdown, and b) if it does, what is it? This study addresses these issues through
exploring driver’s behavior in response to a speed disturbance from a vehicle in the
front and defines the upper limits on the magnitude of a speed disturbance that a single
space ( or multiple spaces) can absorb, without causing a breakdown, and the conditions
for local or asymptotic stability.

1.2 Literature Review
Car-following theories were first introduced in 1950’s and still provide a good

way to model speed and space interaction between vehicles. Most engineering-inspired

car-following studies model car following behavior first and then analyze the conditions
for model’s stability. As Boer (2000) comments, those models presumed that driving is
equivalent to the continuous application of a single control law in a series manner.

The earlier car-following models discuss speed-spacing relationships from a
Newtonian kinetic point of view. For example, the car-following model in the Highway
Capacity Manual (FHW A 1950) suggests a minimum spacing between two vehicles as
the sum of a vehicle length, the distance that a following vehicle travels during the
driver’s reaction time and the stop distance of the following vehicle. The car-following
models proposed by Pipes (1953) and Forbes (1958) define a minimum headway
between two consecutive vehicles as the sum of reaction time plus the time required for
the following or lead vehicle to travel a distance equivalent to its length.

A breakthrough in car-following theories took place when human factors were taken
into account in stimuli-response car following models first proposed by the General
Motor (GM) researchers (Chandler et al. 1958, and Herman et al. 1959). Chandler
considered the car-following law to be

x (t + At) = xiii... (t) - 59.0)] 1-1
where in (t + At) = the acceleration of the following vehicle at time r+ At ,

5cm(t) , in (t) = speeds of vehicle n+1, vehicle n, and/i = a constant.

Model l—l implied a velocity-headway relation of v = 2(k - k m) where k is the

concentration and kid," the 1am concentration. Thrs shows that for xiAt > 2 any rnrtral

disturbance of an equilibrium state will grow in time as it passes down a line of vehicles

and, for a sufficiently long line of vehicles, will create stop-and-go conditions.

However, this stability criterion was “slightly violated” in some cases when traffic
should have been stable. Ferrari (1994) showed that when the flow is unstable,
instability may take too long to manifest itself and the flow to break down, which partly
demonstrates Hall’s conclusion (1993) described in the Section 1.1.

Herman et al. (1959) continued the investigation of the stability of Chandler’s

model and concluded that: for 11A! < e", an initial disturbance is non-oscillatory and

. - 71' . . . . . . .
exponentially damped; for e ‘ < 11A: < —2-, the Initial disturbance is oscrllatory wrth
exponential damping; for Mt = 3, the initial disturbance lS oscrllatory wrth constant

amplitude; and for MI > 2’ the initial disturbance is oscrllatory wrth increasmg

amplitude. Figure 1-1 Shows the stability of Chandler’s model for three values of C

 

th
20 —“ - I,
18 -l / n
.’ 5.6
16 - 34 c =- 11. a 0.333

 

 

 

L I
14— "2 'ZUMM\
0510326233635“

TIME(SEC)
Figure 1-1 Stability of Chandler’s Model (Herman et (11.1958)

 

 

Following the GM pioneers, Gazis et al. and Edie explored the sensitivity parameter
in the GM models (Gazis er al. 1959, Edie 1961, and Gazis er al. 1961) and proposed
non-linear car-following models. To represent quicker reactions for denser traffic, Gazis

et al. suggested that

x (t + At) = ’1 [xn,,(t) — in(t)] 1-2.

x" (t) — x"+1 (t)

 

k.
This equation leads to the velocity-headway relationship v = ziln(—];’-) , which is

identical to Greenberg’s flow-concentration curve (Greenberg, 1959). Edie found that
another amendment should be made to the sensitivity constant, namely, the introduction

of the velocity dependent term. This produced a new model of the form

 

56,0 +At) — Axel“)

_ [x (0% (012 [gum—5c, (01 1-3
n n+1

which can be integrated to give a velocity-headway relationship as v = V] exp(-7k-)

where vf = free flow speed and km = density at maximum flow. Further, Gazis et al.
introduced the general scaling constant m and l to investigate the sensitivity of their
macroscopic relationships to variations in the magnitude of the v and the x" (t) - x“, (t)

terms respectively

.. 15c" (t) . .
x,(t + At) = [an (t) — x" (1)] 1-4.

[19.0) - x,“ (01'

 

l-m

l
_ l—m T
,ia 1)b + c]! 1

The integration of 1-4 produces a velocity-headway function v =[

where c = a constant of the solution of the differential equation 1-4. The stability

condition of this model is the same as for Chandler’s model i.e., for xl'At < % , the

system is stable (Holland, 1998).
Newell (1961) considered consequences of non-linearity and postulated that the

velocity of vehicle n+1 is some nonlinear function of the headway
. xi.
xn+1(t + At) = V[l — exp[—V(xn (t) — Jr"H (t)) - dj] 1-5

where V = maximum velocity or free flow speed of vehicle n+1, and
d = minimum headway of vehicle n+1.

Equation (5) implies a velocity-headway relationship of the form

xi
v — V[l - CXMV (k — k M H]

Newell analyzed the stability of this system and found that for xiAt > $— , the system

was unstable and the behavior was diffusive under stable conditions.

After Newell’s work the engineering-inspired car-following models were less
and less investigated until 1995 when Bando et at. (1995) suggested a new car-
following law

55,,(t + A) = a[V(xMl (t) — x" (2)) -— xn(t)] 1-6
where a = an acceleration constant,

V = a legal velocity which is a function of following distance of the preceding

vehicle.

Equation 1-6 leads to a velocity-headway relationship of v = tanh(k - 2) + tanh(2) and
the similar stability condition for the system a > 2V'(k0) , where tanh(k) represents V(k)
and k0 a concentration at steady state flow.

With the exception of the recent work of Bando, all other models imply different

velocity-headway relationships, but conclude the same criterion AAI <% for traffic

stability. The definition of A in these models vary, but basically it is regarded as a
driver-behavior-related parameter, which makes it difficult to envision what the 2.
exactly represents in car-following theories. As mentioned at the beginning of this
literature review, all of the models previously reviewed presupposed that driving is
equivalent to the continuous application of a single control law in a series manner.
Therefore it is not surprising that there is a single criterion for traffic stability for all
models. However, it is commonly acknowledged that drivers use different criteria in
different situations when carrying on a single-lane car following task, which inspired
researchers to think beyond engineering models.

In 2000, Boer’s commentary pointed out that that some important issues that
characterize driver behavior are still ignored in engineering car-following models. This
position ignited a debate between researchers. The issues raised by Boer include the
following: (1) car following is only one of many tasks that drivers perform
simultaneously and receives therefore only intermittent attention and control, (2) drivers
are satisfied with a range of conditions that extend beyond the boundaries imposed by
perceptual and control limitation, and (3) in each driving task drivers use a set of highly

informative perceptual variables to guide decision making and control. On the basis of

these three issues, Boer proposed a framework that seeks to depart from determinism
and presents two approaches to further develop the area, namely satisfying behavior and
task scheduling. The problem with this approach, as Brackstone and McDonald pointed
out (2000) is obtaining data with which to calibrate the model. Since it is a new concept
in modeling car-following behavior, more development is expected as the development
of the conjunctive field of traffic psychology progresses.

In the debate, Van Winsum (2000) showed how psychological knowledge about
car following behavior by human drivers could be applied in a mathematical model that
can be used in traffic engineering. Based on a wide variety of behavior studies, (Van
Winsum 1998, Heino 1996, Van der Horst 1999, Fuller 1981, Steven 1957, Schiff and
Detwiler 1979, McLeod and Ross 1983, Cavallo et al.1986, and Van der Horst 1990),
Van Winsum established his human element-based model that describes the relation of
psychological factors (time headway, distance headway, and deceleration) used by the
driver in car following. Van Winsum’s model is established upon three rationales: (1)
drivers use time headway as a safety margin; (2) if distance to the lead vehicle is larger

than the preferred distance headway that drivers try to maintain (D p ), there is no

safety-related reason for the driver to decelerate until the preferred distance headway is
reached; and (3) the deceleration initiated by the driver is a function of the Time-to-
Collision ('I'I‘ C) parameter,

55",, = ch‘Cm + d + a, with d < 0, 1-7

where: TIC“, =1.04T1'C°'72 , the TTC as estimated by the driver,

c and d, constants; and E , a random error term.

Back in the late 1950’s, Helly (1959) used a criterion similar to the second
rationale of Van Winsum to develop a model conceptually close to the risk-threshold
model of driving behavior. Both models have flaws in their methods to estimate actual
distance headways. Helly’s model compares the driver’s estimation of spacing with the
actual spacing, presumably not directly available to the driver. The truthfulness of the
assumption of constant time headway proposed in Van Winsum’s model, which is the
key variable in calculating distance headway, was also questioned. Similar to Boer’s
model, Van Winsum’s model has not been tested with data.

The debate between traffic engineering and psychology researchers finally
focused on the question posed in the discussion of Hancock (2000) of whether car
following is the real question and if equations are the answer. The answers to both
questions by Brackstone (2000) are negative, however it has been acknowledged that
the car following concept provided an acceptable tool for the job and equations are the
most suitable answers for the task at hand. Brackstone’s statements concur with
Ranney’s (2000) who also suggests that the development of better models of car-
following behavior that are applicable to a well-defined situation is probably a more
worthwhile approach to follow.

The review of the literature clearly shows that both psychological and
Newtonian kinetic principles play important roles in car-following theories. Although,
the needs of traffic psychologists and traffic engineers have not yet met in a satisfactory
manner, both groups agree that an effort should be made to consider both aspects in

developing more understandable car-following models under well-defined situations.

1.3 Objective

The thesis presented here is an attempt to consider both the psychological and
engineering aspects in modeling car-following behaviors. This effort explores driver’s
behavior in response to a speed disturbance from a vehicle in the front and defines the
upper limits on the magnitude of a speed disturbance that single space or multiple
spaces can absorb, without causing a breakdown, and the conditions for local or

asymptotic stability.

1.4 Contributions and Contents

The major contributions of this work are the findings of the upper boundary
theory of speed-disturbance absorption and the formulas to calculate this upper
boundary. The thesis focuses on the boundary conditions when a speed disturbance
occurs. This work is different from previous studies in that it starts from Boer’s
psychological hierarchical framework to develop a well-defined problem questioned in
Section 1.1, and then uses the Newtonian principle in the defined problem to establish
equations that lead to the calculation of the boundary conditions of speed disturbance at
the levels of either local or asymptotic stability.

The thesis consists of five parts.

Chapter 1 provides an introduction and a discussion of relevant literature.

Chapter 2 discusses salient performance aspects of the human elements in
carrying out the car following tasks in the defined problem. By exploring the discrete
components of Boer’s hierarchically psychological model, including monitoring and

controlling processes, perceptual variables and expected state in car following, the

10

chapter discusses the transiting chain of expected states and then obtains three scenarios
in car following for further analysis. The chapter focuses on the characteristics of
population and propounds a Normative Behavior Hypothesis, and also discusses the
error of human beings in the assessment of the perceptual variables and control skills.

Chapter 3 discusses the localized behavior of the following vehicle in response
to a change in speed of the vehicle immediately in front and its direct impact on the
dynamic spacing between them. Minimum dynamic spacing for each scenario defined
in Chapter 2 is obtained, which establishes the fundamental work for the next chapter.

Chapter 4 develops the concepts of speed disturbance absorption, and
establishes an upper boundary theory of speed disturbance absorption. It presents the
speed disturbance absorbed by a single spacing, and speed disturbance propagation and
cumulative absorption by multiple spacing in a traffic stream. Mathematical models are
developed that give descriptions on the upper limit of speed disturbance that a single
spacing or multiple spacing can absorb. Furthermore the conditions for local and
asymptotic stability are determined.

Chapter 5 summarizes results from the use of Simulation data to test the
validation of the models obtained in Chapter 4.

Chapter 6 briefly states the most important findings from the study and offers

recommendations for further research.

References
Athol, P. Interdependence of Certain Operational Characteristics Within a Moving

Traffic Stream, Highway Research Record 72, HRB, National Research Council,
Washington, DC, 1965, pp. 58-87.

11

Banks, J H Freeway Speed-Flow-Concentration Relationships: More Evidence and
Interpretations, Transportation Research Record 1225, TRB, National Research
Council, Washington, DC, 1989, pp. 53-60.

Boer, E. R. Car Following from the Driver’s Perspective. Transportation Research Part
F, Vol. 2, No. 4 (1999). Pp. 201-206.

Brackstone M. and M. McDonald. What Is the Answer? And Come to That, What Are
the Questions? Transportation Research Part F, Vol. 2, No. 4 (1999), pp. 221-224.

Brackstone M. and M. McDonald. Car-Following: A Historical Review. Transportation
Research Part F, Vol. 2, No. 4 (1999), pp. 181-196.

Chandler, R.E., R. Herman, and E.W. Montroll. Traffic Dynamics; Studies in Car
Following. Operations Research 6, 1958, pp. 165-184.

Edie, L.C. Car Following and Steady State Theory for Non-congested Traffic.
Operations Research 9, 1961, pp. 66-76.

Elefteriadou, L., R. P. Roess, and W. R. Mcshane. Probabilistic Nature of Breakdown at
Freeway Merge Junctions. Transportation Research Record 1484, 1995, pp. 80-89.

Ferrari, P. The instability of Motorway Traffic. Transportation Research Part B, Vol.
28, No. 2. 1994, pp. 175 — 186.

FHWA. Highway Capacity Manual. US. Government Printing Office, Washington
DC. 1950.

FHW A. Highway Capacity Manual. US. Government Printing Office, Washington
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2000.

Forbes, T. W., H. J. Zagorshi, E. L. Holshouser, and W. A. Deterline. Measurement of
Driver Reactions to Tunnel Conditions. Highway Research Board, Proceedings, Vol.
37, 1958, pp. 345-357.

Gartner, N., G. C. Messer, and A. K. Rathi. Traffic Flow Theory. Chapter 4 of R. W.
Rothery, Car Following Models. Transportation Research Board Special Report 165,
Transportation Research Board, 2000, Page 4-6.

Gazis, D.C., R. Herman, and RB. Potts. Car Following Theory of Steady State Traffic
Flow. Operations Research 7, 1959, pp. 499-505.

Gazis, D.C., R. Herman, and R. Rothery. Non-linear Follow-the-leader Models of
Traffic Flow. Operations Research 9, 1961, pp. 545-567.

12

Greenberg, H. An Analysis of Traffic Flow. Operations Research 7, 1959, pp. 79-85.

Hall, F. L., A. Pushkar, and Y. Shi. Some Observations on Speed-Flow and Flow-
Occupancy Relationships under Congestion Conditions. Transportation Research
Record 1398, 1993. PP. 24-30.

Hancock, P. Is Car Following the Real Question - Are Equation the Answer?
Transportation Research Part F, Vol. 2, No. 4 (1999), pp. 197-199.

Helly, W. Simulation of Bottlenecks in Single Lane Traffic Flow. In Proceedings of the

Symposium on Theory of Traffic Flow. Research Laboratories, General Motors, 1959,
pp. 207-238. New York: Elsevier.

Herman, R., E. W. Montroll, R. B. Potts, and R. W. Rothery. Traffic Dynamics:
Analysis of Stability in Car Following. Operations Research, E. 17, 1958, pp. 86-106.

Holland E.N. A Generalized Stability Criterion for Motorway Traffic. Transportation
Research, Part B, Vol. 32, No. 2, 1998, pp. 141-154.

Homberger, W. S., L. E. Keefer, and W. R. McGrath. Transportation and Traffic
Engineering Handbook, 2“d Edition, Englewood Cliffs, NJ: Prentice-Hall, 1982.

Koshi, M., M. Iwasaki, and I. Ohkura. Some Findings and an Overview on Vehicular
Flow Characteristics. Proceedings 8‘“ International Symposium on Transportation and
Traffic Theory, 1983, pp. 403-426.

Lorenz, M., and L. Elefteriadou. A Probabilistic Approach to Defining Freeway
Capacity and Breakdown. Fourth International Symposium on Highway Capacity
Proceedings, Transportation Research Board, 2000.

Newell, G.F. Nonlinear Effects in the Dynamics of Car Following. Operations Research
9, 1961, Pp. 209-229.

Pipes, L. A. An Operational Analysis of Traffic Dynamics. Journal of Applied Physics,
Vol. 24, No. 3, 1953, pp. 274-287.

Ranney, T.A. Psychological Factors that Influence Car-Following and Car-Following
Model Development. Transportation Research, Part F, Vol. 2, No. 4 (1999), pp. 213-
219.

Van Winsum, W. The Human Element in Car Following Models. Transportation
Research Part F, Vol. 2, No. 4 (1999), pp. 207-211.

13

CHAPTER 2 HUMAN BEHAVIOR IN SINGLE-LANE CAR FOLLOWING

Drivers generally engage in multiple tasks on roads, such as lane changing, car
following, over-taking, etc. In 1999, Boer attempted to consider elements of human
decision-making and action-taking in the development of a hierarchical driver-modeling

framework as shown in Figure 2-1.

/\

 

 

 

 

 

 

 

Strategic Strategy Selection
6
Route
w “
r
Situatio

 

Tactical

 

 

Task Scheduler/Attention Manager

Operational Car Following

Figure 2-1. Hierarchical Driver Modeling Framework (Boer 1999)

 

t

Each vehicle control task consists of two processes, namely monitoring and
controlling. The monitoring process once initiated by the attention manager is
decomposed into several stages: (1) pay attention to the perceptual variables that
characterize a particular task in the form of the present state, (2) use those variables to
evaluate current and expected future states, (3) assess whether the present performance
will most likely remain acceptable for a certain amount of time, and if so (4) determine

when attention should be given to the task again. If the present performance is not

14

satisfactory, then the task scheduler initiates the appropriate control process that
employs a suitable skill to bring the vehicle within a limited time into an acceptable
state that is expected to remain acceptable for some time (Boer 2000).

Consistently, the conventional control theory describes the car-following task as
a process where drivers dynamically and interactively carry out perception and
information collection, which is equivalent to the aforementioned stage (1), as well as
decision making and execution tasks while following another vehicle in a single lane,
which is equivalent to the stages (2) and (3). A detailed generalization of car following
from the perspective of a conventional control theory is illustrated in Figure 2-2

(Gartner et al. 2000).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Error Driver Output commands Followin
. w _ g
. . . : Vehicle vehlde
Lead 7 Perception & Decrsron ; Dynamics State
Vehicle \ Information Making & >
State { Collection Execution
(Feedback Loop)

 

 

 

 

 

 

Figure 2-2. Detailed Generalization of Car Following from Conventional Control

Theory

2.1 Perceptive Variables and Expected States in Single-Lane Car Following
When carrying on a single-lane car following task, drivers pay attention to two
major perceptual variables: spacing and speed. Drivers evaluate their current and
expected future states by using these perceptual variables to assess whether they are
acceptable. During the assessment process drivers do not always give an equal priority

to each of them. If the degree of unacceptability of the spacing perceptual variable is

15

greater than the target risk level, which gives drivers a signal of an oncoming collision,
this signal overwhelms speed perceptual variables and triggers the task scheduler to
initiate an immediate brake-control action without evaluating speed satisfaction. For
example, when following a vehicle in close proximity and observing a brake signal
from the vehicle in the front, the driver of the following vehicle responds by applying
the brakes immediately without thinking about speed. Sometimes the spacing perceptual
variable is unacceptable but not at the target-risk level, and drivers apply comfort
deceleration instead of reacting to an emergency. In this case, a new standard of
satisfactory speed is adopted. The spacing perceptual variable is improved by trading
off the satisfaction of speed. When the spacing perceptual variable is acceptable, drivers
adopt a satisfactory speed and keep the current state until a new state emerges. Figure 2-

3 illustrates this interactive process.

 

 

 

Evaluate the perceptual variables

 

 

 

Unacceptable at
target risk level

  
     
   

   

Spacing
perceptual variab ‘
acceptable?

Acceptable

 

    

not at t get risk level Una ceptable Acceptable

V L i 1 1
Immediate Acceleration Comfortable Monitoring

control action deceleration, or process
’ adoption of new
satisfactory speed

I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2-3. Interactive Perception and Reaction Process of Single-Lane Car Following

16

The assessment of the perceived spacing variable produces three expected states
that call for different mental models and control skills. An unacceptable state (at the
target risk level) calls for an emergency-response model and skill. An unacceptable
state, but not at the target risk level uses a mental model that predicts its new state based
on its current state and decides when to take an action to achieve the new state. An
acceptable state does not invite any controlling actions, just the monitoring process. The
three expected states of the spacing perceptual variable can be represented by the
following three spacing limits: a) minimum safety spacing between two consecutive

vehicles, SW, , that requires an emergency-response model, b) a spacing that calls for a

comfortable control skill, and c) a spacing that invites only the monitoring process.
Unlike the spacing perceptual variable, the speed perceptual variable is assessed

by different criteria in different States. A satisfactory speed in one state may not be

satisfactory in another, as drivers adopt different satisfactory speeds based on their

situation.

2.2 Expected State-Control Action Chains in Single-Lane Car Following

Process

For convenience, the discussion in this section starts at the time point to . At t0
the lead vehicle is traveling at a speed it" (to) and starts to reduce its speed. This vehicle
needs a period of time, denoted as T, to recover its speed (it, (to + T) = 5C, (10)) as shown

as in Figure 2-4. During T, there exist two phases: the speed disturbance development

phase that extends from 10 to t], and the recovery phase, from II to to + T.

17

Spefid 5c

   
 

 

 

x."(to)9xn(to+T) '-------—-----'T
5 E 456..
amt-z- --------- . “"~--...3'C.(t)i i

to t, tO + T time

Figure 2-4. Speed Diagram of Lead Vehicle and Its Two Phases of Speed Disturbance

Evidence indicates that speed choice is fairly consistent over time for an
individual driver (Haglund and Aberg, 2000). If the original Speed that the driver of the

lead vehicle adopts is 5c(t0) in a given situation before a speed disturbance, the desired
speed after the speed disturbance is recovered (in (t0 + T) ) is most likely to be identical
to in (to) in the same situation. Thus to simplify the discussion, we assume that the

speed of the vehicle that develops a speed disturbance will be recovered to its original
level (Figure 2-4).

At tO when the speed disturbance is initiated, the following vehicle could be at
any of the three states previously described. The states of the following vehicle could
either change or remain unchanged during T as the two consecutive vehicles interact
with each other, which forms state-control chains. On one hand, if at to the initial state
of the perceptual variables of the following vehicle is acceptable and it remains
acceptable during T, then the following vehicle keeps its original speed unchanged, as
illustrated in Figure 2-5. On the other hand, it is possible that at t0 the initial state of the

perceptual variables is acceptable and during T the state of the spacing perceptual

variable becomes unacceptable due to the speed drop of the lead vehicle (Figure 2-6).

18

In this case, a control action of deceleration is initiated to bring the unacceptable state
of the spacing perceptual variable back to an acceptable level. The perception of
acceptable or unacceptable for the spacing perceptual variable varies from driver to
driver. The drivers who would like to use controlled deceleration may adopt a larger
spacing as an acceptable threshold than the drivers who choose to hit the brake pedals
harder. The braking action of the driver of the following vehicle can start as early as
when the driver perceives the speed drop of the lead vehicle to as late as when he/She
has reached the minimum braking distance to avoid collision. Moreover, the adoption
of deceleration rates would differ by drivers. Some may just release the gas pedal to
reduce the vehicular speed in response to the speed drop of the lead vehicle when a
large enough spacing is available. Other drivers may apply “controlled braking” to
control their speed and the spacing between the two vehicles, whereas some drivers
may choose to brake at the last minute with “maximum braking force.” The reaction of
the following vehicle to the speed disturbance from the lead vehicle varies with respect
to time from the earliest to the latest possible action time, the region for the speed
reduction, from the minimum to maximum, is shown by the dash-dot lines in Figure 2-

6. If at to the initial state of the perceptual variables of the following vehicle is

unacceptable and at target risk level, an appropriate control action is immediately taken
to bring them away from the target risk. In this case the interactive speeds of the two
vehicles are shown in Figure 2-7.

Through the above analysis, there are three basic Expected State-Control Action
Chains: acceptable-acceptable, acceptable-unacceptable-control action-acceptable, and

unacceptable at target risk level-control action-acceptable. Single—lane car following is a

19

process consisting of any combination-or repetition of these three basic chains. The
basic Expected State-Control Action Chains are summarized in Table 2-1 along with the

actions of the following vehicle in response to a speed reduction from the lead vehicle.

Speedi ........................ xn+1(t)
Aan (to):x.,,+l (to) ‘1 ........

  

 

r
I
|

in (t1 )

'O
‘-
~-
'-
n"
.....
‘5
'0
0.
'-
u
I-

i L x (t)

 

)----J---------

t0 tl t0 + T time
Figure 2-5. Interactive Speeds of Following Vehicle with the State of Perceptual

Variables Acceptable from t0 through T

Speed 5:

A
in (t0 )’ ‘x.II+I (t0)

MSH i

we:
ii

 

 

 

V

tl 6'“, IO+T time

Figure 2-6. Interactive Speeds of Following Vehicle with the States of Perceptual

Variables Acceptable at t0 and Becoming Unacceptable at Sometime between time t0

and to+T

20

Spegl 5c

in(to),*n+i(to)

 

 

------—-_£-

  

-
.a
a
.n
o

 

: ' : .0
: : : AX...
= E 5 it")

to r tI to+T time

t : reaction time of the following driver

Figure 2-7. Interactive Speeds of Following Vehicle with the States of Perceptual

Variables Unacceptable at t0

Table 2-1 Basic Expected State-Control Action Chains and Three Scenarios

 

Basic Expected State-Control Action Chains

Action of the following vehicle

 

Control action

Control action time

 

Acceptable-Acceptable

No action

N/A

 

Acceptable-Unacceptable-Control action-
Acceptable

Minimum < 56;“ <
maximum

r < 5 < latest
possible action
time

 

 

Unacceptable at target-risk level-Control
action-Acceptable

 

5c” 5 maximum

n+1

T

 

 

2.3 Normative Behavior Hypothesis

Drivers have the tendency to react to speed disturbance when following another

car. The predisposition of reacting behavior is assumed to be determined by individuals’

attitudes towards that behavior (Parker et al., 1992; Rothengatter, 1993). The

personalities of human beings result in a variety of attitudes towards car following

behaviors. Not every driver follows a vehicle in front in accordance to normative

behavior nor tends to willingly deviate from that behavior. Although the modem

21

 

attitude theories assume a causal relation between attitudes and behaviors, only
normative behavior is possibly describable for every given situation and every given
interaction (Mcknight et al., 1970; Mcknight et al., 1971). In addition, normative
behavior represents the behavior of the majority of the drivers. Therefore, it is assumed
in this work that the drivers respond to a speed disturbance in single lane car following
in a normative manner and have the tendency to react to specific situations predictably

(Rothengatter, 1999).

2.4. Deviance of Assessment and Control Skill

The normative behavior hypothesis establishes a ground for the further
discussion. However, the hypothesis does not mean that the exact same values are
produced in the perceptual variable assessment and control actions. Errors in both the
assessment and skills are unavoidable due to the difference of physical and intelligence
capabilities of human beings.

Drivers estimate the distance and relative speed from the vehicle ahead through
the visual angle change. Human visual angle transition changes from near linear to
geometric in magnitude as an object is approaching at a constant velocity. As the rate of
change of the visual angle becomes geometric, the perceptual system triggers a warning
that an object is going to collide with the observer, or, conversely, that object is pulling
away from the observer. This looming phenomenon is a function of distance from the
object. If the rate of the change of visual angle is irregular, that provides information to
the perceptual system that the object in motion is moving at a changing velocity (Schiff,

1990).

22

Human visual perception of acceleration of an object in motion is very gross and
inaccurate; it is very difficult for a driver to discriminate a speed change from constant
velocity unless the object is observed for a relative long period of time — 10 or 15
seconds (Boff and Lincoln, 1988).

The delta speed threshold (i.e., change in relative velocity of the lead vehicle
and the following vehicle) for detection of an oncoming collision or pull-away has been
studied in collision-avoidance research. Drivers can detect a change in distance between
the vehicles they are driving and the one in fi'ont when it has varied by approximately
12% (Mortimer, 1988). Mortimer notes that the major one is rate of change in visual
angle. This threshold was estimated in one study as 0.0035 radians/sec (Gartner, 2000).
This would suggest that a change of distance of 12 percent in 5.6 seconds or less would
trigger a perception of an approach or pulling away. Mortimer concludes that: “. ..
unless the relative velocity between two vehicles becomes quite high, the drivers will
respond to changes in their headway, or the change in angular size of the vehicle ahead,
and use that as a cue to determine the speed that they should adopt when following
another vehicle.” This study implies that the perception time 2" of the following drivers
during which the driver becomes aware that the distance between his/her vehicle and
the vehicle in front is decreasing, depends on the spacing between them and the relative
speed of the two vehicles. If we assume that the speed of the following vehicle is kept in

a stable state during 1' the relative speed is determined by the speed of the lead vehicle.

I
The speed of the lead vehicle at time point t e [t0,r'] is mic) + I)?” (t).

Rothery (1968) revisited the speed curves obtained by Ohio State University in

their steady—state car following study (Todosiev, 1963) and concluded that the

23

acceleration/deceleration rate can be considered roughly to be constant (Gartner, 2000).
Therefore, the integration equation above can be rewritten as

I
it,' (to) + If" (t) = x" (to) + ant , At = t — to. The dynamic distance S(t) between two

‘0

consecutive vehicles is as follows:
S(t) = S(to) + J'cn(t0)At + i—k‘nazt — 5an (t0)At.

The relative spacing change

S(t.)—S<t>_ 1
S(to) ‘ S(t.)

 

[-i.(to)At-%5i..(to)42t+inn(to)At] 2-1.

Based on Mortimer’s results, drivers can detect a change in distance between their own
vehicle- and the one in front when it has varied by approximately 12%. Based on this
observation we set Equation 2-1 equal to 0.12. The time that allows the driver of the
following vehicle to become aware the speed change of the lead vehicle can be obtained

as follows

 

T._ (we) —5c..,<t..)) : Jeni.) —x..,(z.»2 - 2x 0.122,,300)

 

 

 

.. 2-2.
xn(to)
If we assume that at t0 , it" (to) = x,,,(t,) ,
—2 0.12" S t
7': J x .. x" (0) = 0.4899 —S(ff’) 2-3-
x" -Jr,l

Driver’s input to a planned braking situation approximates a “ramp” function
with the slope determined by the distance to the desired stop location or steady-state
speed in the case of a platoon being overtaken (Gartner, 2000). The driver applies pedal

pressure to the brakes until a desired deceleration is obtained. The maximum

24

“comfortable” braking deceleration is generally accepted to be in the neighborhood of -
0.30 g, or around 3 m/sec2 (ITE, 1992). Wortman and Matthias observed a range of -
0.22 to - 0.43 g with a mean level of - 0.36 g (Wortman and Matthias, 1985). Hence,
“comfortable” braking performance that yields a g force of about - 0.2 would be a
reasonable lower level, i.e. almost any driver could be expected to change the velocity
of a passenger car by at least that amount, while a “typical” level would be around -
0.35 g. If a driver responds to the speed disturbance by removing his or her foot from
the accelerator pedal, dragging and rolling resistance produce deceleration at about the
same level as “unhurried” acceleration, approximately 1 m/secz. Drivers use controlled
braking skills to respond to an unexpected braking situation or an anticipated one when
they are unsure of when it would happen. The empirical data collected on drivers in
their own vehicles Show a wide variation ranging from —0.46 g to —0.7 g with a mean
level of —0_.55 g for “rmexpected” situations and —0.45 g for “anticipated but unsure
when” situations (Fambro, 1994). In general, the deceleration rates can be from 1
m/sec2 to 3.6 m/sec2 with a reasonable low rate of 2.2 rn/sec2 for a “planned” situation,
from 4.6 m/sec2 to 7.0 m/sec2 with a mean rate of 5.5 m/sec2 for an “unexpected”
situation, and from 2.4 m/sec2 to 6.6 rn/sec2 with a mean rate of 4.5 rn/sec2 for
“anticipated but unsure when” situation. Therefore the deceleration rates of a following
vehicle could be from the minimum of 1.0 m/sec2 to the maximum of 7.0 m/secz. The
driver’s braking behaviors under the three situations discussed above are summarized in

Table 2-2.

25

Table 2-2. Driver’s Braking Behaviors under Three Situations

 

 

 

“Anticipated but
Situations “Planned situation” unsure when” “Unexpected”
Deceleration l- 3.6 iii/sec2 2.4 — 6.6 rn/sec2 4.6 — 7.0 m/secT
Rate Range (3— 12 fi/secz) (8 — 22 ft/secz) (15.3- 23.3 fi/secz)
Mean of 2.3 m/sec2 4.5 m/sec2 5.5 In/sec2r
Deceleration (8 ft/secz) (15 ft/secz) (18 ft/secz)
Rate

 

 

 

 

 

 

The amount of time required to make the braking input movement (M T) was
also investigated in three studies that proposed mean times ranging from 0.20 — 0.26
sec. Research indicates that the relationship between perception-reaction time and MT is
weak to nonexistent. That is, a long perception time does not necessarily predict a long
MT (Berrnan, 1994). Thus, the earliest possible time at which the speed of the following
vehicle is expected to be reduced is the sum of the perception time, 1' plus the braking-

input time as follows

r = r'+MT = 0.4899 w + MT 2-4.

-X

II

The values of r as a function of ( S(to),5c'n) for the spacing from 100 m to 10 m

and deceleration rate from 1.0 m/sec2 to 7.0 iii/sec2 with MT = 0.2 seconds are

calculated in Table 2-3.

26

Table 2-3. Earliest Perception-Reaction Time as a Function of Spacing and Deceleration

Rates

 

Spacing (m) r

 

Original Reduced x' = 1.0 x' = 2.0 .‘r = 3.0 x = 4.0 x‘ = 5.0 it = 6.0 x‘ = 7.0
Spacmg Spacmg m/sec2 m/sec2 m/sec2 m/sec2 m/sec2 m/sec2 m/sec2
100.0 88.0 5.10 3.66 3.03 2.65 2.39 2.20 2.05
90.0 79.2 4.85 3.49 2.88 2.52 2.28 2.10 1.96
80.0 70.4 4.58 3.30 2.73 2.39 2.16 1.99 1.86
70.0 61.6 4.30 3.10 2.57 2.25 2.03 1.87 1.75
60.0 52.8 3.99 2.88 2.39 2.10 1.90 1.75 1.63
50.0 44.0 3.66 2.65 2.20 1.93 1.75 1.61 1.51
40.0 35.2 3.30 2.39 1.99 1.75 1.59 1.46 1.37

30.0 26.4 2.88 2.10 1.75 1.54 1.40 a g.
20.0 17.6 2.39 1.75 1.46 E— a" a fi
10 8.8 ms tad TEETETET -

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

When following very closely, the driver of the following vehicle is alerted,
because their assessment of the perceptual variable has been near the threshold of the
target risk level. Therefore, a braking signal of the lead vehicle could trigger the driver’s
reaction to the signal almost immediately. There is no specific study of the response lag
to a braking signal. However, studies at signalized intersections suggested that the
average driver’s response lag to a signal change (time of change to onset of break
lamps) is 1.3 seconds and is somewhat inelastic with respect to distance from the signal
at which the signal state changed (Chang, 1985 and Wortman, 1983). If we assume a
value of brake response lag to a braking signal of the lead vehicle similar to the
response lag to achange in signal indication at intersections, then the earliest time that
the following vehicle starts to reduce its speed afier receiving a brake signal would be

1.3 seconds. It is worth noticing that the times shaded in Table 2.1 are times needed by

27

a driver to recognize a spacing change under certain circumstances. Some of these times
are shorter than 1.3 seconds, the time lag for a driver to respond to a signal change. In
those cases, the drivers could use their brake at anytime when they recognize the
distance change, which could be less or more than 1.3 seconds.

Drivers control their vehicular speeds by the skills of speed keeping,
accelerating and decelerating. Under steady-state car following conditions, the range of
speed-control error was estimated to be no more than +/- 1.5 m/sec (Evans and Rothery,
1973). The acceleration rates, particularly in a traffic stream as opposed to a standing
start, are typically much lower than the performance capabilities of the vehicle. Drivers
under “unhurried” circumstances use approximately 65% of the maximum acceleration
for the vehicle, or approximately 1 nr/sec2 (ITE, 1992). AASHTO places the nominal
range for “comfortable” acceleration at speeds of 48 km/h and above to 0.6 to 0.7
m/sec2 (AASHTO, 1990). In steady-state car following circumstance, the arcs of
relative speed versus spacing are approximately parabolic implying that
acceleration/deceleration can be considered roughly to be constant (Barbosa, 1961).
Overall, the acceleration/deceleration rates for an individual vehicle could be constant,
for the vehicles in a traffic stream however, the rate could range from a maximum of 1.5
m/sec2 to 0.6 m/secz.

2.5 Summary

Through exploring the hierarchical driver modeling framework proposed by
Boer, this chapter developed the concepts of Expected State-Control Action Chains and
further obtained the three scenarios of the car-following process when a lead vehicle has

a speed disturbance. On the basis of previous studies on driver’s behavior in perceiving

28

and reacting to a speed change of the lead vehicle, we proposed the possible control-
action time and control strategies that could be used by the driver of the following
vehicle to respond to a speed disturbance from the lead vehicle. They are summarized in

Tables 2-2 and 2-3 and will be used in Chapter 5.

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Gartner, N., G. C. Messer, and A. K. Rathi. Traffic Flow Theory. Chapter 4 of R. W.
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Gartner, N.G., C. Messer, and A. K. Rathi. Traffic Flow Theory. Chapter 3 of R. J.
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Haglund, M.and L. Aberg. Speed Choice in Relation to Speed Limit and Influences
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Hoffman, E.R. Acceleration to Brake Movement Times and a Comparison of Hand and
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31

CHAPTER 3 SINGLE SPEED DISTURBANCE AND DYNAMIC SPACING

Error and deviation of speed in manual controlled vehicles is common. The
speed of a human—controlled car usually fluctuates around desired value that the driver
adopts. Therefore, speed disturbance is a natural phenomenon of traffic flow. The speed
disturbance could be large or small in value, long or short in duration, could occur
under various traffic conditions and has various degrees of impact on traffic stability.
The response of the following vehicle to a speed disturbance of a lead vehicle can be
very different depending on the characteristics of the disturbance.

This chapter discusses the localized behavior of a following vehicle in response
to a fluctuation in the motion of the vehicle directly in front and the dynamic spacing
between the two vehicles. Formulas to calculate minimum dynamic spacing for each of
the scenarios of speed disturbance defined in Chapter 2 are obtained in this chapter.

For the readers convenience, the notations used in the following discussion are
defined as:

n: lead vehicle

n+1: the first following vehicle

to : beginning time point of the scenario,
t: time variable ,
t,: time point at which vehicle 11 starts to accelerate,

At: time interval variable,
6 : time point before which vehicle n+1 travels at its original speed,

6': time point at which vehicle n+1 completes its speed reduction,

32

T: time period during which vehicle 11 completes its speed recovery,

51,0) : speed of vehicle 11 (lead vehicle) at time t,

51",,(2) :speed of vehicle n+1 (following vehicle) at time t,

x; : acceleration rate of vehicle 11, roughly to be constant (Rothery 2000),

if; : the deceleration rate of vehicle it, roughly to be constant (Rothery 2000),

it" the deceleration rate of vehicle n+1, roughly to be constant (Rothery 2000),

n+1 '

S minimum safe distance headway for following vehicle, and

safe I
S (t) : distance headway between vehicles n, and n+1 at time t.

As assumed in Section 2.3, vehicles 11 and n+1 are in an acceptable car-
following mode which is considered to be the beginning state of the system, and the

original speeds of the two vehicles are roughly the same. At t0 , vehicle 11 develops a
speed disturbance and starts to recover it at t,. At t0 + T the vehicle It completes its

speed recovery. During the development and recovery of a speed disturbance the
following vehicle may not reduce its speed to respond to it or may respond to it with
different rates of deceleration depending on the assessment of perceptual variables and

the Expected State of the following vehicle as seen in Figure 2-5, 2-6 and 2-7. In the

period of 110,10 + T] the dynamic spacing between the two vehicles can vary greatly but

will follow the law

S(t) = S(to) + Ixn(t)dt - jx,,,(x)dr 3-1.

‘0

33

3.1 Dynamic Spacing with Acceptable-Acceptable Expected States
As illustrated in Figure 2.5, during the lead vehicle speed disturbance the
following vehicle does not need to take any control action to respond to the speed

disturbance, therefore, 56",,(t) = 0 for te [t0,t0 + T].
In the time interval [to,t,] vehicle 11 reduces its speed and a speed disturbance is

developed. According to Rothery’s conclusion (Gartner et al. 2000) in reexamining the
data obtained in the studies by Todosiev (1963) and Rothery (1968), the deceleration

rate of an individual vehicle can be assumed as almost constant. Thus Equation 3.1

 

becomes:
S(t) =S(t0)+::-ir'; x(t—t0)2 where if; <0,te [to,t,] 3-2.
. dS (t) .._ . . .
Since t = x" (t — to) < 0, the S( t) in the equation 3-2 monotonically decreases

and has its minimum value as shown in Equation 3-3 below:
Minimum S(t) = S(t,) = S(to) +éii; x(t1 -to)", at I: t, 3-3.

In the time interval (t,,t0 + T] vehicle 11 increases its speed and the speed

disturbance is being recovered. The dynamic spacing between the two vehicles

becomes:

S(t) = S(r,) + jinmdt - Ixn+l(t)dt t e [t,,tl + T) 3-4

where 50,) = $0,) + é": x (tl -to)’.
2,0,) = 31,004.55; x0, —i,),

x.rH1(t) : x.H+1(t0) ’ and

34

inn) = xn(i,)+ Jin(t)dt = goo) + 55; x0, —i,) + Imam

= xn(to)+5c'; ><(tl —to)+5c‘; x(t—t,)

Equation 3-4 can be reorganized and simplified as follows:

 

 

 

S(t) = S(to) THEE; ><(tI «10)2 + 56; x(tI -to)x(t —i,)+-;—5i: X(t —t,)2 3-5.
2
Since dS(t) = J'r';x(tl —to)+5r';(t—t,) and d 32(1) = x'; >0, the S(t) has a
dt dt
minimum value in (t,,to + T]. Let fl = 0 , we solve
t —t, = —f—:(t, -t0) = x"(t°)_:x"(t‘) = (t0 +T) -tl 3-6.
x

II II

The S( t) has a minimum value:

x"
00+
"

 

Minimum S(t) = S(t0 +T) = S(to)+%5i;x(tl —t0)2x[l— ], at t = to +T 3-7.

x"
ou+
II

 

Comparison of Equations 3-3 and 3-7 shows that, since [1 - ] > 1 and it; (t) <

0, S (t0 + T) < S (1,) . Thus in this scenario during the entire period of [t0,t0 + T] S(t) has

a minimum value of

Minimum S(t) = S(t0 +T) = S(t0)+-;-.ié; x(rl --t0)2 x[1-%] 3-8.

3.2 Dynamic Spacing with Acceptable-Unacceptable-Acceptable Expected
States
In this scenario, as illustrated in Figure 2-6, during the lead vehicle speed

disturbance the expected states of the following vehicle changes from acceptable to

35

unacceptable, and it would take a speed control action from the following vehicle to
keep a safe spacing. A control action can be taken as early as at time t0 + Perception-
Reaction Time to the time at which the driver of the following vehicle is required to
reduce its speed to avoid a collision. The deceleration rate as analyzed in Chapter 2 can
be as small as 1 m/sec to a 7 m/sec. If 6 represents the beginning time of a control

action and 6' its ending time, there exist three cases to be discussed: Case 1) 5 < t1 and

§'> t1, Case 2) 6 < t1 and 5'< tl , and Case 3) 5 > t1 and 6'> tl . They are illustrated in

      
 

 

 

Figure 3-1.
Speed
A ’, xn+1
...... 1‘ _----- -....-|
g i
: 4141mm- 5c" 1: Case 1
: , l : 2:Casc2
: . g l : 3:Case3
I l I l r
I l t ' 1
: : . i : 1 :
1 1 = u i l
4 - +
to 6 6' 1, 6'6 6' to+T time

Figure 3-1. Possible Control-Action Strategies that Following Vehicle Can Take with
Expected States of Acceptable-Unacceptable-Acceptable

Case 1. 6 < t1 and 6'> tl
In [5,6] , the following vehicle keeps its original speed and the dynamic spacing

between it and the vehicle in front follows the same way as described in 3-2, and
Minimum S(t) = 5(6) = S(to) +éir'; x(§ - to)2 3-9.

In (§,t,] , the following vehicle reduces its speed to keep a safe distance from

the vehicle in front. The dynamic spacing between them is:

36

S(t) = 3(6) + jx,(i)di - [imam 3-10.
6 6
where S(6) = S(to) +le-x'; ><(6—t0)2 ,
Emu) = 59,45) + [iguana = 56....00) + 3;” x (r - 6), and
6

inc) = x,(6)+jx,(i)di = 51,00) +55; x(5-z,)+jx;di = 5c,(:,)+ie; x0 -i,).
6 6

Equation 3-10 can be rewritten and simplified as:

S(t) = S(to)+-:—5é;x(t—to)2 gig, ><(i--§)2 3-11.

dS(t)

dt x (t — 6) < 0, which means that 3-11 monotonically

=x;x(i—i,)-it;

+1
decreases and has its minimum value at t = t, as:
. . 1 .._ 2 1 .... 2
Minimum S(t) == 50,) = S(to) + ~2—xn (tl —to) — -2-xm,l(tl - 6) 3-12.

In (t,,6'] , the following vehicle n+1 keeps reducing its speed and the lead

vehicle 11 recovers its speed. The dynamic spacing between them is:

S(t) = S(i,)+ Iin(t)dt - jx,,,(i)di 313.

If substitute S(t,) , x" (t) = 12,00) + if; x(tl —to) + if: (t —t1), and

x,,,(i) = EH00) + 56;“ x (t‘ — 6) + ii;+,(t — t,)into Equation 3-13, it becomes:
1 ...... 2 H. l ..+ 2
S(t) = S(to)+-2—xn x(tl -to) +xn ><(t1 —to)x(t-t,)+5xn x(t—t,)

- fix. x11. —6)2 —5r';.. x0. —6>><(t 40556;” ><(t -t.>’ 3-14.

Since in the time interval of t e (il ,6'] the differentials of Equation 3-14 are:

37

flaw x(i -t0)+x'; x(t—t)—

“Tm-1X01" 6-) xn+lx(t_ t)=x,."(t) xn +510) 0
dt

dS2 t ,,
d E ) = x; —- 56;“ > 0, and S (t) is continuous on t, Equation 3-14 has its minimum
t

 

value at t = 6' as:

S(6') = $0,) + é—x'; x(tl -t0)2 + x'; x(tl —to)X (6'—-t,) +—:-x; x(6'—-t,)2— - x (6'-—-6)

xn+1

=S(t0)+-:-x°;><(tl-to)(6'—t0)——25&;+1X(6'-§-)(t —6) 3-15.

At t = 6 ' , the speed of the vehicle n+1 is reduced to roughly the same speed as
that of the vehicle 11 and in the period (6 ',to + T] this vehicle keeps following the lead
one at roughly the same speed. Therefore the spacing in this time interval can be
roughly regarded as constant.

If subtract Equation 3-9 from 3-12, we have:

S(t,)-S(6) = x; x(tl —6)x(6—t0)+-:-x';x(tl -6)2— — x(i —6)

in+1

_(r.—6)

 

[xnl—(t) xn+,(t)+x; x(6— to),]<0

which means that in the time interval of (t0,t,] Minimum S(t) = 50,).

Again, if subtract Equation 3-12 from 3-15, we have:

x;+lx(t 6—)(6’—t,)—— x;,lx(6' —t)]

S(6')—S(t,)=(6'-t,)[x‘;x(t —to)+;x'; x(6'—i)—

. Since in (6 ') = xn+,(6') as assumed in the definition of this scenario, i.e.,

xn0(t)+x; x(t,o—t)+x'; x(6'—t)=x x,,,(i,‘)+x' x(t,— 6x”)+ x(6'—t,),and

n+1 xn+1

x (t0 )= x M,(t ), S (6 )— S (t ) can be rewritten and simplified as.

38

S(6') — 50.) = $164012, 0.) — 5r,..(t.)] < 0.

which means that in the time interval of (t0,6'] Minimum S( t) = S (6 ’) . Thus in this
scenario

Minimum S(t) = S(6') 3-16.

Case 2. 6 < t1 and 6'< tl

In [t0,6] , the following vehicle keeps its original speed and the dynamic spacing
between it and the vehicle in front follows the same way as described in 3-2. In this
time interval

Minimum S(t) = 8(6) = 50,) +—:-5e; x(6—to)2 3-17.

In (6, 6 '] , the speed disturbance is continuously developed and vehicle n+1

reduces its speed starting at t = 6 to keep a safe distance space in front until t = 6 ' at
which time its speed is reduced to about the same as that of vehicle 11. The dynamic

spacing between them is:

S(t) = 5(6) + jxnuyiz — jx,,,(i)dt 3-18
6

6
where S(6) = S(to)+%x'; x(6_,0)2 ’
xn+1(t) = in+l(6) + jin+101dt = xn+1(t0)+ 2;“ X0 ‘- 5) , and
6

inc) = x,(6)+jx,(i)dz = = xn(t0)+5c';X(6-to)+5€;x(t-6).
6

If substitute them into Equation 3-18, we have:

S(t) = 5(6) + x; X(6-t0)x(t —6)+%x'; X(t —6)2 —%x;,, x(t — 6)2 3-19.

39

Since in the time interval of IE (6 ,6 '] the differentials of Equation 3-19 are:

111(1). = x;x(5_zo)+x;x(i—6)—x;,,x(:—6) = x,(t)—x,...(t)50.

1152 t .._ ..- . . . . . .
( ) = x — x"+1 > 0 , and S (t) is continuous over t, Equation 3-19 has its rrnnimum

dt2 "

 

value in the time interval of (6,6‘] as:
Minimum S(t) = S(6') = S(to) +éx; x(6'—to)2 --;—x;,, X(6'—6)2, at t = 6' 3-20.

In (6 ',t0 + T], the following vehicle keeps roughly the same Speed profile as that
of the vehicle directly in front. The dynamic spacing between them can be regarded as
constant. Therefore, the minimum value of dynamic spacing in this scenario is:

Minimum S(t) = S(6').

Case 3. 6 > t1 and 6'> II

In [t0,t,] , the following vehicle keeps its original speed and the dynamic spacing
between it and the vehicle in front follows the same way as described in 3-2 and reaches

its minimum value at t = t1, i.e.,
MinimumS(t) = S0,) = S(t0)+—;-x';><(tl -to)2 3-21.

In [t1,6] , vehicle 11 recovers its speed. The Expected States of vehicle n+1 have

been kept acceptable and the following vehicle travels at its original speed until t = 6 .

The dynamic spacing between the two vehicles is:

S(t) = 301 ) + j x, (t)dt - j 51",, (t)dt 3-22.

‘1 '1

40

As we did above, substitute in (t) = 51,00) + it; (t — to) and x,,,(i) = 62",,00) + ii;+,(t - to)

into Equation 3-22, we have:

S(t) =S(t0)+%x; x(tl —t0)2 +56; x(tl —to)(t—tl)+-;—x'; x(t—t1)2 3-23.

 

Since dS(t)
dt

= if; x (tl — to) + if; x (t - t,) < 0 the S( t) in this time interval
monotonically decreases and has its minimum value at t = 6 as :

. . 1 u. 2 n. 1 ..+ 2
Minimum S(t) = S(to)+3xn x(tl -to) + x" x(t1 —t0)(6—t1)+§xn x(6—t1) 3-24.

In (6 , 6 '] , vehicle 11 continuously recovers its speed and vehicle n+1 reduces its

speed to keep a safe distance headway from the vehicle in front. The dynamic spacing

is:
S(t) = 3(6) + j x, (t)dt — j 5c“, (t)dt 3-25.
6 6

Since x,(i)=x,(6)+5e; x(z-6)=x,(io)+x; x(tl —i,)+x; x(t—tl), and

x"+1 (t) = x,” (to) + 56;“ x (t — 6), the above equation can be rewritten and simplified as:
1 .., 2 1 ..- 2 .._ 1 .._ 2
S(t): S(t0)+-2-xn x(t—t,) +2x" x(tl -to) +xn x(tl —t(,)(t-t,)----2-x,,+l x(t—6) 3-26

It is:

d8 (1‘)

dt =i;x(t—t,)+x;x(tl—to)-jc" x(t—6)=x;x(t—6')—x‘ x(t-6')

n+1 n+1

 

= (6'—i)(ie;,, — it; ) < 0.

The S(t) monotonically decreases and reaches its minimum value at t = 6 'as:

Minimum S(t) = S(6') =

41

=S(t,)+%x; x(6'--t,)2 +éx; x(tl —t0)2 +x;x(t1—t0)(6'-t,)-—:-x;+lX(6'-§)2 3-27.

In (6 ',t0 + T], the two vehicles keep the same speed profile and the spacing
between them is roughly constant.
If subtract Equation 3-21 from 3-24, we have:

S(6)—S(t,) = (6—:,)[x; x(t, 4.0-€55; x(6—i,)

< (6-:,)[x; x(tl —i,)+x; x(6—t,) <0, i.e., S(6)<S(t,).

Subtract Equation 3-24 from 3-26, then we have:

S(6)-S(6')
= (6-t,)[x'; x(t‘l —t0) +%x'; x(6-tl)]+ (6'-—t,)[5r';+1 X(6'—6)]—%x': x(6'—tl)

<(6 — i, )[x; x (z, - to) + it; x (6 - i,)] + (6'—-t, )[xg+1 x (6'-6)] — éx: x (6'—t,)

< 0, i.e., S(6') < S(6) 3-28.

Thus in this scenario, Minimum S (t) = S (6').

3.3 Dynamic Spacing with Unacceptable Expected State at Target-Risk Level
Under this scenario, as described in Figure 2-7, the driver of the following
vehicle is alert at all times and would take control action when receiving any stimulus,

such as brake signals from the vehicle in front. In this case the control action occurs as

soon as t = t0 + 2'. There are two cases in this scenario: 6'> t1 and 6'< tI , shown in

Figure 3-2.

42

Speed

   

--------- ...........

1:Cmml

    

2: Case 2

 

------_---.T----
H

 

:
E ' - ' >
to r 6' t1 6' to+T time

Figure 3-2. Possible Control-Action Strategies that Following Vehicle Can Take with
Expected States of Unacceptable at Target-Risk Level
Case 1. 6'> tl
This case is similar toCase 1 in Section 3.2 except for the beginning time of a
control action of the following vehicle. In this case the beginning time of a control
action ist = r , while in the Case 1 of Section 3.2 the beginning time of a control action
is t = 6 . If 6 is replaced by 2' , the equations in the Case 1 of Section 3.2 are

applicable in this case. Therefore the minimum spacing in this case becomes:

Minimum S(t) = S(6') = SUCH-:46; x(tl —t.,)2 +56; x(tl —to)><(6'-t.)+-12-5r'.’.’ ><(6'-t.)2

"£55.11 x (5-7)2 329.

Case 2. 6'< t1

This case is similar to the Case 2 in Section 3.2. As pointed out in the above
case that the beginning time of a control action of the following vehicle in this case is
t = 1' , in stead of t = 6 . If 6 is replaced by T , the equations derived for the Case 2 in
Section 3.2 are applicable in this case. Therefore the minimum spacing in this case

becomes:

43

Minimum S(t) = S(6') = S(to) + ix; x (6'—to)2 —%x';,, x (6'—r)2 at t = 6' 3-28.

3.4 Summary

Localized behavior of the following vehicle, in response to a speed disturbance
of the vehicle directly in front, changes the spacing in a continuous and dynamic
manner. Minimum values of the dynamic spacing for each case analyzed in Sections
3.1, 3.2 and 3.3 have been derived and are summarized in Table 3-1. These values are
of interest in the discussion of upper boundary conditions for stabilities. This discussion

is presented in detail in Chapter 4.

References

Gartner, N. Traffic Flow Theory. Chapter 4 of R.W. Rothery, Car Following Models.
Transportation Research Board Special Report 165, Transportation Research Board,
2000. PP.

Todosiev, EP. The Action Point Model of the Driver Vehicle System. Report No.

202A-3. Ohio State University, Engineering Experiment Station, Columbus, Ohio,
1963.

 

 

 

 

 

 

 

 

 

 

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45

CHAPTER 4 SPEED DISTURBANCE AND STABILITY

During the interaction of the following vehicle in response to a speed
disturbance the minimum dynamic spacing is a key restrictive parameter to safely carry
out a single-lane car-following task. Therefore it becomes of interest to discuss the
upper boundary conditions for speed-disturbance absorption and stability. On the basis
of the Chapter 3, the existence of an upper boundary will be demonstrated and the
magnitude of a speed disturbance that a single spacing can absorb will be quantified.
Moreover, the conditions for local and asymptotic stability will be discussed.

The following definitions will be found useful for the discussion that follows.

Definition 4-1: The value of a speed disturbance is expressed as

Aft" = 2,0,) -5cn(t1) =

 

 

x; x (tl — to).

Definition 4-2: A generated speed disturbance is defined as the speed reduction
of the following vehicle in response to the speed disturbance of the vehicle direct in
front. It can be quantified by

At

11+

x(6'—6).

 

 

1 = 55.3.1
Definition 4-3: The amount of a speed disturbance absorbed by the spacing

between the lead vehicle 11 and the following vehicle n+1 is defined as

A2,, -Ax,,,, = |x;|x(il —to) -ix‘ x(6'—6).

n+1

 

Obviously, Ax" — Ax"+1 is a function of the speed change of the two vehicles,

and the control-action time of the following vehicle.

46

Definition 4-4: The upper boundary on the magnitude of a speed disturbance

that a single spacing can absorb is defined as

Aif,’ = Maximummx, — An 2;,

n+1

x(6'-6)}.

 

 

 

 

) = Maximum{ 5:; x (tl — to) —
4.1 Value of Upper Limit of A Speed Disturbance that a Single Spacing Can
Completely Absorb

As illustrated in Figure 2-5, when a speed disturbance from a lead vehicle
occurs. the following vehicle may not need to reduce its speed in response to it during
the period of the speed disturbance development and recovery. In this case, the speed
disturbance does not have any impact on the speed of the following vehicles, but
reduces the spacing between them. In other words, the speed disturbance is completely
absorbed by the single spacing. The minimum dynamic spacing in this scenario is at
t=5+Ta

Minimum S(t) = 500 +T) = S(to) +%x; x(t1 -t0)2 x[1-if"T].
x

n

The above equation has to be equal to or greater than a minimum safe distance,

SW, , for safety reason, i.e.,

4-1.

safe

S(t,)+%x°; x(il -t0)2 x[1—%—]2 s
x"

Equation 4-1 can be rewritten as

S(to)—Smf, 2 -%x; x(tl -—t0)2 ><[1— f1]

’1

 

47

=%Axnx(tl —t0)><[1—x

 

].

n
00+
II

1 . x
sup-SW, = Em, x(z,-i,)x[1—_—j';].

fl

Solving the above equation, we get

S 2[S(IO) — Smfe]

 

 

 

 

 

 

 

 

 

 

 

456.. 4-2.
x
(t. -to)[1- 73-]
xn
Since tl — t0 = _._" , we can substitute into 4-2 Equation and obtain
455.. S — 2x,I [S (‘01‘ S We] 5 2[S (if) — Slmfe] 43.
x
N [1 - {:1 V _ it: + 7,-
x" xn xn
One can easily find that the upper boundary on the magnitude of a speed
disturbance that a single spacing can completely absorb, denoted as Air: , can be
obtained when
M0 = 2[S(t0) - Ssafe] 4-4
" l 1
l - ~—— + E
xn xi!

Equation 4-4 gives the upper boundary on the magnitude of a speed disturbance
that a single spacing can completely absorb. It confirms that when the acceleration and

deceleration rates are chosen, a larger spacing absorbs greater speed disturbance.

48

4.2 Value of Upper Limit of A Speed Disturbance that A Single Spacing Can

Partially Absorb
When a speed disturbance is larger than Ax: in Equation 4-3, the spacing

between the two vehicles cannot completely absorb the speed disturbance. The
following vehicle reduces its speed at sometime t = 6 to maintain an acceptable

spacing in front. When taking a control action, the driver will decide when and how to

do this, i.e., will decide on 6 and 56;, . If a control-action strategy is denoted by 52 it

can be described by Q( 56’ 6 ).

n+1 ’

Definition 4-5: A control-action strategy 9( 56;, , 6 ) is feasible if its
corresponding dynamic spacing defined in Section 3.2 is equal to or greater than Sm.
Theory 4-1: For any control-action strategy 9( 56;, , 6 ) there exists an

equivalent control-action strategy 5 (52;, , 6 ) where 12;, = 56; such that the
an“, = Aft".
Proof: By definition, Ax;, =1 56;, |x(6,'-6,) . When S2( 56;, ,6) is given,

Ax;, =| 55;, |><(6,'—6,) is determined. For a given Ax

n+1

due to the contiguity of
Ax;, on both I and the deceleration rate 56;, , one can always find a ‘6 (56;,- , 6 ) such that
| it}, |x(6'—6) =| x; |x(6'—6).

I!

Now we come to prove the feasibility of :2- (56; , 6 ), i.e., its Minimum S(t)

safe '

49

Case 1 in Section 3.2

a. First we consider the case when 57;, = |5i; |> |5c' ;.,|

The minimum dynamic spacing for the control strategy fi (56" , 6 ) can be

determined by Equation 3-15 as

Minimum S( t)

80,) fix; x(t, —t0)(6'—t0)— - 56;, x (6'— 6)(t — 6)

S(t,)+%x;x(t,-t,)(6'—t,)—2—2 x;,,x(6'—6')(i,— —6')

=S(t0)——Ax x(6'—to)+21Ax; ,,x(t —6) 4-5.

Likewise, the minimum dynamic spacing for the control strategy £2 (56,,+1 , 6 ) is

Minimum S(t)= S(to)+%5r';x(t,—t0)(6'-to)--:- 5&;,x(6'— —6)(t, —6)

= S(to)+%5t';x(t, —t,,,,)(6'-t)— in; ,x(-6' 6)(t, —6)

= S(t0 )——Ax x(6'-t,, )+% Ax;,x(t,— —6) 4-6.

If the control-action time of 5 (x'; , 6 ) starts at 6 = 6 its ending time will be

6' < 6'. Then Equation 4-5 can be rewritten as

Minimum S(t)=-S(to)—-1-Ax x(6' —t0)+% Ax;, x(t, —6)

>S(t0 )——Ax,, x(6'—t0 )+— Ax;,><(t,- 6—) 4-7.

Since the control- action strategy £2 (56

x;, , 6 ) is feasible Equation 4-6 2 SW. Thus,

Equation 4-5 2 Equation 4-5 > S

— safe '

50

b. Secondly consider the case when 56;, = |56; | < |56;, I.
If we let the control-action E2— (56; , 6 ) end at 6' , the starting time 6 < 6 . The

minimum dynamic spacing for E (56; , 6 ) is

Minimum S(t) = S(to) —%Aic,l X(6'—t,,) + gm“, x (t, - 6)

=S(t,,) €132; x(6'—t,,) +%Ax;, x(t, —6') 4-8.

Since 6 < 6 , (t, — 6 ) > (t, — 6). Equation 4-8 2 Equation 4-72 SW; , i.e.,

Minimum S(t) = sag—$1316; x(6'-t,,) +£13th x(t, -3) 2 S

safe ‘

Thus, the existence of 32- (56; , 6 ) is true for the situation of Case 1 in Section 3.2.

Case 3 in Section 3.2

If we replace the 6 in Equation 4-6 by 1' , the Proof that Theory 4-1 is true for
the Case 3 is similar to that for Case 1.

Case 2 in Section 3.2

In Case 2, all of the control-action strategies start and end before 1,. If
Q( 56;, ,6) is a control-action strategy defined in Case 2 and 56;, > 56; , its minimum

dynamic spacing can be determined by Equation 3-20 as shown below
Minimum S(t) = S(6') = S(to) +%x; x(6'—t,,)2 —-;—56;, x(6'—6)2, at t = 6' 4-9.

Equation 4—9 must be greater than SW, , otherwise, the speed reduction of the following

vehicle, Ax;, should equal the speed reduction of the lead vehicle, Ax; , which is not a

case defined in this scenario.

51

Now let us keep 56;, unchanged but increase 6. As 6 increases, the 6' and the
(6' - 6) increase, but Equation 4-9 is monotonically decreases if 6' < t,. During the

increase of 6 Equation 4-7 cannot become equal to S before 6' becomes greater

safe
than t, since it is not a case defined in this scenario. Therefore, the increment of 6 until
6 ’ becomes greater than t, results in either Case 1 or Case 3. For both of these cases it
has just been proven that there exists an equivalent control-action strategy 5 (56; , 6 )
such that |x°;,, |x(6'—6) =| 56; |x(6'—6) .

Thus, Theory 4-1 is true for the all of cases in this scenario.

Definition 4-6: For any given 56;, , a control-action strategy £2 (56;, , 6 i) is

superior than another control-action strategy S2,- (56;, , 6 ,-) if 25,553,, < A .x and

j n+l’

subject to Minimum S(t) 2 S

safe

Lemma 4-1: For any given 56;

+1

there exists a control-action strategy
£20( 56;, , 60) such that
A563,, 3 {Aim In, (56;, .69, i = 1,2,3 ........ Minimum S(t) .2 SW, }.

Proof:

Case 1 and Case 3 in Section 3.2

As illustrated in Figure 4-1, for the given 56;, , the speed reduction of the

=| J6 |x(5,'—6,)=x; x(t,, +T-6,') , where i = 1, 2, 3, .......

n+l

following vehicle is A,x

13+]

52

‘ 51 52 53
*nUoMnqu)

 

 

 

 

I

I

I

, I

to t. 6;

Figure 4-1 Absorption of speed disturbance with a certain 56’ but different 6

n+1

Since the 56’ and 56; can be regarded as constants (Rothery 2000) and as 6,’

n+1

increases, the (6,'—6,)decreases, and Ad: =| 56;, |x(6,.'—6,) monotonically decreases.

n+1
The minimum dynamic spacing for the any control-action strategies in Case 1 and Case
3 discussed in Section 3.2 is continuous on 6 and decreases as 6 increases and

(6,. '-6,. ) decreases. Thus there must exist a 60 such that
Minimum S( t) = Sm]; , and

-o
Axn+l +1 v

2 Minimum { Ax;, | £2( 56; 6,), i = 1,2,3, ...... , Minimum S(t) = Sm], }.

Similar to the analysis in Theory 4-1 for Case 2, any control-action strategies in Case 2
can be either transferred to an equivalent one in either Case 1 or Case 3. Therefore
Lemma 4-1 is true for all of the cases in this scenario.

Definition 4-7: A control-action strategy is optimal if its

A56;, = Minimum { A6,,“ }.

there will be a £2( 56’

n+1 ’

According to Lemma 4-1, for any given 56’ 6 0) such

n+1

that Ax°

M, = Minimum { 1356;, I52( 56;, ,6,), i = 1,2,3, ...... , Minimum S(t) = S }. Thus,

safe

53

an optimal control-action strategy must be one among the control-action strategies of

S2( 56'

n+1 ’

6 o), and the A563,, of which is minimal of the all A563,, . If we denote an optimal
control-action strategy as 90( 56;, , 6 o) and its speed reduction in response to a speed

disturbance from the lead vehicle as A0560

n+1

respectively, then

- 0
on

. . .0
n,, = Minimum { Ax;, }.

Theory 4-2: If $20( 56"

n+l ’

6 o) is optimal, its equivalent control-action strategy
62’ (56; , 6) is also optimal and | 55;, |x(6’—6) =| 56; |x(6'—6) =on3,, .

Proof: We use the method of reduction to absurdity. We assume that the

equivalent control-action strategy 5 (56; , 6 ) is not optimal.
If thefi (56; , 6 ) is not optimal, the-S3 (56; , 6 ) is not a control—action strategy of
$2( 56; , 6 0) that belongs to {£2( 56;, , 6 0)] for any given 56;,, }. Otherwise, the assumption

at the beginning of the proof is not correct, and the Theory 4-2 is true.

If thefi (56; , 6 ) is not a control-action strategy of $2( 56; , 6 0) that belongs to
{Q( 56;, , 6 o)| for any given 56;,, }, according to the Lemma 4-1 there exists an
fi (56; , 6 0) such that
A563,, 2 | 56; |x(6,,'—6,,) = Minimum { 1356;, | for all 52 (56; , 6 ) and subject to Minimum

S(t)? S }. Thus there exists a 5 (56;,60) such that I56; |x(6,,'—60) < |56; |x(6'—6) .—.

safe
| 56;,, |x(6'—6) , which is contradictory to the assumption at the beginning that the

equivalent control-action strategy 5 (56; , 6 ) is not optimal. Thus, the equivalent

control-action strategy 33 (56; , 6 ) is also optimal and the Theory 4-2 is true.

54

According to Definition 4-4,
Ax: = Maximum(A5cn - Aim)
= Air" - Minimum {Airn+l }.
Theory 4-1, Lemma 4-1 and Theory 4-2 demonstrate not only the existence of

an upper limit on the magnitude of a speed disturbance that can be absorbed but also

indicate that this upper boundary condition can be obtained by discovering an optimal
control-action strategy of £2( 56; , 6 o) . Therefore the discussion regarding the finding of
an upper boundary on the magnitude of a single speed disturbance that spacing can
absorb is simplified to the condition when if; = 56;“

Case 1 in Section 3.2

If it"

n '- xn+l’

the minimum dynamic spacing given in Equation 3-15 can be

simplifiedas
Minimum S(t)= S(to)+%ir';x(t —t 00)(6'—t)— ix; H'x(6 —6)(t 6—)

=S(t,)— ’°

 

MW ..1 + A1,.) 4-10.

As shown in Figure 4-2, Line cd is parallel to the Line hg and Line bd is parallel

to Line eg. Thus, the length of Line bd is equal to the length of Line eg which is equal

to Air" — Ax“. Since it; 2 5g?“ , it is easy to calculate that the length of Line c- -—b -
Ax—i and the length of Line b- a =w. Therefore we get
In xn
6— t0 =(b— a)+(c— b)= (—__—-——)x(Ax —Ax+,) 4-11.

I: n

55

 

 

 

 

 

Aa lb c h
Ax" — Mn“ I”: --------------------
_. a E
E ,g “““
1 J ’
t0 5 t,

C.-

Figure -4-2 control-action strategy of the following vehicle with it; = m
If we substitute Equation 4-11 into 4-10, we get

Minimum S(t) = S(to) — ‘5 ’ ’0

 

x (Axn+1 + Air")

 

=S(t0)——;-x(%—_.i_)x[(AJ'rn)2 —(A5cn+,)2] = Sm], 4-12.
By solving Equation 4-12, we obtain
2 S t —S
(Ax'n)2 = [ (10) Isa/t] + (Mn+1)2 4_13.
3'? ”if

Equation 4-13 suggests that square of the single speed disturbance produced by
the lead vehicle 11 is composed of two parts, one of which is absorbed by the single
spacing and another that is transmitted backward through the speed reduction of vehicle

n+1. If we let the Ax"+1 = 0 , the speed disturbance that the single spacing absorbs is

 

[2[S(t0) — Ssafe]

 

0= ..
AX. 3“}— 414
x; x;

which is concurrent to the Equation 4-3.

56

Case 3 in Section 3.2
If x; = 516;” , the minimum dynamic spacing given in the Equation 3-27 can be

simplified as follows

Minimum S(t)

=S(to)+%5c'; ><(6'—t1)2 +%Jr'; x(t1 -to)2 +56; x(tl —t0)(6'-tl)-%5c" x(6'—6)2

n+l

= S(to) + £55; x (tl — to)(6'-to) — gig” x (6'—6X:1 — 6)

6—t0

 

=S(to)— x(Aich +Axn) 4-15.

This is exactly the same as in Equation 4-10. Therefore its solution is the same

as the Equation 4-13.

4.3 Local Stability
According to the definition of Rothery (Gartner, 2000), local stability is
concerned with the response of a following vehicle to a fluctuation in the motion of the

vehicle directly in front. Equation 4-3 shows that an initial speed disturbance

 

Mo< [2[S(t0)_ssafe]
""1 L__1__

 

ou+ on—

x x

n n

will be absorbed by the spacing and the localized interaction of car-following is stable.

57

4.4 Asymptotic Stability for a Single Speed Disturbance

Asymptotic stability is concerned with the manner in which a fluctuation in the
motion of any vehicle, say the lead vehicle of a platoon, is propagated through a line of
vehicles (Gartner, 2000).

When a speed disturbance cannot be absorbed, it is propagated backward
through the speed reduction of the following vehicles. This generated speed reduction
has an impact on the vehicle following the following vehicleas shown in Figure 4-3.

n+2 n+1 n

L_) _> __>

 

 

 

 

 

 

Sn+l (to) 5:100) I

l
r I

Figure 4-3 Illustration of car following of the next-nearest coupling vehicle

According to the analysis in Section 4.2 if the generated speed disturbance is

 

. 2[Sn+l(to) — Ssafe]
<
Axnfl — l 1 ’

..+ — ..—
V xn+l xn+l

 

 

 

the spacing between vehicle, n+1, and its following vehicle, n+2, can absorb the rest of
the speed disturbance originally produced by vehicle n and propagated through the

speed reduction of the vehicle n+1, Afc Otherwise, the vehicle n+2 has to reduce its

n+1 °

speed to respond to the generated speed disturbance Aim, and in this way the original

speed disturbance produced by the vehicle n is further transmitted backward through
speed reductions of vehicle n+1 and its next nearest coupling vehicle n+2. The
propagation of the generated speed disturbance is similar to that analyzed in Section

4.2, thus

58

2[SIM] (t0) — Ssafel+
l 1

03+ 00—

x X

n n+1

(A&M!)2 :

 

+(Axn +2)2 4'16.

 

If we substitute Equation 4-16 into 4-15, the original speed disturbance produced by the

vehicle n is

 

 

ZSt -S S t -S
(Mn)2= [ (10) lsafe]+2[ n+ll(0)l safe]+ +(Axn +2)2 4-17.
E7;— E_E

Likewise, if the spacing behind vehicle n+2, Sn+2(t0) , cannot absorb the speed

disturbance propagated by vehicle n+2, the transmitting process will be can'ied on

backward resulting in

2[S(t0)—Ssafe] + 2[Sn+l(t0)-Ssafe] + 2[Sn+2(t0)—Ssafe]+

 

 

 

 

- 2_ -
(Ax,)- _1-__1_ __1___1__ i_l ...... 418.

If we assume that there are N vehicles in a vehicular platoon, the upper
boundary on the magnitude of the single speed disturbance produced by the leading

vehicle that a vehicle platoon can absorb is

21$ (1: )— 155...] 449

 

N
(Ax) =Z(Ax.°) )=2
1

x1? x.-

Therefore, Equation 4-19 gives the upper boundary condition of asymptotic

stability for a single speed disturbance.

4.5 Summary
The existence of the upper boundary on magnitude of a speed disturbance has

been demonstrated by means of the minimum dynamic spacing obtained in Chapter 3.

59

here, we obtain the formulas required to calculate the upper boundary of the magnitude
of a speed disturbance that a single spacing (or multiple spacings) can completely

absorb. Moreover, the conditions for local and asymptotic stability are provided.

References

Gartner, N., G.C. Messer, and AK. Rathi, Traffic Flow Theory, Chapter 4 of R.W.
Rothery, Car Following Models, Transportation Research Board Special Report 165,
Transportation Research Board, 2000, pp. 4-6, and 4—9.

Todosiev, ‘E.P. The Action Point Model of the Driver Vehicle System. Report No.

202A-3. Ohio State University, Engineering Experiment Station, Columbus, Ohio,
1963.

60

CHAPTER 5 MODEL VERIFICATION

The results obtained in Chapter 4 can be tested by field, laboratory or
simulation experiments. The verification of the model requires dynamic data
including individual vehicular speed, spacing, and acceleration/deceleration data. In
order to capture the entire process of the interaction of a following vehicle to a
speed disturbance of its lead vehicle, data need to be collected over a sufficient
length of a segment of a roadway and over a sufficient time period and on a group
of vehicles that are following one another without lane-changing. To meet the data
requirements of a model demonstration, a field or a laboratory experiment would
require specialized equipment and be very costly and time consuming to perform.
Therefore, a simulation approach is used.

5.1 Description of the Experiments

The simulation software used for the experiments was Traffic Simulation
Integrated System (T SIS) Version 5.0 published by the FHWA in March 2001.

Two experiments were conducted. All input parameters for the two
experiments were the same except the ratio of Driver Type.

The experimental site was a one-mile long, straight, level, single-lane basic
freeway segment with dry concrete or dry asphalt pavement. A passenger car was
selected as the test vehicle type with an average occupancy of 1.3 persons per car.

Simulated drivers were grouped into 10 types by their driving behavior from
the least aggressive (ranked as 1) to the most aggressive (10). A variety of car-

following sensitivity multipliers were assigned to different types of drivers with less

61

sensitivity and larger following headways assigned to less aggressive drivers and
more sensitivity and smaller following headways assigned to more aggressive
drivers. The first experiment used the default values of the sensitivity factors, which
assigned a ratio of 10% to each of the 10 Driver Types. To obtain a better
observation of car-following behavior at the boundary conditions discussed in the
previous chapters, the second experiment was conducted with the percentage of
aggressive drivers increased. The distributions of Driver Types in the two
experiments are summarized in Table 5-1.

Table 5-1 Percentage of Driver Types in Two Experiments

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Percentage of Driver Types

Lease Egressive More aggressive
Driver Types 1 2 3 4 5 6 7 8 9 10
Experiment 1 10 10 10 10 10 10 10 10 10 10
Experiment 2 0 0 0 0 10 10 10 20 20 30

 

 

Vehicles were generated by the simulation program with a Pearson I
distribution and discharged into the experimental segment through an entry link.
The hourly vehicle discharge rate was 2400 vph and the free flow speed was set at
88 ft/sec (60 mph).

Vehicle performance in acceleration/deceleration was determined according
to the values for driver braking behavior under three situations (planned, anticipated
but unsure when, and unexpected situations) given in Table 2-2 in Section 2.5.
When the driver of a following vehicle anticipates its Expected State is unacceptable

and takes a control action, the driver is most likely to perform an “anticipated but

62

unsure when” control action. Therefore according to Table 2-2 the average
deceleration rate of 15 ft/sec2 for the “anticipated but unsure when” is used as the
deceleration rate of a following vehicle in response to a speed disturbance. When a
vehicle develops a speed disturbance of its own, the driver is most likely to perform
a “planned” braking. Thus, according to Table 2-2 a deceleration rate 8 ft/sec2 for a
“planned” situation was assigned.

To obtain as-detailed-as-possible dynamic data, the time interval for a single
step was set as 0.3 sec in the first experiment. Since the program updates speed and
acceleration/deceleration data every second, to record more vehicular data the second
experiment uses one second as the time interval of a single step. The total time of the
simulation for each experiment was 15 minutes (900 sec).

5.2 Data

TSIS does not provide step-by-step individual vehicular speed, spacing, and
acceleration and/or deceleration data in the reports. To meet the data requirement
for the model verification, data were recorded manually.

5.2.1 Experiment One

During the 15 minutes of the simulation period 519 vehicles entered the
experimental site. Each vehicle had at least 150 step-by-step records. Each record
has three data values, namely speed, position, and acceleration/deceleration rate. If
all of the vehicles are recorded the total data that needs to be recorded would be at
least 519* 150*3 = 233,550, which is difficult to record by hand. Therefore, a
vehicle ID (number 166) was randomly selected. The dynamic movement of vehicle

166 and its following vehicles were monitored. The observation of the vehicle

63

movements showed that vehicle 167 responded to a speed disturbance of vehicle
166. Moreover, vehicle 168 responded to the speed disturbance of vehicle 167 and
so on until vehicle 171. Vehicle 171 was not in a car-following state with vehicle
170 and never responded to the speed change of vehicle 170. To obtain a picture of
the car-following behavior of the vehicles following vehicle 171 (that are not in a
car-following state) with the vehicle in front, three additional vehicles (172 through
174) were also monitored. Therefore, the speed, position and
acceleration/deceleration of vehicle 166 through vehicle 174 were recorded in a
step-by-step manner (Appendix 1).

The step-by-step spacing between each pair of vehicles, the step-by-step time
headway of each vehicle, the minimum time headway of each vehicle during the 15-
minute simulation, and the minimum safe distance headway maintained by the
vehicles were calculated by:

Spacing = position of the lead vehicle - position of the immediately following
vehicle (ft) 5-1

Time headway = spacing lspeed (sec) 5-2

Minimum time headway for each vehicle during the simulations

= Minimum {Time headways of the vehicle} (sec) 5-3

Minimum distance headway maintained by each vehicle during the simulations

= the minimum time headway * the speed 5-4.

The driver type and the minimum time headway of each vehicle during the
first experiment are listed in the Table 5-2. The speed interactions between each pair

of adjacent vehicles are illustrated in Figure 5-1 at the end of the chapter.

Table 5-2. Driver Type and Minimum Time Headway of Each Vehicle in Experiment

 

 

 

 

 

 

 

 

 

 

 

 

 

 

One
VehicleID
166 167 168 169 170 171 172 173 174
Driver Type 7 7 7 5 10 2 2 10 7
Minimum Time
Headway during the Unknown 0.88 0.90 1.15 0.61 2.31 1.48 0.65 0.92
Simulation (sec)

 

5.2.2 Experiment Two

During the 15 minute period of the experiment two, 597 vehicles entered the
experimental site. Since the experiment two took 1 second as a single step, each
vehicle took approximately 50 steps to travel through the site. A sample size of 30
vehicles was selected starting from a randomly selected vehicle II) 69. The speed,
position, acceleration/deceleration rates of vehicle 69 through 98 were recorded in a
step-by-step manner and the spacing between each pair of adjacent vehicles was
calculated by using Equation 5-1 (Appendix 2).

The observation of the 30 vehicle movements showed that vehicles 69, 70, 77,
81, 90, 93, 94, 96, and 97 had roughly stable speeds, kept relatively large spacing from
the vehicles in front, and did not respond to any speed disturbances from the leading
vehicles. Their approximately constant speeds, Driver Types, and minimum time

headway during the simulation are listed in Table 5-3.

65

 

Table 5-3 Driver Types, Approximately Constant Speeds, and Minimum Time

Headway of Vehicles 69, 70, 77, 81, 90, 93, 94, 96, and 97 during the Simulation

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Vehicle ID

69 7O 77 81 9O 93 94 96 97
Driver Types 7 7 7 7 8 9 8 8 7
Approximately 86 86 86 86 89 94 89 89 83

Constant Speed (ft/sec)

Minimum Time Unknown 2.28 4.06 2.42 1.10 0.86 1.28 1.57 1.40
Headway during
Simulation (sec)

 

 

 

Vehicles 71 through 76 followed one another during the simulation. Each of
them kept small gaps from the vehicle immediately in front once it entered a car-
following mode and responded to a speed disturbance from its leading vehicle. The
driver types and the minimum time headways during the simulation are listed in Table
5-4.

Table 5-4 Driver Types and Minimum Time Headways of Vehicles 71 through 76

 

 

 

 

 

 

 

 

 

 

 

Vehicle ID
71 72 73 74 75 76
Driver’s types 9 9 9 10 9 9
Minimum time headway 0.70 0.76 0.76 0.64 0.76 0.73
during the simulation (sec)

 

 

Vehicle 78 developed speed disturbances of its own and vehicle 79 responded
to them, which transmitted the speed disturbances from vehicle 78 backward. Vehicle
80 responded to these generated speed disturbances from vehicle 79 when the spacing

between them became small. Vehicle 82 followed Vehicle 81 closely and developed

66

 

speed disturbances that were transmitted backward through vehicles 83, 84, 85, 86 and
87. Vehicle 88 maintained a stable speed and large spacing from vehicle 87 until step
66 when the spacing became small. Vehicle 89 developed a speed disturbance when it
moved closely to vehicle 88. The driver types and the minimum time headways of
vehicles 78 through 80 and vehicles 82 through 89 are listed in Table 5-5.

Table 5-5 Driver Types and Minimum Time Headways of Vehicles 78 through 80 and

Vehicles 82 through 89

 

Vehicle ID

 

78 79 80 82 83 84 85 86 87 88 89

 

Driver Types 8 9 8 9 10 10 10 10 10 9 10

 

Minimum Time
Headway during 0.86 0.75 0.85 0.76 0.66 0.50 0.59 0.62 0.52 0.71 0.64
Simulation (sec)

 

 

 

 

 

 

 

 

 

 

 

 

 

Vehicles 91 and 95 developed speed disturbances of their own. Vehicle 92
responded to the speed disturbance from vehicle 91when the spacing between them
was reduced to less than 75 feet, but vehicles 96, 97 and 98 did not respond to the
speed disturbances from vehicle 95 because of a large spacing. The driver types and
the minimum time headways of vehicles 91, 92 and 95 are listed in Table 5—6.

Table 5-6 Driver Types and Minimum Time Headways of Vehicles 91, 92, and 95

 

 

 

 

 

 

Vehicle ID
91 92 95
Driver Type 9 9 9
Minimum Time 0.73 0.75 0.73
Headway during
Simulation (sec)

 

 

 

67

 

The speed interactions between each pair of the adjacent vehicles except
vehicles 69 and 70, and vehicles 93 and 94 are illustrated in Figure 5-2 at the end of
the chapter.

5.3 Case Study

By examining Figure 5-1 and Figure 5-2 thoroughly, one can find some cases
in the experiments in which speed disturbances were absorbed completely or partially.
They are identified and listed in Table 5-7 and will be discussed in the case study
sections of 5.3.1, 5.3.2, and 5.3.3.

Table 5-7 Cases in which Speed Disturbances Were Absorbed Completely or Partially

 

 

 

 

Case for Scenario 1 Case for Scenario 2 Case for Scenario 3
Experiment One Figure 5-2 c Figure 5-2 a Figure 5-2 d
Experiment Two Figure 5-3 k Figure 5-3 i Figure 5-3 (1
Figure 5-3 y Figure 5-3 q Figure 5-3 e
Figure 5-3 11

 

 

 

 

 

5.3.1 Cases for Scenario 1 - Acceptable-Acceptable Expected State

This scenario describes a car-following behavior under which the lead
vehicle has a speed reduction and its nearest following vehicle does not take any
control action to respond to it. Thus a reduction of the spacing between the two
vehicles takes place. Therefore, the speed disturbance is completely absorbed by the
spacing between the first two vehicles.

In Experiment One, Figure 5-1 e shows that vehicle 170 developed speed
disturbances at steps 66-76, 82-93 and 111-118 and vehicle 171 did not respond to

any of them. In Experiment Two, Figures 5-2 k and y show that vehicles 80 and 95

68

developed speed disturbances at steps 34-37, 43-47, 51-55, and at steps 56-58, 65-
67, and 78—80. Vehicles 81, 96 did not reduce their speed to respond to them. Those
speed disturbances were completely absorbed by the spacing between the vehicles.
Table 5-8 summarizes the values of the speed disturbances developed by vehicle
170 in Experiment One, and by vehicles 80 and 95 in Experiment Two, the average
acceleration/deceleration rates in each speed disturbance, and the spacing between
vehicles 170 and 171, 80 and 81, and 95 and 96 at the beginning of the speed
disturbances, the minimum distance headways during the experiments, and the
upper boundary condition for speed absorption in Scenario 1 obtained by Equation
4-16. From Table 5-8 one can confirm that the speed disturbances discussed in this

section are less than the upper boundary conditions calculated by Equation 4-16.

69

Table 5-8 Case Study for Scenario 1 - Acceptable-Acceptable Expected State

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Experiment, Vehicle ID, and Speed Disturbance
Summary of Fata
1 2 3
Steps in which a speed disturbance took 66 — 76 82 - 93 111 -
place 1 18
Minimum distance headway of vehicle 198 198 198
o 171 (ft)
5 Acceleration (ft/secz) 2.0 3.0 1.0
E Deceleration (ft/secz) 1.5 4.0 1.3
O
E 170
§ Initial spacing between 170 and 171(ft) 467 467 467
K
m Speed reduction (ft/sec) 4.0 5.0 1.0
Upper speed reduction boundary 22.0 30.0 17.4
condition for speed absorption (ft/sec)
Steps in which a speed disturbance took 34 — 37 43 - 47 51 — 55
place
Minimum distance headway of vehicle 208 208 208
81 (ft)
Acceleration (ft/secz) 4.0 3.5 4.0
Deceleration (ft/secz) 2.0 2.0 1.7
80
Initial spacing between 80 and 81(ft) 334 334 334
Speed reduction (ft/sec) 4.0 4.0 4.0
o
[E Upper speed reduction boundary 29.0 29.0 28.1
‘5 condition for speed absorption (ft/sec)
D
.g Steps in which a speed disturbance took 56 - 58 65 — 67 78 — 80
8 place
[33 Minimum distance headway of vehicle 139 139 139
96 (ft)
Acceleration (ft/secz) 1 2 1.0
Deceleration (ft/secz) 1.5 2 1.5
95
Initial spacing between 95 and 96 (ft) 181 181 181
Speed reduction (ft/sec) 1.0 1.0 1.0
Upper speed reduction boundary 7.0 9.0 7.0
condition for speed absorption (ft/sec)

 

70

 

5.3.2 Cases for Scenario 2 - Acceptable-Unacceptable-Acceptable Expected State

As defined in Chapter 2 this scenario describes a car-following behavior
such that when the lead vehicle has a speed disturbance (speed reduction), the
Expected State of its following vehicle is acceptable, but during the recovery of this
speed the Expected State of the following vehicle becomes unacceptable and the
following vehicle takes a control action until the Expected State of the following
vehicle returns to acceptable. In this speed interaction, the following vehicle
generates a speed disturbance in response to the speed disturbance of the lead
vehicle. In this case a speed disturbance of the lead vehicle can only be partially
absorbed by the spacing between the first and second vehicle.

Figure 5-1 a of Experiment One shows that vehicle 166 developed a speed
disturbance at steps 49-63. After a two-second delay vehicle 167 responded to it at
steps 55-66. Figures 5-2 i and q of Experiment Two also show that vehicle 79 and
vehicle 87 developed speed disturbances at steps 37 —40, and 63-70 respectively.
After a time delay vehicle 80 and 88 responded to them respectively at steps 40-42,
and 65-72. From the figures one can see that the speed disturbances identified in this
section were partially absorbed and partially transmitted backward. Table 5-9
summarizes the acceleration/deceleration rates, the minimum distance headways
during the experiments, the spacing, the speed reductions and the upper boundaries
on the magnitude of a single speed disturbance that a single spacing can absorb
obtained by Equation 4-17. For reader convenience in locating the cases, Table 5-9
also gives the steps during which the speed disturbances took place and the steps

during which they were responded to by the following vehicles.

71

Table 5-9 Case Study for Scenario 2 — Acceptable-Unacceptable-Acceptable Expected

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

State
Experiment, Summary of Data
Vehicle ID
Steps in which a speed disturbance took place 49-63
Steps in which the speed disturbance responded 55-66
8 6 Minimum distance headway of vehicle 167 (ft) 77
16
% Acceleration (ft/secz) 1.5
E Deceleration (ft/secz) 2.0
'c
8. Initial spacing between vehicles 166 and 167 (ft) 84
><
m Speed reduction of 166 (ft/sec) 2.0
Upper speed reduction boundary condition for 4.0
Partial absorption of speed disturbance (ft/sec)
Steps in which a speed disturbance took place 37-40
Steps in which the speed disturbance responded 40-42
79 Minimum distance headway of vehicle 80 (a) 73.1
Acceleration (ft/secz) 4.0
Deceleration (ft/secz) 3.5
Initial spacing between vehicles 79 and 80 (ft) 77
Speed reduction of 80 (ft/sec) 4.0
o
[E Upper speed reduction boundary condition for 4.0
{3 Partial absorption of speed disturbance (ft/sec)
{8: Steps in which a speed disturbance took place 63-70
é Steps in which the speed disturbance responded 65-72
B1
87 Minimum distance headway of vehicle 88 (ft) 60.0
Acceleration (ft/secz) 3.3
Deceleration (ft/secz) 4.3
Initial spacing between vehicles 87 and 88 (ft) 80
Speed reduction of 88 (ft/sec) 13
Upper speed reduction boundary condition for 13.3
Partial absorption of speed disturbance (ft/sec)

 

 

 

72

 

From Table 5-9 one can find that in each case the upper boundary on the

magnitude of a speed disturbance which can partially be absorbed by a single

spacing obtained by Equation 4-17 is larger than the corresponding speed

disturbance. The values of the speed disturbances developed by vehicles 166, 79,

and 87, the values of the speed disturbances absorbed by the spacing, the values of

the speed disturbances propagated through their immediately following vehicles,

and the upper boundary conditions for each case calculated by Equation 4-17 are

listed in Table 5-10.

Table 5-10 Values of the Developed, Absorbed, Propagated, and Calculated Speed

 

 

 

 

 

Disturbance
Vehicle Speed Disturbance
ID Developed Propagated Absorbed Calculated
(ft/sec) (ft/sec) (ft/sec) (ft/sec)
166 l 2.0 2.0 0.0 4.0
79 4.0 1 3.0 4.0
87 13.0 11.0 2.0 13.3

 

 

 

 

 

 

5.3.3 Cases for Scenario 3 - Unacceptable Expected State at Target-Risk Level

Scenario 3 defined in Chapter 2 describes a situation where a vehicle follows

its lead vehicle at a minimum headway and the driver of the following vehicle is

always alerted and can be triggered to take an immediate control action at anytime

once receiving a brake signal from the vehicle in front.

By observing Figures 5-1 and 5-2, one can find that when a vehicle is

following its leading vehicle at a minimum headway, most of the time it responded to

73

 

a speed disturbance from its leading vehicle with a larger speed reduction than that of
the leading vehicle. In those cases Equation 4—17 can always be met.

Four cases in the simulations were found where the following vehicles
followed the leading vehicle at approximately minimum time headway and
responded to a speed disturbance with one-second perception-reaction delay and
their speed reductions were either less than or equal to that developed by their
leading vehicles. In Experiment One, Figure 5-1 d shows that when vehicle 169
developed a speed disturbance at steps 64-75 vehicle 170 maintained a small
spacing from it and responded to it at steps 67-76, with a one-second perception-
reaction delay. In Experiment Two, Figures 5-2 (1, e, and n show that vehicles 74, 75
and 84 developed speed disturbances at steps 33-36, 40-49, and at steps 35-39 and
43-47. Vehicle 75 kept the spacing from vehicle 74 almost at its minimum safe
headway and responded to the speed disturbance at steps 34-37. Vehicle 76 and
vehicle 85 kept small spacing and responded to the speed disturbances at steps 42-
50, and at steps 36-41 and 44-49. The perception-reaction delay for vehicles 170,
75, 76, and 85 was one second. The features of the speed disturbances and the speed
interaction between vehicles 169 and 170, 74 and 75, 75 and 76, 84 and 85 and the
boundary conditions calculated by Equation 4—17 are listed in Table 5-11. From
Table 5-11 one can find that in each case the speed reduction is less than the

corresponding upper boundary conditions calculated by Equation 4-17.

74

 

Table 5-11 Case Study for Scenario 3 - Unacceptable Expected State at Target-Risk

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Level
Vehicle ID Summary of Data
Steps in which a speed disturbance took place 64-75
5 Steps in which the speed disturbance responded 67-76
‘5 Speed reduction of 169 (ft/sec) 3.0
E 169 Acceleration (ft/secz) / Deceleration (ft/secz) 2.0 / 3.0
g Initial spacing between vehicles 169 and 170 (ft) 58
133 Minimum distance headway of vehicle 170 (ft) 53
Upper boundary condition for Speed absorption (ft/sec) 4.6
Steps in which a speed disturbance took place 33-36
Steps in which the speed disturbance responded 34-37
74 Speed reduction of 74 (ft/sec) 4.0
Acceleration (ft/secz) / Deceleration (ft/secz) 3.0 / 4.0
Initial spacing between vehicles 74 and 75 (ft) 68
Minimum distance headway of vehicle 75 (ft) 65
Upper boundary condition for Speed absorption (ft/sec) 4.5
Steps in which a speed disturbance took place 41-48
Steps in which the speed disturbance responded 42-50
75 Speed reduction of 75 (ft/sec) 7.0
Acceleration (ft/secz) / Deceleration (ft/secz) 1.8 I 3.0
0 Initial spacing between vehicles 75 and 76 (ft) 77
E? Minimum distance headway of vehicle 76 (ft) 63
E Upper boundary condition for Speed absorption (ft/sec) 7.0
'1: Steps in which a speed disturbance took place 35-39
1% Steps in which the speed disturbance responded 36-41
Speed reduction of 84 (ft/sec) 6.0
84 Acceleration (ft/secz) / Deceleration (ft/secz) 3.2 / 3.2
Initial spacing between vehicles 84 and 85 (ft) 64
Minimum distance headway of vehicle 85 (ft) 53
Upper boundary condition for Speed absorption (ft/sec) 7.7
Steps in which a speed disturbance took place 4347
Steps in which the speed disturbance responded 44-49
Speed reduction of 84 (ft/sec) 8
84 Acceleration (ft/secz) / Deceleration (ft/secz) 4.0 / 3.7
Initial spacing between vehicles 84 and 85 (ft) 66
Minimum distance headway of vehicle 85 (ft) 53
Upper boundary condition for Speed absorption (ft/sec) 9.2

 

 

 

75

5.4 Summary

Two experiments of car-following behavior were simulated by using TSIS.
The dynamic speed, position and acceleration and/or deceleration data of randomly
selected lines of individual vehicles were collected in a step-by-step manner. The
cases in which speed disturbances were absorbed completely or partially by spacing
were identified from the experimental data and studied. The upper boundary on the
magnitude of the speed disturbance that the spacing can absorb in each case was
calculated by either Equation 4-16, or 4-17. The cases studied verified the models

obtained in Chapter 4.

76

Figure 5-1 Speed Interaction between Vehicles in Experiment One

 

 

speedWs) Veh. 166 Veh. 167

 

110

 

1 21 41 61 81 101 121

 

 

 

5-1 3 Vehicles 166 and 167

 

 

speed (ft/S) ——veh. 167 —veh. 168
110

 

 

5-1 b Vehicles 167 and 168

77

 

Figure 5-1 Speed Interaction between Vehicles in Experiment One (continued)

 

speed (ft/s) —veh. 168
110

Veh 169

 

 

1 21 41 61 81 101 121

 

 

 

5-1 c Vehicles 168 and 169

 

 

speed(ft/s) Veh 169 —veh 170
110- - - - ,
100
90
80 7 step
1 21 41 61 81 101 121

 

 

 

5-1 d Vehicles 169 and 170

78

Figure 5-1 Speed Interaction between Vehicles in Experiment One (continued)

 

 

 

 

 

 

 

speed(fi/S) ——veh 17o -—\eh 171
110
100
9° H1
{'1 til
30 _ _. . step
1 21 41 61 81 101 121
5-1 e Vehicles 170 and 171
spee(fl/s) —\eh 171 ———\eh 172
110
100
90
JL
U \_l
ste
80 p
1 21 41 61 81 101 121

 

5-1 f Vehicles 171 and 172

79

 

 

Figure 5-1 Speed Interaction between Vehicles in Experiment One (continued)

 

speed(ft/s) ————veh 172 —veh 173
100
90
80 _ 7 7 ,, ,,_ 7_ 7‘ step
1 21 41 61 81 101 121

 

 

 

5-1 g Vehicles 172 and 173

 

speed(ft/s) —veh 173 —veh 174
90
80 .. step
1 21 41 61 81 101 121

 

 

 

5-1 h Vehicles 173 and 174

80

Figure 5-2 Speed Interactions between Vehicles in Experiment Two

 

 

speed (ft/sec)

105

95

75

—veh70 —veh71

BSA/\AAAAAAAAVM

vVVVVVVVVV V‘

 

 

 

 

 

 

step
1 1 1 21 31 41
5-2 a Vehicles 70 and 71
veh 71 web 72
105
A 95
6
E
i
85
75 steps

11

21 31 41 51

 

5-2 b. Vehicles 71 and 72

81

 

 

Figure 5-2 Speed Interactions between Vehicles in Experiment Two (continued)

 

veh 72 ten 73

 

 

105

 

 

 

 

 

 

 

8
0)
g
i
75 , steps
1 11 21 31 41 51
5-2c Vehicles 72 and 73
veh 73 veh 74
105....., W-, A” ., n. _, ,, A , ,,
A 95
.9,
g 85
steps
75 ,
1 11 21 31 41 51

 

 

 

5-2 (1 Vehicles 73 and 74

82

Figure 5-2 Speed Interactions between Vehicles in Experiment Two (continued)

 

 

 

 

 

 

 

 

 

 

veh 74 veh 75
105
A 95
6
E,
”’ 85
steps
75 - , -
1 1 1 21 31 41 51
5-2 e Vehicles 74 and 75
veh 75 van 76
105
A 95 l
:3,
g" 85
steps
75
1 1 1 21 31 41 51

 

 

 

5-2 f Vehicles 75 and 76

83

Figure 5-2 Speed Interactions between Vehicles in Experiment Two (continued)

 

veh 76 veh 77

 

 

105

95

 

speed (ft/sec)
/

85

75 steps

 

 

 

5-2 g Vehicles 76 and 77

 

 

 

 

veh 77 veh 78
105
g 95
«E-L
i x \n.
85
steps
75
4 14 24 34 44 54

 

 

 

5-2 h Vehicles 77 and 78

84

Figure 5-2 Speed Interactions between Vehicles in Experiment Two (continued)

 

 

 

 

 

 

 

 

 

veh 78 veh 79
105
A 95
6
e
g 85
75 ,, , - ~ , - - - ~
5 15 25 35 45 55 steps
5-2 i Vehicles 78 and 79
veh 79 veh 80
105 7 - - - - - - - -

speed (ft/sec)

 

75

6 16 26 36 46 56 “995

 

 

 

5-2 j Vehicles 79 and 80

85

Figure 5-2 Speed Interactions between Vehicles in Experiment Two (continued)

 

veh 80 web 81

 

 

105

i. \AMA/‘x,
.5 VVVV

 

 

 

 

 

 

7s
7 17 27 37 47 57 67 steps
5-2 k Vehicles 80 and 81
veh e1 veh 82
105
g 95
g

g 85
75

9 19 29 39 49 59 69 Steps

 

 

 

5-2 1 Vehicles 81 and 82

86

Figure 5-2 Speed Interactions between Vehicles in Experiment Two (continued)

 

 

 

 

 

 

 

 

 

veh 82 veh 83
105
g 95
E,
g- 85
75 , , , r ,
1 1 21 31 41 51 61 steps
5-2 m Vehicles 82 and 83
veh 83 veh 84
105 7 , -
l
g 95
g .
1 x *
(I)
75 steps
13 23 33 43 53 63

 

 

 

5-2 11 Vehicles 83 and 84

87

Figure 5—2 Speed Interactions between vehicles in Experiment Two (continued)

 

 

 

 

 

 

 

 

 

veh 84 veh 85
1 15
105
i 95
. /
85 \
75 ’ ’ ' ' steps
14 24 34 44 54 64
5-2 0 Vehicles 84 and 85
veh 85 veh 86
1 10
100
g
g 90
(I)
80
70 step
15 25 35 45 55 65

 

 

 

5-2 p Vehicles 85 and 86

Figure 5-2 Speed Interactions between Vehicles in Experiment Two (continued)

 

 

 

 

 

 

 

 

 

 

veh 86 veh 87
1 10
100
61
g 90
i I
80
7O __ I I, step
1 8 28 38 48 58 68
5-2 q Vehicles 86 and 87
veh 87 veh 88
1 10
100
3:; 90 V
i
80
70 step
19 29 39 49 59 69

 

 

 

5-2 r Vehicles 87 and 88

89

Figure 5-2 Speed Interactions between Vehicles in Experiment Two (continued)

 

 

 

 

 

 

 

 

 

 

veh 88 veh 89
105
A 95
8
m
e
"’ 85
75 ~ ~ -
20 3O 40 50 60 70 step
5-2 8 Vehicles 88 and 89
veh 89 veh 90
105 -
A 95
6
s
\ . -
6 u v
U’ 85
75
21 31 41 51 61 71 step

 

 

 

5-2 t Vehicles 89 and 90

90

Figure 5-2 Speed Interactions between Vehicles in Experiment Two (continued)

 

 

 

 

 

 

 

 

 

 

 

veh 90 veh 91
105
g 95
5
"’ 85
75
22 32 42 52 62 72 step
5-2 u Vehicles 90 and 91
veh 91 veh 92
105
95 ‘
g I
(D
E.
g 85
75
23 33 43 53 63 73 step

 

 

 

5-2 v Vehicles 91 and 92

9|

Figure 5-2 Speed Interactions between Vehicles in Experiment Two (continued)

 

 

 

 

 

 

 

 

 

 

veh 92 veh 93
105
A 95
8 /
(I)
9
g" 85
75 , -
25 35 45 55 65 75 step
5-2 w Vehicles 92 and 93
veh 94 veh 95
105
g 95
s
"’ 85
75 step
28 38 48 58 68 78

 

 

 

5-2 x Vehicles 94 and 95

92

Figure 5-2 Speed Interactions between Vehicles in Experiment Two (continued)

 

 

 

veh 95 veh 96

105

A 95
6
e

g 85

75

30 40 50 60 70 80 step

 

 

 

 

5-2 y Vehicles 95 and 96

93

CHAPTER 6 FINDINDS AND RECOMNIENDATIONS

6.1 Findings

The study discussed the single-lane car following behavior when a speed
disturbance from a lead vehicle occurs. A concept of Expected State-Control Action
Chains and three scenarios of single-lane car-following behavior situations were
developed (Table 2-1). The minimum dynamic spacing for each scenario was analyzed
and defined (Table 3-1). The existence of an upper boundary of a speed disturbance
that a spacing can absorb was proved through Theory 4-1, Lemma 4-1 and Theory 4-2.
The mathematical models were obtained to calculate the upper boundary of the
magnitude of a speed disturbance that a single spacing or multiple spacings can absorb
(Equations 4-4, 4-13. Moreover, the conditions for local and asymptotic stability
were determined (Equations 4-16 and 4-17). Two simulation experiments were
conducted and studied. The cases identified from the experimental data verified the
models obtained in Chapter 4. The most important findings of this work are
summarized in Table 6-1.

The findings of this work demonstrate that speed fluctuations of individual
vehicles do play a role in traffic stability, and can be used in car-following analysis to

find upper boundary conditions for the stability at the microscopic level.

94

Table 6-1 Summary of Findings

 

Findings

Reference Numbers of Figures,

Tables, or Equations in the Contents

 

 

Concept of Expected State-Control Action Table 2-1
Chain (Car-Following Prcess)
Minimum Dynamic Spacing Table 3-1

 

Existence of an Upper Boundary of 3 Speed

Disturbance that a Spacing Can Absorb

Definitions 4-1 through 4-6

Theories 4-1 and 4-2, Lemma 4-1

 

Mathematical Models to Calculate an Upper
Boundary of the Magnitude of a Speed

Disturbance that a Single Spacing Can Absorb

Equations 44, 4-13

 

Conditions for Local and Asymptotic Stability

Equations 4-16 and 4-17

 

 

Model Verification

 

Sample Cases from the Two

Simulation Experiments

 

6.2 Recommendations for Further Research

This work discussed a single speed disturbance and its absorption at

microscopic level. The findings established a basic framework for the research on

multiple speed disturbances and their absorption. As we pointed out in Chapter 2,

errors and deviation from the assessment or control skill of human beings are

unavoidable. When a following vehicle, n+1, reduces its speed in response to a speed

disturbance from the lead vehicle, 11, vehicle n+1 may create a speed disturbance of its

95

 

own because of the errors. In other words the speed reduction of a following vehicle
includes two parts: speed reduction in response to the speed disturbance from the lead
vehicle and speed disturbance from its assessing and/or controlling errors. The errors
of assessment and/or control can be regarded as a speed disturbance produced by the
following vehicle n+1. Therefore, further research is needed to understand the
phenomenon of multiple speed disturbances and their absorption in'a group of vehicles
that follow one another. The upper boundary conditions of multiple speed disturbances
that can be absorbed by a vehicular platoon will result in a macroscopic model for

traffic stability.

96

APPENDICE

97

APPENDIX 1 DATA OF EXPERIMENT ONE

98

 

 

 

 

 

 

 

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mm mm o FmNN m3 3 v- momN 3 cm v- NFmN 5 mm N SN 3 m- mmom mm
mm mm o NNNN X: No v- owNN mm vm v- vmmN 3 mm N NmmN S m- vmom no
mm mm o mmmN 2: 3 F.- NmNN Nm 3 v- mmmN mm mm N mmmN 3 m- Now we
mm mm m- mooN moF om o NNNN Fm mm v- NNmN mm mm F momN mm F SN 8
mm mm m- mmmN moF om o NmoN Fm mm v- max-N mm 8 F mNmN an F meN No
mm mm m- vooN BF cm 0 NomN Fm mm v- max-N mm 8 F ommN an F mmmN Fm
mm Nm N- mFmN SF Fm o FmoN mm om N max-N 3 8 N- FNmN 3 m momN om
om Nm N- Nva BF Fm o FooN mm om N wouN mm mm N- mmNN 5 m QBN mm
mm: :9, as: 395 g a: :9, as: was: E a: :9, 35 295 a we :9, is 655 cu Ea 395 a: 8%
EB. 86% moo< .38 60: 693,-... 084. LEE So: 08% moo< .38 E0: 8QO wood. 3:: peace. .moo< .EE a.an
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36:53:03 95 EmEtmaxw - cone-.385 0526.6; .50 mag-6.9% No Ema

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33 33 3 333 33F 33 N 33N3 33 33 3- 3333 33 33 3 3333 33 N 3N33 3:
33 33 3 333 32 33 N 3NN3 33 33 F1 N333 33 33 3 NF33 33 N 3333 3:
33 33 F- 323 33F 33 N 333 N3 33 3 N333 33 33 3- 3333 33 N 3333 3:
33 33 F- 333 32 33 N 333 3 33 3 N3N3 33 33 3- 3333 33 N 3333 3:
33 33 F- F333 33F 33 N 333 33 33 3 N33 33 33 3- 3333 33 N :33 NFF
33 33 F N333 33F 3 F- 333 33 33 3 NFN3 33 33 3 33N3 33 F- N333 FFF
33 33 F 3N33 N3F 3 F- F333 33 33 3 333 33 33 3 33N3 33 F- 3333 3:
33 33 F 3333 SF 33 F- 3333 33 33 3 33 33 33 3 33N3 33 F- 3N33 33F
33 33 3 3333 N2 33 N- 3N33 33 33 N- 83 33 33 N 3FN3 33 N- 33N3 33F
33 33 3 3333 32 33 N- 3333 33 33 N- 3333 33 33 N 33 33 N- 33N3 33F
33 33 3 3333 33F 33 N- 3333 N3 3 N- 3333 33 33 N 33 33 N- 33N3 33F
33 33 3- 3333 32. 33 3 3333 N3 33 3. F333 33 33 N 3NF3 33 3 3FN3 33F
33 33 3- F333 32 33 3 3333 N3 33 3- NF33 33 33 N 3333 33 3 33 33F
33 33 3- NN33 33F 33 3 3333 F3 33 3- 3333 33 33 N 3333 33 3 $3 33F
33 33 F- N333 33F 33 F- 3333 N3 33 3 3333 33 33 N- 3333 33 3 NNF3 N3F
33 33 F- N333 33F 33 F- 3N33 33 33 3 3N33 33 33 N- 3333 33 3 3333 SF
33 33 F- 3333 33F 33 F- F333 33 33 3 3333 33 33 N- 3333 33 3 3333 33F
33 3 F N333 33F 33 F- F333 33 33 3 3333 33 33 N- F333 33 3 3333 33
F3 F3 F F333 33F 33 F- N333 33 33 3 3333 33 33 N- NN33 33 3 3333 33
N3 3 F F333 33F 33 F- 3333 33 33 3 3333 33 33 N- 3333 33 3 3333 33
33 33 3 333 32 33 N 3333 33 33 F 3333 33 33 N 3333 33 N- 3333 33
33 33 3 3333 33F 33 N 3333 33 33 F 3333 33 33 N 3333 .33 N- FN33 33
33 33 3 3333 33F 33 N 333 33 33 F 3N33 33 33 N 3333 33 N- 3333 33
N3 33 3 FN33 33F 33 N 3333 33 33 3- N333 33 33 N 3333 33 N- 3333 33
F3 33 3 N333 FFF 33 N 3333 N3 3 3- 3333 33 33 N 3333 33 N- 3333 N3
3 33 3 3333 NFF 33 N 3N33 F3 33 3- 3333 33 33 N 333 33 N- 3333 F3
33 3 F 3333 NFF 33 N 3333 F3 33 3- 3333 33 33 3 3333 33 N 3333 33
33 3 F 3333 NFF 33 N 3333 N3 33 3- 3333 33 33 3 3333 33 N 3333 33
33 3 F 3333 FFF 33 N 3333 33 33 3. 3333 33 33 3 N333 33 N 23 33
33F 33> 33212333 93 33F 33> 333313333 33 33F 33> 3333 3333 33 33F 33> 3333 33133 93 3323 @333 33 33.3
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Anmaczcoov ch chEtmaxm - 5:33:35 0:330:03. :30 mam-3-3.935 30 £30

102

C3233 32333533 3395 3 2 C325 32333333 .333: F Ea: 95:33 3333 3.3325 n be ”362

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

N3 33 N 333 33F 33 N- 3333 F3 33 N 3333 33 33 3- 33 33 N 33N3 F3F
33 33 N 3333 33F 33 N- 3333 N3 33 N F333 33 33 3- 333 33 N 83 33F
F3 33 N 3333 32 33 N- 333 3 33 N NN33 33 33 3- 333 33 N 333 33F
33 33 3 3N33 33F 33 N 3333 3 3 N 3333 3 33 N- 3333 33 F- 33 33F
33 33 3 3333 33F 33 N 3333 33 3 N 3333 33 33 N- 3333 33 F- 333 33F
33 33 3 3333 32 33 N 3333 33 33 N 3333 33 33 N- FN33 33 F- 333 33F
33 33 3 3333 32 33 N 3333 33 33 3- 3333 33 33 N F333 33 3- 3333 33F
N3 33 3 3333 3:. 33 N 3333 N3 33 3- 3333 33 33 N N333 33 3- 3333 33F
F3 33 3 3333 FFF 33 N F333 3 33 F7 N333 33 33 N 3333 33 3- FN33 33F
33 F3 3 N333 FFF 33 N NF33 F3 33 3- 3N33 33 33 F 3333 33 3 F333 N3F
33 F3 3 3N33 FFF 33 N 3333 F3 33 3- 3333 33 33 F 3333 33 3 N333 E
33 F3 3 3333 FFF 33 N 3333 F3 33 3- 3333 33 33 F 3333 33 3 3333 33F
33 N3 3- 3333 38 33 N 3N33 N3 33 3 3333 33 33 N- 333 33 N 3333 3NF
33 N3 3- 3333 33F 33 N 3333 33 33 3 3333 33 33 N- 3333 33 N 3333 3NF
3 N3 3- 333 33F 33 N 3333 33 33 3 3333 33 33 N- F333 33 N 3333 3NF
33 33 3- 3333 33F 33 3- N33 33 33 3 3333 33 33 F- N333 33 3 333 3NF
33 33 3- 3333 NS 33 3- 333 33 33 3 333 33 33 F- 3333 33 3 3333 3NF
33 33 3- F333 32 33 3- 3333 33 33 3 3333 33 33 F- 3333 33 3 3333 3NF
33 33 3 F333 32 33 3- 3333 33 33 3 3333 33 33 N 3333 33 N- 3N33 3NF
33 33 3 F333 38 33 3- 3333 33 33 3 3333 33 33 N 333 33 N- N333 NNF
33 33 3 N333 33F 33 F1 N333 33 33 3 N333 33 33 N 3333 33 N- 3333 FNF
33 33 3 NF33 N2 33 F N333 N3 33 3- 3333 33 33 N 3333 33 F- 3333 3NF
33 33 3 N3N3 33F 33 F N333 F3 33 3- 3333 33 33 N 3N33 33 F- 333 3:
33 33 3 33N3 33F 33 F 333 33 33 3- 333 33 33 N 3333 33 F- 3333 3:
33 33 3 3NN3 33F 33 N 33N3 33 33 3- 3333 33 33 3 3333 33 N 3333 3:
33F 33> 3333 333333 33 33F 33> 3% 333333 33 33F 33> 3335 333333 a? 33F 33> 3333 3333 33 3333 333333 33 3333
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8335:3033 3:0 E3Et3axm - 823.385 9:26:03 ..30 3:33.935 30 $30

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3FF mm o 333 No mm F- vooF mNF mm o 33oF mom mm o mFNF mN
mFF mm o 333 mm mm F- 333 3NF mm o mmoF own 33 o 3mFF 3N
mFF mm o NNm mm 33 F- va 3NF mm o mNoF mmm mm o meF 3N
FNF mm o 333 om 33 F- oFm mNF mm o oooF omm mm o mNFF 3N
NNF mm o 333 Nm vm F- 333 mNF mm 0 F33. 333 mm o ooFF mN
mNF mm o 3N3 mm mm F- 333 mNF mm 0 N33 va mm o F3oF 3N
3NF mm o me mm mm F- mFm mNF mm o mFm can on o NvoF mN
mNF mm o 333 ooF mm F- 333 3NF mm o 333 N33 33 o mFoF NN
mNF mm o 3N3 voF mm F- F33 mNF mm o 333 on mm o 333 FN
3NF mm o 333 33F 33 F- an 3NF mm o 3N3 FNm mm o 333 ON
3NF mm o F33 oFF 33 F- 333 mNF mm o 333 oFm mm o on 3F
3NF mm o 3N3 vFF 33 F- 333 3NF mm o 333 OF” 33 o 333 3F
3NF mm o 333 mFF mm F- FNm mNF mm o mm3 3cm mm o 333 3F
3NF mm 3 N33 NNF mm F- mam 3NF mm o FF3 mmN mm o ovm 3F
3NF mm F 3N3 3NF mm F 333 3NF mm 0 N33 FmN mm o FFm mF
3NF mm F mam FmF mm F NNm mNF on o mmm mmN mm 0 N33 3F
mNF 33 F vow mmF mm F mmv 3NF mm o 3N3 m3N mm o mmn mF
mNF 33 N me mmF mm N 333 mNF on o mmm N3N mm o 3N3 NF
va 33 N mmN va mm N 3N3 mNF mm o com owN mm 0 mac FF
3NF mm N ooN 33F mm N can 3NF mm 0 3mm omN on o 333 oF
mNF mm N va FmF mm N 3mm mNF mm o mom mmN mm 0 3mm 3
mNF mm N NON 33F om N mNm mNF mm o 333 va mm 0 mom 3
mNF mm N 03F 3mF mm N mmN mNF mm o 033 OVN mm o 333 3
mNF mm N me 03F 33 N FmN 3NF mm o FNv mmN mm o 033 o
mNF mm N 30F 33F 33 N mNN ONF mm o Nmm mmN mm o NFm m
NNF no N 33 30F 33 N 33F mNF mm o 333 mFN mm o 333 3
mmF Fm 3 me mNF on o mmm NFN mm o 333 m
F3F Fm v mmF mNF ow 0 won mON mm o mmv N
M3F F3 3 33F mNF mm o 33N mmF mm o mow F
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3333:3303 3:0.Fc3E333xm - 3033385 9:26:33 330 3:33.335 30 3:30

104

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

33 33 F 333F 33 33 3 333F 3NF 33 3 3N3F 333 33 3 333N 33
33 33 3 333F 33 33 F 333F 3NF 33 3 333F 333 33 F- 3N3N 33
33 33 3 FN3F 33 33 F 333F 3NF 33 3 333F 333 33 F- 333F 33
33 33 3 333F 33 33 F 333F 3NF 33 3 333F F33 33 F- 333F 33
33 33 F- 333F 33 33 3 333F 3NF 33 3 333F. 333 33 3 333F 33
33 33 F- 333F 33 33 3 3N3F 3NF 33 3 333F 333 33 3 333F 33
33 33 F- 333F 33 33 3 N33F 3NF 33 3 N33F 333 33 3 F33F N3
33 33 3 333F 33 33 N- 333F 3NF 33 3 3N3F N33 33 3 N33F F3
33 33 3 333F 33 33 N- 333F 3NF 33 3 333F 333 33 3 3N3F 33
33 33 3 3F3F 33 33 N- 333F 3NF 33 3 333F 333 33 3 333F 33
33 33 F- 333F 33 33 F- 333F 3NF 33 3 333F 333 33 3 333F 33
33 33 F- 333F 33 33 F- 333F 3NF 33 3 333F F33 33 3 333F 33
33 33 F- 333F 33 33 F- 3F3F 3NF 33 3 333F 3N3 33 3 333F 33
33 F3 3- 333 F3 33 3 333F 33F 33 o 333F 333 33 o 333F 33
33 F3 3- 333F N3 33 3 333F 3NF 33 3 3N3F NN3 33 3 333F 33
33 F3 3- 333F N3 33 3 3N3F 3NF 33 3 F33F 3F3 33 3 3N3F 33
33 33 3- 333F 33 33 3 333F 3NF 33 3 N33F 3F3 33 3 F33F N3
F3 33 3- 33NF 33 33 3 333F 3NF 33 3 333F 3F3 33 3 N33F F3
N3 33 3- 33NF 33 33 3 333F 3NF 33 3 333F 333 33 3 333F 33
33 33 3 3FNF 33 33 3 333F 33F 33 3 333F 333 33 3 333F 33
33 33 3 N3FF 33 33 3 33NF 3NF 33 3 333F 333 33 3 333F 33
33 33 3 33FF 33 33 3 33NF 3NF 33 3 3F3F 333 33 3 333F 33
F3F 33 3 3FFF F3 F3 3 3FNF 3NF 33 3 33NF 333 33 3 3F3F 33
33 F 33 3 339 N3 F3 3 33F F 3N F 33 o 33N F 333 33 o 333 F 33
33F 33 3 N33F 33 F3 3 33FF 3NF 33 3 F3NF 333 33 3 333F 33
33F 33 3 3F3F 33 N3 3 3NFF 3NF 33 3 N3NF N33 33 3 F33F 33
3FF 33 3 333 33 N3 3 333F 3NF 33 3 33FF 333 33 3 N33F N3
3FF 33 3 333 33 N3 3 333F 3NF 33 3 33FF 333 33 3 33NF F3
3FF 33 3 3N3 33 33 F- 333F 3NF 33 3 3FFF 333 33 3 33NF 33
3: 53> 35 332? 33 N: 33> $33 433323 33 Ft 33> is @323 a 3: 53> 35 3333 33 33.3
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3333:3303 3:0 E3Et3axw - 30:33:35 3:56:33. 330 3334-2353 30 3330

105

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

33 33 3 3F3N 33 33 N- 333N 3NF 33 3 N33N N33 33 3 F33N 33
33 33 3 333N 33 33 N- 333N 3NF 33 3 333N 333 33 3 N33N 33
33 33 3 333N 33 33 N- 333N 3NF 33 3 333N 333 33 3 333N 33
33 33 3 3N3N 33 33 F 3F3N 3NF 33 3 333N 333 33 3 333N 33
33 33 3 F33N 33 33 F 333N 3NF 33 3 333N. 333 33 3 333N 33
33 33 3 333N 33 33 F 333N 3NF 33 3 3F3N 333 33 3 333N N3
33 33 3- 333N F3 33 3 3N3N 3NF 33 3 333N 333 33 3 3F3N F3
N3 33 3- 3F3N F3 33 3 333N 3NF 33 3 333N 333 33 3 333N 33
F3 33 3- 333N F3 33 3 333N 3NF 33 3 F33N 333 33 3 333N 33
F3 33 3- 333N F3 33 F- F33N 3NF 33 3 N33N 333 33 3 F33N 33
N3 33 3- F33N 33 33 F- 3F3N 3NF 33 3 333N 333 33 3 N33N 33
33 33 3- N33N - 33 33 F- 333N 3NF 33 3 333N 333 33 3 333N 33
33 33 3 F3NN 33 33 3- 333N 3NF 33 3 3F3N 333 33 3 333N 33
33 33 3 F3NN 33 33 3- 3N3N 3NF 33 3 333N 333 33 3 3F3N 33
33 33 3 FFNN 33 33 3- 333N 3NF 33 3 333N 333 33 3 333N 33
33 33 3 F3FN 33 33 3 33NN 3NF 33 3 3N3N 333 33 3 333N N3
33 33 3 N3FN 33 33 3 F3NN 3NF 33 3 333N 333 33 3 3N3N F3
33 33 3 3NFN 33 33 3 NFNN 3NF 33 3 F3NN 333 33 3 333N 33
33 33 3 333N 33 33 3 N3FN 3NF 33 3 N3NN 333 33 3 F33N 33
33 33 3 333N F3 33 3 N3FN 3NF 33 3 3FNN 333 33 3 N33N 33
33 33 3 333N F3 33 3 3NFN 3NF 33 3 33FN 333 33 3 3F3N 33
33 33 3- 3F3N N3 33 3 333N 3NF 33 3 33FN 333 33 3 33NN 33
33 33 3- N33F F3 33 3 333N 3NF 33 3 3NFN 333 33 3 33NN 33
33 33 3- 333F 33 33 3 333N 3NF 33 3 333N 333 33 3 3NNN 33
33 33 3 3N3F 33 33 N- 333N 3NF 33 3 333N 333 33 3 33FN 33
33 33 3 333F 33 33 N- F33F 3NF 33 3 333N 333 33 3 33FN N3
33 33 3 333F 33 33 N- 333F 3NF 33 3 FF3N 333 33 3 33FN F3
33 33 F 333F 33 33 3 3N3F 3NF 33 3 N33F 333 33 3 3FFN 33
33 33 F 333F 33 33 3 333F 3NF 33 3 333F F33 33 3 F33N 33
3: 53> A333 3333 9W N3F 53> A323 3333 333 Ft 53> 333 333335 33 32 53> A333 3333 33 333.3
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333:3:83 3:0 E3E_:33xw - 533.253 3:350:33 :30 353-2353 30 3330

106

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

33 33 3- 3333 33 33 F- F333 33F 33 3 F333 333 33 o 3333 3FF
33 33 3- 3333 33 33 F- 3F33 33F 33 o 3333 333 33 3 F333 3FF
33 33 3 3333 33 33 3. 3333 33F 33 o 3333 333 33 o 3333 3FF
33 33 3 3333 33 33 3- 3333 33F 33 o 3F33 333 33 o 3333 3FF
33 33 3 3333 33 33 3- 3333 33F 33 o 3333. 333 33 o 3F33 NFF
33 33 3 3F33 33 33 o 3333 33F 33 o 3333 333 33 o 3333 FFF
33 33 3 F333 33 33 o 3333 33F 33 o 3333 333 33 o 3333 3FF
33 33 3 3333 33 33 3 3333 33F 33 3 3333 333 33 o 3333 33F
33 33 3- 3333 33 33 3 3F33 33F 33 o 3333 333 33 o 3333 33F
33 33 3- 33F3 F3 33 3 3333 33F 33 3 F333 333 33 o 3333 33F
33 33 3- 33F3 F3 33 3 F333 33F 33 3 «F33 333 33 3 F333 33F
33 33 3- 33F3 F3 33 o 3333 33F 33 3 3333 333 33 o NF33 33F
33 33 3- 3FF3 F3 33 o 33F3 33F 33 o 3333 333 33 o 3333 33F
33 33 3- 3333 33 33 o 33F3 33F 33 o 3333 333 33 o 3333 33F
33 33 3 3333 33 33 3- 33F3 33F 33 o 33F3 333 33 o 3333 33F
33 33 3 «333 33 33 3- 33F3 33F 33 o 33F3 333 33 3 3333 FoF
33 33 3 3333 33 33 3- 3333 33F 33 o 33F3 333 33 o 3333 33F
33 33 3 3333 33 33 F- F333 33F 33 o 33F3 333 33 o 3333 33
33 33 3 3333 33 33 F- «~33 33F 33 o 3333 N33 33 o 3333 33
33 33 3 3333 33 33 F- 3333 33F 33 o 3333 333 33 3 F3F3 33
33 33 3- 3333 33 33 N 3333 33F 33 o 3333 333 33 o mmFm 33
33 33 3- 3333 F3 33 N 3333 33F 33 o 3333 333 33 o mmFm 33
33 33 3- F333 F3 33 N 3333 33F 33 o 3333 333 33 o 3333 33
33 33 3- 3333 F3 33 N 3333 33F 33 o 3333 333 33 o 3333 33
33 33 3- 3333 F3 33 N 3333 33F 33 3 3333 333 33 o 3333 N3
33 33 3- 3333 F3 33 N 3F3N 33F 33 o 3333 333 33 o 3333 F3
33 33 N 3333 33 33 N- 3333 33F 33 o 3333 333 33 o 3333 33
33 33 N 3333 33 33 3. F333 33F 33 o 3333 333 33 o 3333 33
33 33 N 3333 33 33 - 3333 33F 33 3 F333 333 33 o 3333 33
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63:53:83 3:0 E38333xm - 833.3333 3:350:03 :30 3:33.353 30 3330

107

C026 3335mm 625 2. 2 C025 «239.com .mmmc — Ea: 95cm. 3%... 9525 u .5 ”202

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Nm on v onv mm mm m- mme NNF mm o Nva vmv mm o Fmvv .vF
mm mm c mvrv mm on m- «NNV NNF mm o NNNV now an o Nva ow?
mm on v NvPv mm mm m- moNv on? on o vav Fov mm o vmmv NNF
mm «N v mmov mm mm o NNFV oar mm o mva omv mm o momv NNF
mm vm v mmov mm mm o mv.¢ on? mm o oONv mmv mm o ommv NNF
mm vm v omov mm mm o mrpv NNF mm o NNFV mmv mm o Nomv oar
Na Na m- Noov on No N mmov NNF mm o mvvv mmv NN o NNNV mmr
mm Nm m- «Nam- Po N» N mmov mN? mm o ONVV mmv Nm 0 NVNV «NF
mm No m- Nvmm .0 Na N omov mm_ mm o Fmov vmv Na 0 oNNv NNF
mm on v- NFNN Po «N P Foov va mm o Noov Pow mm o Fmpv NNF
NN om v. ommm Fm V» w NNmm NNF on o mmov new on o Nmpv Fm?
Nm om v- Noam Po Va P vvmm NNF on o moov mmv mm o NNF? on?
cm mm o NNNN om «N N- cram NNF mm o onN mow mm F vorv NNF
mm mm 0 mean mm vm N- amen NNF mm o Nvmm Nov mm F mNov NNF
mm mm o VNNN an «N N- omen NNN mm o NFNN ONV mm P mvov NNF
Nm mm v «VNN am No N- FNNN NNF mm o ommm ONQ mm P- Npov oN.
mm mm v «FNN an N» N- Noam NNF mm o roam ONv mm P. mmmm mNP
mm mm v mmom mm N» N- VNNN NNF mm o Nmmm ONV mm P- Foam «NF
mm vm N omen mm mm P «VNN NNF ea c Noam mow mm o Nmmm NN—
Nm vm N NNQN om mm P VFNN NNF mm o VNNN new mm o moan NNF
mm em N mama Nm mm . vmmm NNF mm o oVNm New on o mNmN FNF
mm mm v- onN Nm on v omen NNF on o NFNN oov mm o mvmm ON?
vm mm v- Nwmm No mm v mNom NNF mm o moon mow mm o NFNN me.
NN mm v- mrmm N@ we v Nmmm NNF mm o amen new on o mmNm NF.
mm Na v- mmvm Fe vm .- mmmm NNF mm o omen mow mm o mmNm NF.
at :9 3:: 33:: E N: :9, a»: 3%: g E 50> 35 3&5 8% at cm; as 3%: a: am;
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Smaczcoov ch EmEtmaxm - co=m_:E_m 9.56:0”. .mo mag-0.9% B San

108

APPENDIX 2 DATA OF EXPERIMENT TWO

109

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

No 3 v- ovvm mm em o 5mm 8 mm N 23 3F ow o 3mm mm o 33 Nm
I. mm o mem mm mm F.- NNvm on mm N 33” SF on o Nmmm mm o vam Fm
ms mm v NmNm mm mm N- mmmm so vm F- nova 3F mm o ouvm mm o Noam on
we vm F 9.5 Nu mm v mVNm 5 mm m- NFNN SF on o vmmm mm o mem mN
mm mm v- Fmom NN mm v SFn mm an F mNNm SF on o FmNm mm o vmvm 3
ON mm o voom mm mm m- 3.8 on mm m ova NmF mm o oFNm mm o Novm NN
NN no v mFmN No S m- mme we cm o mmom 5F mm o mNFm mm o onm oN
mm vm o ommN 2. an N mmwN no mm m- onmN BF mm o nmom mm o vam mN
mm mm v- mEN 2. mm v FFmN mm mm o NmmN SF on o 88 mm o Nva .N
mm mm F- mmmN mm vm F- oNNN mo 3 o vaN 5F mm o mmmN mm o coon mN
NN mm v momN mm mm m- FvoN mm mm F BNN 5F 8 o oNNN mm o mNmN NN
mm no N NNVN on on F NmmN No on N- NNmN mmF mm o mmmN mm o NmmN FN
mo vm v- mmmN NN on v EN 5 5 F- ommN 3F mm 0 Saw. mm o oomN 8
8 mm m- NFmN mo- vm o 28 8 mm N vaN 3F mm o mFmN mm o NFNN mF
NN mm m NNNN mm on v- vaN mm mm N ommN 3F mm o mNVN mm o mNmN mF
I. ma v mmFN mm mm o moNN 3 cm F- mNNN me mm o Nva mm o 33 NF
no mm m- FmON FN no v mFFN Fm .3 N- mmFN NON mm o omNN mm o mmVN mF
B S v- mmmF mm vm o NNON mm an F FoFN 3F mm o mmFN mm o ommN mF
E. Fm m- mBF mm mm v- Nva mm on N 98 3F mm o NNON mm 0 EN 3
ex- vm o max-F mm an F- mmwF we 3 o NNmF mmF on o mmmF mm o mmFN mF
Fm mm N mmoF NN mm v ONNF 5 mm m- Nva 5F 8 o momF mm o moFN NF
mm Fa o mmmF NN mm v NmmF mm mm o 3NF 5F 8 o NNmF mm o mFON FF
vm Nm m- momF no no N- mmmF mm 5 N momF 3F mm 0 mm: mm o NmmF oF
voF 3 o FFvF mm on v- mFmF mm mm o ommF me mm o mva on o mva m
oFF vm 0 NSF mm an N- FNF-F so mm N- mva 3F mm o NomF mm 0 mm: m
vFF vm o mNNF FN mm m nmmF no 3 F- 83 3F mm o vaF mm o NNoF N
NNF vm o mNFF mm mm o omNF mm an N mFmF 5F 8 o mme on o mmmF 0
OFF cm o 39 mm mm n. 5: mm on F NmNF mmF mm o FomF mm o mva m
omF vm o mmm NN om m- 9.3 mm mm F- 3FF SF on o mFNF mm o NFvF v
mmF vm o 9% K Na N- 35 B B N- FooF 5F mm o mNFF mm o mNmF m
2F 3 o Fm» 8 vm o own mm mm F m3 SF on o FvoF mm o mmNF N
03 vm o mmo mm mm o @2- mm on N mom NON mm o 0mm mm o NmFF F
NN :9 wejmag a: E :9, wajmug a: 2 :9 35 $35 a 8 :9, Es 3&5 E was 3&5 as New
EP: cmmqm moo< 5:: E0: 3on wood. 3:: ES. 8QO moo< 3:: Eat ommnm m8< 3:: 88m .moo< 3:: Fan
028% 3&0va :9 058% anFeNN :9, 9:83 8&9 E :9, 868m. $82: :9, :uFovmo cm)

 

 

02F EmEtmaxm - cozm_:E_w @5326“. :No mam-70.9.5 .0 ENG

110

 

 

VO

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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NO

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Om

NO

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ON O O OOOO OO F OOFO FO

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OO NO v- OFov OO OO F- 53 OO OO O ovom OO F- OFFO 9.

ON OO N ONOv OO OO v- OOOv ON OO N NOov OO o NOOO Ov

FN OO N OONv OO OO o OOOv NO O O- ONOv OOF OO o Ovov OO o FvFO Nv

NO OO O- 89- NN OO v ONNv OO NO O- NONv NOF OO o OOOv OO O OOOO O.»

OO OO F- OOOF. ON O o OOOF. OO OO N OONv NOF OO o NNNv OO O OOov Ov

ON OO N ONE. OO OO v- ovmv ON OO O OFOv NoF OO o OOOv OO o NOOv vv

ON OO v OOOF. OO OO F- FOE. OO O N- on? NoF OO o OOOF. OO o OONF. OF.

OO OO o vomv ON OO v NNOv OO OO F.- Ovvv OOF OO o :3 OO o OONv Nv

OO O v- ONNv FN OO F. OONv OO OO F OOOv NoF OO o ON: OO o NNOF. Fv

NO OO N- vav NO OO O- FONv oN NO v OONv NoF OO O -OOOv OO O OOOv ov

FN OO v Ovov NO NO v- OFFv OO O O OOFv NOF OO O FONv OO O Ovvv on

OO OO F OOOO FN OO F NNov OO OO v- OOov OoF OO O VOFF. OO O NOOv OO

NO OO v- NNOO FN OO v OOOO OO OO O 99. NoF OO O ONov OO O ONNF. NO

OO OO F- OONO NO O F- vOOO ON OO O FNOO NOF OO O FOOO OO O OOFv Om

ON OO v OOOO NO OO O- OONO OO OO F OOOO NOF OO O coon OO o 53 OO

NN O V OOOO ON OO F OOOO NO .O O- OONO OOF OO o NFOO OO o OFov vm

NO OO v- ONOO NN OO v NOOO NO NO N- VOOO NOF OO o FONO OO o ONOO Om
NN :9 32133 g E :2 35 3%? g S :9, 33 323 g 8 cm; 35 535 E 334mg; a 3%
E0: ommom moo< 3 _E E0: 823 mou< 3 :: Eo: ommom mood. 3 _E E0: 95% moo< 3 _E 80% .894 3 :: Bu

8.8% ANNEX-N :2 958m auFofiNN :9, 9.8% auEO I :9, 9.8% ANHFOOE cm) :uFoflwO :9,

 

 

335308 03: 395598 - COOOSEE 9:30:03 60 83.29% B BOD

111

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

NN O F- OOON NOO OO o OOON OvF VO o NOFO FO OO N OONO FO OO o ONOO NO
ON NO N- NNVNOOO OO o OOvN NOF O o OOOO OO NO N oFNO FO OO O OONO FO
NN OO F vOON NOO OO o FFF-N OOF O o OOON OO OO N ONFO OO OO F- OONO OO
ON OO O OVNN OVO NO o VNON OOF O O OOON NN OO N OOOO FO NO 0 OFFO ON
ON O o FOFN OOO OO o NONN FOF "O o ONNN FN FO N OOON O NO F NNOO ON
ON OO N- ONON ONO OO F- FOFN OOF O O OOON OO FO v- ONON O OO v OVON NN
ON NO o OOOF FNO OO F- OOON OON O o OOON OO OO v- FONN OO OO o OOON ON
ON OO F FOOF OFO OO o ONOF NFN O o NOvN OO OO O voNN NO O V. ONNN ON
NN OO o OFOF OOO OO o NOOF OFN O o NOON NN OO N OFON OO NO N NOON VN
ON OO F- ONNF OOv OO o OOOF ONN O o OOON ON OO o ONON OO OO F OOON ON
NN NO o NVOF OOF. OO o OFNF OON O o OONN OO NO O- FVVN OO OO o OOON NN
ON NO F VOOF NOv OO o NOOF OON O o vFFN OO OO F- NOON NO NO o FNVN FN
NN OO o OOvF ONF. OO o OVOF OVN O o ONON ON O F OONN OO NO F OOON ON
ON OO F- NOOF NOF. - OO o OOvF 3N O o ONOF NN OO F OOFN OO OO v OVNN OF
NN OO N- OONF OOv OO o NNOF OVN O o FOOF NO OO F- OOON OO NO F NOFN OF
ON OO o OONF FOv OO o OONF OON O o OONF FO NO v- OOOF OO NO v- OOON NF
NO OO o OFFF 33 OO o OOFF OON O o NVOF OOF OO N- OOOF OO OO v- OOOF OF
OO OO o ONOF NOF. OO o FFFF OVN O o OVOF oFF OO o NONF OO OO v- NOOF OF
OO OO o NOO ONF. OO o ONOF OVN O o OOvF OFF OO o OOOF NO OO o vFOF VF
FO OO O NO FNv OO o OOO NVN O o OOOF OFF OO o FOOF OO OO N ONNF OF
OO OO o OON 3: OO o FOO OON O o OONF VNF OO o OOOF NO NO o NNOF NF
OO OO o OOO OOF. OO O vON OON O o oNFF OOF OO o 33 FO FO o OOOF FF
ooF OO o ONO OOO OO o ONO OON O o ONOF NOF OO o OOOF NO NO F- O3; 9
NOF OO o OOv OOO OO o FOO NNN O o FOO NvF OO o OONF FO O N- OOOF O
OOF OO o OOO OOO OO o vOO ONN O o NOO vvF OO o OFFF OO OO F- vONF O
NOF OO o oFO ONO OO o NFv OFN O O OON OvF OO o NFoF OO NO o OOFF N
FFF OO o ONN NOO OO o FOO OFN «O o OOO OvF OO o vFO OO NOO o OOOF O
vFF OO F OOF OOO OO o 3N FFN O o FOO NvF OO o OFO NN OO o NOO O
OOO NO F- NOF NON O O OFO NvF OO o NFN ON OO o VOO v
ON 3 o OFF. OvF OO o OFO OO OO o OON O
OON O o FNO 3: OO o FNO OO OOF o OOO N
OOF O O NNN FvF OO o ONF. NO FOF o VOO F
NN :9, 35 3%: a ON :9, 35 338 g O22 3a 3&9 E E :9 35 via a NN :3 33 40.3% o: Osv-
Eo: vomamduud. 3:: E0: 80% moo< 3 _E E0: 80% moo< 3 _E So: 03% moo< 3:: E0: 80% mou< 3:: hon
9:83 OHEO ON :2, 9:83 NNHEO NN :9 38% 8&9 ON :9, 9:83 SHE ON cm) 9.83 SWFQ E :9

 

 

 

Smoczcoov 03: 3955me - :o:m_:E_w Oanzou 60 33.29% B SOD

112

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

OO
OO N- OOFO OO
ON NO N- OFFO OO O OOFO NO
ON OO F FNOO OO O OOOO FO
ON OO N OOOO OO O NFOO OO
NN OO F- OOOO OO O ONOO OO
ON OO O- OONO OO O OOOO OO
NN OO O ONOO OO O _ NONO NO
ON NO O NOOO OO O OOOO OO
NN OO O FOOO OO O ONOO OO
ON OO O- OFOO OOO OO O FOOO OO OO O- OOFO OO N OONO OO
ON OO O- ONOO NOO OO O OOOO FO OO O- NOOO OO OO O- ONFO OO O OOFO OO
ON OO N OONO OOO NO O OFOO OO OO O- FOOO OO OO O- OOOO OO NO F OFFO NO
ON OO N OOFO OOO - OO O FONO FN NO O OOOO OO NO O- OOOO OO NO O- ONOO FO
ON OO F- OOOO OOO OO F OOFO ON FO N NONO ON FO F- ONOO OO OO O- OOOO OO
ON OO N- OOOO OOO OO F- OOOO FN OO N NONO NN FO N ONNO OO OO F- OOOO OO
NN NO O OOOO OOO OO O ONOO OO NO N OFOO ON OO N NOOO FO OO F OONO OO
ON NO N OOOO OOO OO O OOOO NO OO O NOOO ON NO N OOOO OO OO O ONOO NO
NN OO F NNNO OOO OO O OONO OO OO F- NOOO ON OO N NFOO NO OO O OOOO OO
ON OO F- OOOO OOO OO O NFNO NO OO O OOOO ON OO N NNOO OO NO O NOOO OO
ON NO F- OOOO NOO OO O ONOO FN NO O ONNO OO FO F OOOO OO OO O NFOO OO
NN OO F NOOO NOO OO O OOOO NN OO O- OOFO NO FO N- OONO OO FO N- OOOO OO
ON OO N ONOO OOO OO O NOOO OO NO O- OOOO NO OO O- OOFO NO OO O- NONO NO
ON OO O OONO NOO OO O OOOO NO OO O NOOO NO OO O- OOOO OO NO N- FOFO FO
ON OO N- OONO ONO OO O ONNO OO OO O NOOO ON FO O OOOO FO OO F ONOO OO
NN OO O OFFO FNO OO O NOFO OO OO O OFOO ON OO N NFOO OO NO O OOOO OO
ON NO N NNOO OFO OO O OOFO OOF OO O OFNO ON OO N ONOO FO OO O NOOO OO
NN OO O FOON OOO OO O OFOO NFF OO O ONOO NO OO N OONO OO OO O OOOO NO
ON OO N- OOON OOO OO O NOON ONF. OO O OOOO ON OO F OOOO OO NO O ONNO OO
ON OO F- OONN OOO OO O OOON FOF OO O OOOO OO OO O- OOOO FO OO N OOOO OO
ON OO N OOON OOO OO O OONN OOF OO O FOOO NN OO F1 OOOO OO NO O- NOOO OO
ON OO N- OOON ONO OO O FNON NOF OO O NONO ON FO F OOOO OO OO O- NOOO OO
NN Om) 321$an OOO ON :9, 55 $25 OO ONO? 35 335 E ON O9, ES 33% OO ON :2, Es OOO-O OOO 3%.
E0: 303.894 3 :: Eo: 30% 804. 3:: So: 25% moo< 3 _E E0: Boom 80¢. 3 _E E0: OOOOO mood. 3 :: 3».
9.3% NOHNOO ON Om) OEOOOO NNuNOO NN O9, 9:83 3qu ON O? OOBOOO NOu-Fo ON :9 9:83 AOFnNOO ON :9,

 

 

 

Nomzczcoov 03H EmEtmaxm - :o:m_:F:_m 9.56:0”. 30 23.29% No BOO

113

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

FO NO N FOOF OO OO o NOOF ONO OO o OOON OO OO N OOON OO OO o OOON NO
OO OO o OFOF ON NO O ONOF NNO OO o OOOF ON NO o FNNN ON OO O OOON FO
5 NO F- NNNF OO OO o OONF NNO OO o NOOF ON NO O- OOFN FN OO O OONN OO
OO OO N- OOOF NO OO O- OONF ONO OO o ONNF OO OO N- OOON OO OO O- OOFN ON
OO NO o NOOF NO OO o OFOF ONO OO o OOOF NO NO o OOON OO OO O- OOON ON
NN FO F OOOF ON NO O NNOF OFO OO o NOOF OO NO o OFOF ON OO F OOON NN
NN OO o OOOF OO OO F FOOF OOO OO o OFOF ooF NO o. OFOF FN OO O OFOF ON
OO NO O- NNNF NO OO N- OOOF NOO OO o ONOF OOF NO o ONNF OO OO o OOOF ON
FO OO F- ONFF NO NO N- OONF NON OO O OOOF OFF NO o OOOF OO OO O- OONF ON
OO OO o NOOF OO OO N FOFF OON OO o OONF ONF NO o OOOF OO OO O- OOOF ON
NOF OO o OOO ON OO N OOOF OON OO o OOFF ONF NO o OOOF OO OO O ONOF NN
OFF OO O- OOO OO OO N- OOOF ONN OO o ONoF OOF NO o OOO F OO OO O OOOF FN
OOF FOF o OON OO OO O- FNO ONN OO o OOO OOF NO o OONF OO OO F- OOOF ON
OOF FOF F NOO NN -FO o FOO NON OO o OOO OOF NO o ONFF OO NO O- OFOF OF
OOF 03 F OOO NN NO o OON NON OO o OFO OOF NO o ONOF FN FO O- ONNF OF
FOF OO N OOO NO NO F- NOO NON OO o ONN OOF NO O OOO ON OO o FOFF NF
OOF OO N OOO OO OO F- OOO FON OO o NOO OOF NO o OOO ON OO o NOOF OF
NOF OO N NON NO OO o OOO OON OO O OOO OOF NO o FOO OO OO O FOO OF
OOF NO N OOF OOF OO F OOO OON OO o OOO NOF NO o OON NO OO O OOO OF
OOF OO N OOF NFF OO N ONN OON OO o NOO OOF NO o OFO OO OO o OON OF
NFF FO N ONF ONN OO o OON ONF NO o ONO OOF OO o NOO NF
FNF OO O OO NNN OO o OON ONF NO o FOO NFF OO o OOO FF
OFN OO o NNF NNF NO o OOO OFF OO o OOO OF
FFN NO 0 OO NFF NO o OON ONF OO o OOO O
OFF NO F- OOF OOF OO N OON O
FFF OO o OO OOF OO N FNF N
NOF NO N ON O
O
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NO 50> 35 325 E FO50> Oefiwgico OO 50> 35 33% E ON 50> 39 E3: 9% ON 50> 38 3%: g O0;
So: 30% wood. 3 :: So: 25% moo< 3 .F: 80: 803.0. moo< 3 :: So: 39% 08¢. 3:: E0: 93% moo< 3 :: BO
950% NOFuNOOOO 50> 960% NOuNOO NO 50> O5_000O ANHNQFO 50> 9:83, FOu-FOOOO 50> 8.00%. AOuNoO ON 50>

 

 

AuszEoov 02F EmE:m3xw - co:m_:_:_m 9:523“. 50 mag-EOEO .0 9mm

114

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

OO OO N FNOO OO OO O- OONO ONO OO O OOOO NO N NOFO OO
OO OO N OOOO NO NO O- OOOO ONO OO O ONNO NO OO N OOOO OO OO O NNFO OO
OO NO N FOOO OO OO O OOOO ONO OO O OOOO NN OO O FOOO NN NO O OOOO NO
NO NO O- OFOO ON OO O ONOO FOO OO O OOOO ON OO O- NNOO OO OO O NOOO FO
NO OO 7 OOOO NO OO N- NOOO NOO OO O OOOO NN NO O FONO OO OO O- OOOO OO
OO OO N- NONO NO OO O- OOOO OOO OO O ONOO ON OO O OONO NO OO O- FONO OO
NO FO F OOFO OO OO F OFNO ONO OO O OONO ON OO F. OFOO NN OO O FOOO OO
OO OO N OOOO ON NO O ONFO ONO OO O OOFO ON OO O ONOO NN OO O OOOO NO
OO NO O ONOO OO OO O OOOO ONO OO O NF FO OO OO N OOOO OO OO N- OFOO OO
ON OO N OOOO NO OO O- OOOO NNO OO N ONOO OO OO N OOOO NO OO O- OOOO OO
OO OO N OOOO NO OO F- NNOO ONO OO O OOOO NO OO N OONO FN OO O OOOO O
OO NO N ONNO OO OO O OONO NNO OO O NOOO NN NO F ONFO ON NO O OONO OO
OO NO O- OOOO OO - OO N OOOO NOO OO O OONO ON NO O- NOOO ON OO O ONFO NO
OO OO O- OOOO NO OO O- NFOO OOO OO O ONOO ON OO O- OFOO OO OO O- NOOO FO
OO OO F- OOOO NO NO O- ONOO OOO OO O NOOO NN OO N- ONOO OO NO O- NOOO OO
OO OO F ONOO OO OO N OOOO ONO OO O OOOO ON OO N OOOO ON OO N OFOO OO
NO OO F OONO ON OO O OOOO ONO OO O OFOO 5 OO N OONO FN OO O ONOO OO
OO NO O OOFO OO OO F- OONO ONO OO O NOOO OO OO N OOOO NO OO N- FONO NO
OO OO N FFFO OO OO Y ONFO ONO OO O OONO OO OO N ONOO NO OO O- OOOO OO
OO OO N ONOO OO OO O OOOO ONO OO O OOFO FO NO N OOOO OO OO F NOOO OO
OO NO O NOON OO NO O NOOO OOO OO O FNOO ON NO O- OOOO NN NO O ONOO OO
NO OO O- OOON OO OO O OFON OOO OO O OOON ON OO O- OFOO OO OO O OOOO OO
OO NO O- ONNN NO OO O- FOON OOO OO O OOON ON OO O FONO OO OO O- OOOO NO
OO OO O OOON NO OO F- OONN FOO OO O FFON OO NO N NOFO NO OO N- NNNO FO
FO OO N OOON OO OO O OOON FOO OO O ONNN NN OO F- OOOO FN OO O NOFO OO
OO OO O OOON ON OO F OOON ONO OO O OOON NN OO O NOON FN OO O OOOO OO
NO NO N NFON NO OO N- OOON ONO OO O FOON FO OO N ONON NO OO O- OOON OO
NO OO N OOON NO NO N- NOON ONO OO O OOON OO OO N NONN OO NO O- ONON NO
OO OO N OONN OO OO N OOON OOO OO O NNON ON OO N NONN ON OO F OONN OO
OO OO O- OOFN ON OO N FNNN OOO OO O FONN ON OO O- ONON NN NO O NOON OO
OO OO. O- ONON OO OO F- OOFN OOO OO O OONN ON NO :2 OOON OO OO O NFON OO
FO OO F OOOF OO OO O- FOON FOO OO O NF FN ON OO F OOON OO OO O- NNON OO
NO 50> WE 3.0-EOE 5:2 3E NOO-OE E OO 50> ES 3%? E ON 50> 3:132: E ON 50> OE EOE E 20;
E0: 80% moo< 3 _E Eo: OOOOO moo< 3 _E E0: OOOOO moo< 3 _E E0: 8qu moo< 3 _E Eo: OOOOO muo< 3 _E Oou

9:83 aFuEflOO 50> O5_000O_ NOuNOO NO 50> O5_00OO ANuEO FO 50> 300%. NOuEO OO 50> O5_00Om_ NOnEO ON 50>

 

 

335308 03: EOEFOaxw - co:m_:E_m Oc_26=o.-. OOO mag-QOEO :o Ema

115

 

 

NO
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NOV-EEO 50> 950%

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Eggs“: 02C- EmE:mOxm - co:m_:E_m 9:523”. 50 23.99% :0 San.

116

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

NFN OO O FONF NFF OOF O 83 OOF OOF O OOOF OO O N- OONF OO NO N- OvOF NO
FON OO O OOFFOFF OOF O OOOFOOF OOF O :3 FO NO O- FOOF OO OO F- OONFFO
OOF OO O OOOF OFF OOF O OVNFOOF OOF O NOOF OO OOF O- OOOFOO FO N- VOOFOO
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OOF OO O ONN NFF OOF O ONO VON OOF O OvOF ONF NOF O OONF ON OO O ONOF NN
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OOF OO O NNO OFF OOF O OFN VON OOF O OOO OvF OOF F NOOFOO OO F- OOFFON
NNF OO O FOv OFF NOF O OOO OON NOF O ONN FOF VOF F NOO OO FOF O OOOFVN
OFF OO O OOO OFF NOF F FOO OON NOF O OFO OOF OOF F NNO OOF FOF O NOO ON
NOF OO O OON NFF OOF N OOO NFN OOF N NFO NOF NOF F VNN OOF OOF O FOO NN
OO OO O NOF OFF OOF N OON OFN OOF N OOF. OOF FOF F FNO OOF OOF O OON FN
NO OO O OO NFF - FOF N OOF OFN FOF N OOO OOF OOF F ONO OOF FOF O ONO ON
NFF OO N NO OFN OO N VON NOF OO F ONv OFF FOF O NNO OF

OFN NO N OOF OOF OO F ONO OFF OOF N ONv OF

NOF NO N NNN NFF OO N VNO NF

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EEO Ommam moo< 3 _E SP: 25% .ooo< 3 3. So... nomam moo< 3 _E EEO Ommam moo< 3 _E EEO ommam moo< 3 F: Oct

38% EEOOO :2 38% aFnEONO :9 38% #:qu 8 E, 9:83 SEQ 8 6) 9.68m SWEOOO :2

 

 

 

Emacscoov 02F EmEtmaxm - :oOmSFEm 9.56:0“. .mo mag-035m Oo Ema

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OO OO F- ONFO ONO OO O OONO OOF OO O OOOO NO OO F- OFNO OFO OO O OONO ON
NO OO O OOOO FNO OO O ONFO OOF OO O OFOOO NO OO F- ONOO OFO OO O OOOO OO
OO OO O FOON OFO OO O FOOO OOF OO O OOOO NO OO F OOOO OOO OO O OOOO OO
OO OO O OOON OFO OO O NOON ONF OO O NONO OO OO N OOvO OON OO O OFOO NO
OO OO O ONNN OOO OO O ONON OOF OO O NNFO NO OO O NOOO OON OO O ONOO OO
OO OO O FOON OON OO O OONN FOF OO O NOOO OO OO N- OONO FON OO O OOOO OO
NO :9, EE EOE E OO FE, EE EOE E OO :9, EE EOE E OO :9, EE EOE E OO FE, EEO EOE E 824

30: OOOOO m00< 3 _E 30: 395 m00< 3 _E E0: 93% o00< 3 _E 30: 03% moo< 3 .E 30: ommam m00< 3 _E 3%

95093 Nuhom OO cm> O50O3O ANN—.03 NO zm> O50mam OOH-N03 OO zm> O50OOO ~On-53 OO 50> O50mam NON-N033 cm>

 

 

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