URBAN TRAFFIC SYSTEM SIMULATION
AND CONTROL

Thesis for the Degree of Ph» .D,
MICHIGAN; STATE UNIVERSITY
WAYNE DAVID PANYAN
1969

 

  
      

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This is to certify that the

thesis entitled

URBAN TRAFFIC SYSTEM
SIMULATION AND CONTROL

presented by

Wayne David Panyan

has been accepted towards fulfillment
of the requirements for

Ph. D. degree in E. E.

WET/aw

Major professor

Date August 18, 1969

0-169

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BOOK BIIIIIEIIY INC.

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ABSTRACT

URBAN TRAFFIC SYSTEM SIMULATION
AND CONTROL

by Wayne David Panyan

A simulation model for large urban vehicular traffic
systems is developed in the first part of this thesis. The
model is applicable to systems having signal controlled
intersections and vehicular densities described as light to
medium. Interpreting the system as an interconnection of
smaller components, each exhibiting the same phenomena as
the whole, is the keystone of the development. On each of
these components the behavior of platoons and queues (the
smallest vehicular units considered) are described by a
set of state equations. The variables in these equations
are position and vehicular density. Only one queue can be
present on the component and it can be described wholly by
its position. However, since more than one platoon may
exist on the component and since each requires two variables
in its description, the number of equations required is a
variable, 2 pi + l, where pi is the number of platoons at

any instance.

 

*

Wayne David Panyan

A complete simulation model comprises an inter-
connection of several such components and a set of 2p + q
equations, where p and q are the total number of platoons
and queues, respectively. The structure of the system is
described by a connection matrix which is analogous to the
incidence matrix of graph theory. The inclusion of accel-
eration phenomena, random inputs and turning movements
results in a model which is general enough to simulate most
traffic structures and behavior. A Fortran program based
on the equations was written and used to simulate the
traffic behavior of the central business area of Lansing,
Michigan. Results of this simulation are included in the
thesis as an example.

In the latter parts of the thesis the control prob-
lem is considered. If the vehicular densities are suffi-
ciently low, the steady state control of an urban traffic
system can be effected by a synchronization of the traffic
signals. Such a synchronization, called a progression,
allows vehicles to travel the length of an artery without
having to stop for a traffic signal. Synchronizing the
signals so that progressions are established in the two
directions of a two—way street is simple enough in theory.
However, certain auxiliary strategies can also be applied
to discourage the queuing of vehicles. Further considera-
tions are required when an overall control strategy is to

be instituted on a traffic grid.

 

Wayne David Panyan

The most efficient use of an artery can be achieved
when the progression design is selected in an optimal
manner. An important innovation is the inclusion of the
demands that exist on every part of the artery in a cost
function which is proportional to the total vehicle travel
time. Minimizing this function while satisfying the phys—
ical realizability constraints imposed by the arterial
geometry, fixed signal parameters, and upper and lower
velocity bounds results in the optimal design. The non-
linearities inherent in the problem and the nonconvexity
resulting from the constraints require that an iterative
solution technique be used. A Fortran program to obtain
this optimal design was written and used to find the design
for a typical street. These results are included in the

thesis.

 

URBAN TRAFFIC SYSTEM SIMULATION

AND CONTROL
BY

Wayne David Panyan

A THESIS

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

  

DOCTOR OF PHILOSOPHY
Department of Electrical Engineering

1969

 

 

 

 

ACKNOWLEDGMENTS

The author wishes to express his gratitude to
his major professor, Dr. John B. Kreer, who in the course
of many fruitful discussions suggested many useful ideas
and spurred this work to completion.

The author also wishes to thank the Division of
Engineering Research of Michigan State University under

whose sponsorship this research was done.

ii

 

TABLE OF CONTENTS

Chapter Page

I. INTRODUCTION . . . . . . . . . . . . . . . . . . 1
II. MODELING . . . . . . . . . . . . . . . . . . . . 5
2.1 Previous Modeling Approaches . . . . . . 7
2.2 Present Model . . . . . . . . . . . . . . 11
2.3 Basic Component Introduced . . . . . . . 12
2.4 Selection of State Variables . . . . . . 16

2.5 Velocity-Density Relations for an
Arterial Section . . . . . . . . . . . . 18

2.6 Queues Introduced . . . . . . . . . . . . 20
2.7 Component Equations . . . . . . . . . . . 26
III. SIMULATION . . . . . . . . . . . . . . . . . . . 28
3.1 Acceleration Phenomena . . . . . . . . . 28
3.2 Turning Movements . . . . . . . . . . . . 30

31

3.3 Input and Output Elements
3.4 Traffic Simulation . . . . . . . . . . . 33
3.5 Example . . . . . . . . . . . . . . . . . 40
IV. CONTROL OF URBAN TRAFFIC SYSTEMS . . . . . . . . 51

4.1 Space-Time Diagrams and Traffic Signals 52

4.2 Steady State Queuing . . . . . . . . . . 55

4.3 Progressions . . . . . . . . . . . . . . 60

 

Chapter Page

4.4 One-Way Streets . . . . . . . . . . . . . 68

4.5 Grids . . . . . . . . . . . . . . . . . . 76

4.6 Example . . . . . . . . . . . . . . . . . 81

V. OPTIMAL DESIGN . . . . . . . . . . . . . . . . . 92
5.1 Preliminary Remarks . . . . . . . . . . . 93

5.2 The Mathematical Model . . . . . . . . . 95

5.3 The Optimization . . . . . . . . . . . . 99

5.4 The Search . . . . . . . . . . . . . . . 103

5.5 Example . . . . . . . . . . . . . . . . . 105

VI. CONCLUDING REMARKS . . . . . . . . . . . . . . . 108
6.1 Traffic Model . . . . . . . . . . . . . . 108

6.2 'Control . . . . . . . . . . . . . . . . . 111

6.3 Future Investigations . . . . . . . . . 111

APPENDIX - o o u n o o o O I o o O a o I u I o O o I o 113

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . 127

iv

 

LIST OF FIGURES

Figure

2.1 Traffic model component . . . . . . . .
2.2 Velocity-density characteristic . . . .
2.3 Velocity—platoon density characteristic
3.1 Traffic component symbol . . . . . . .
3.2a Artery with m signalized intersections

3.2b Model of artery (a) . . . . . . . . . .
3.3a Two-way artery . . . . . . . . . . . .
3.3b Model of artery (a) . . . . . . . . . .
3.4 Basic component for grid structures . .
3.5a Traffic grid of one-way arteries . . .
3.5b Model for grid (a) . . . . . . . . . .
3.5c Connection matrix . . . . . . . . . . .

3.6a Intersection of a one-way artery and a
two-way artery 0 O I O O O o l o a o c

3.6b Model of intersection (a) . . . . . . .
3.7a Intersection of two two-way arteries .
3.7b Model of intersection (a) . . . . . . .
3.8 Traffic grid of Lansing, Michigan's CBD
3.9 Model of grid in Figure 3.8 . . . . . .

3.10 Platoon and queue behavior . . . . . .

Page

13
20
24
33
34
34
35
35
35
37
37

37

38
38
39
39
42
43
46

 

Figure Page

3.11 Velocity along Capitol Avenue . . . . . . . . 49
3.12 Velocity along Capital Avenue continued . . . 50
4.1 Typical space—time diagram . . . . . . . . . 53
4.2 Traffic parameters defined . . . . . . . . . 55
4.3 Steady state queuing at an intersection . . . 57
4.4 Queue integral as a function of offset 8' . . 60
4.5 Bandwidth/cycle versus velocity-cycle . . . . 65
4.6 Progression bandwidth . . . . . . . . . . . . 69
4.7 Non-platoon vehicles and queue on one-way

street . . . . . . . . . . . . . . . . . . . 71
4.8 Progression with excess green distributed

to left of band . . . . . . . . . . . . . . . 72
4.9 Illustration for Bij determination . . . . . 73
4.10 Clearing of queues by transient effects . . . 74
4.11 Effect of increasing cycle length . . . . . . 76
4.12 A grid loop model . . . . . . . . . . . . . . 78
4.13 Subdivided grid . . . . . . . . . . . . . . . 80
4.14 Input component velocity . . . . . . . . . . 84
4.15 Downstream velocity . . . . . . . . . . . . . 85
4.16 Downstream velocity continued . . . . . . . . 86
4.17 Downstream velocity continued . . . . . . . . 87
4.18 Downstream velocity continued . . . . . . . . 88
4.19 Downstream velocity continued . . . . . . . . 89
4.20 Output . . . . . . . . . . . . . . . . . . . 90

vi

 

 
  
  
  
  
  

Time spent in queues . . . . . . . .

Flow-density characteristic based on
Greenshields' linear model . . . . .

Velocity—density characteristic . .
Optimal progression characteristics
Flow diagram for TRAFIK . . . . . .

Flow diagram for SEARCH . . . . . .

Page

91

96
97
107
119
124

 

CHAPTER I
INTRODUCTION

The problem of urban congestion that accompanied
the increased use of the automobile has become so great
in recent times that the simple remedies developed in
the past no longer are effective. Observing traffic,
placing a signal here, and posting a speed limit there
are insufficient. Effective utilization of today's
traffic systems demands the use of sophisticated traffic
controls. Such controls can be developed through modern
control techniques. However, a prerequisite is a good
model of the traffic system behavior.

In the past ten or fifteen years some efforts
have been made to explain traffic flow mechanisms and
particular traffic phenomena, but no complete model has
been produced. In part this failure resulted from in-
adequate data. Gathering data is an enormous task be—
cause traffic systems are physically large, and to ob—
serve the propagation of variables within the system
requires many expensive vehicle detectors. More impor-
tantly, a traffic system almost defies macroscopic analy-

sis. The behavior it exhibits is the result of disparate

 

 

 

 

 

phenomena, some of which are little understood in isola—
tion and even less understood within the context of the
system. The system is non-linear, not completely pre-
dictable, and susceptible to small changes of many fac-
tors.

This thesis is concerned with the modeling and
control of an urban traffic system having signal con-
trolled intersections.

In Chapter II a model of signal—controlled
streets with medium traffic density conditions is devel-
oped. These conditions are often encountered during
morning and evening rush hours, periods when improved
control is definitely needed. After examining a set of
possible traffic flow variables, average velocity and
density are selected as most appropriate to describe
vehicular movements. Since the densities encountered
are assumed to be great enough, groups of vehicles,
called platoons and queues, are the smallest vehicular
units considered. It is reasonable to attempt the
study of a complete urban area if platoons and queues
are considered, but the problem becomes too complex and
inefficient to solve if individual vehicles are con-
sidered.

An urban street system can be looked upon as an
interconnection of basic components each displaying the

characteristics of the whole system. The existence of

 

such components is postulated, and for each component a
set of equations is derived which describes the platoon and
queue behavior in terms of the density and velocity
variables.

The problem of simulation is investigated in
Chapter III. It is demonstrated how an interconnection
of a number of basic components can be used to simulate
a variety of traffic systems. The simulation is achieved
by using a Fortran computer program based on the developed
equations.

The main objective in controlling an urban sys-
tem is to minimize the delay that vehicles experience as
they travel through the system. Usually this is achieved
through the use of progressions. A progression is estab-
lished by the settings of the traffic signals which com-
prise the primary control devices. In Chapter IV the
problem of establishing progressions is investigated.
Special attention is spent on one-way street progressions
and on the unique problems presented by grids.

In Chapter V the problem of selecting the opti-
mal progression design for a two-way street having
variable demands along its length is studied. The cri-
terion for this design is a minimization of the total
vehicle-hours of travel time. Since the travel time is

a non-linear function of the arterial geometry and

signal settings, an iterative computer solution is re-
, quired to obtain the optimal settings of the signals.
A simulation of the traffic system of Lansing,
Michigan's central business area is given as a demon—
stration of the versatility of the model developed in
Chapter II. A second example to demonstrate the opti-
mal design procedure on a two-way street is also in-
cluded. 'Flow charts for the simulation and design pro—

grams are given in the Appendix.

 

CHAPTER II
MODELING

The desire to know how a system behaves under a
wide spectrum of conditions when direct experimentation
and observation are not possible (for reasons of economy,
time, safety, system inaccessibility, inadequate instu-
mentation, etc.) is sufficient motivation for generating
a simulation model. If the model accurately describes
the phenomena exhibited by the system, it becomes a
powerful tool for determining the response of the system
to a variety of controls and for investigating the ef—
fects of various parameters.

A traffic system belongs to a huge category of
systems which are difficult to study by direct means.
Physically it is large: in a metropolitan area a traf-
fic system of interest may cover several square miles.

To study such a system requires an extensive instrumenta-
tion network. The vehicle detectors needed for measuring
traffic variables are expensive and usually require
costly installation. Unfortunately, they do not always

prOVide the data in the form needed. Furthermore, many

 

traffic variables cannot be adjusted at will. For ex-
ample, average velocity and density result from the
interactions of many vehicles and are not easily con—
trolled. Finally, even though traffic signals are vari—
able control devices, they should not be indiscriminately
reset under the guise of scientific research.

The analysis situation is made worse by the fact
that an accurate model is difficult to obtain. Certainly
if better data were available, present mathematical de-
scriptions of the system behavior would be more accurate.
Secondly, within the system each driver, while constrained
by the proximity of other vehicles and by legal and phys-
ical limits on speed and maneuvering, operates his ve-
hicle according to his own driving habits. As a conse—
quence of this freedom the system is to a greater or
lesser extent stochastic in nature.

Furthermore, a traffic system exhibits more than
one mode of operation so that under a given set of con-
ditions certain behavioral aspects are dominant and
others are minimal.

As a result of the inherent complexity of a traf-
fic system one might rightfully conclude that its mathe—
matical model needs to be extremely complex. If, however,
a traffic system can be reduced to its essential charac-
teristics, a tractable model is possible.

In defining a traffic system some of the distinct

 

classifications become evident. A difference exists,

for example, between traffic studies on urban streets

and on limited—access freeways since the traffic signals
used to control the flow of vehicles through intersec-
tions force the vehicles into behavioral patterns not
observed on freeways. Moreover, traffic flow on surface
streets exhibits several modes depending on the vehicular
density. In a heavy density mode the queues which occur
at each intersection are sufficiently long that they do
not clear during a single green phase of the signal.
Under these conditions vehicles travel only a short dis-
tance before coming to a stop and each must react instant-
ly to the speed reductions of its predecessor to avoid
collision.

In a medium density mode each vehicle is still
constrained by the action of others, but a group of
vehicles, called a platoon, often can travel through more
than one intersection before stopping. In a light
density mode the speed of an individual vehicle is almost
independent of the speeds of other vehicles; queues and
platoons, existing as random, transient phenomena, do

not comprise a major feature of the flow.

2.1 Previous Modeling Approaches

Over the years many approaches have been taken

to describe traffic flow. If it were a simple task, the

 

work of these previous investigators would have included
a complete simulation model for traffic systems. How-
ever, their efforts have concentrated on very specialized
urban traffic problems and on the peculiar problems of
open highways.

For example, Gazis, et al (GAl) considered in—
dividual vehicles of a line of moving vehicles and pos-
tulated the reaction of the average driver to the braking
and accelerating behavior of the car ahead. Although
this model was used satisfactorily for investigating
local and asymptotic stability of the system of vehicles
and "correctly" simulated car-following data observed in
the Holland Tunnel, it has several weaknesses. Formulated
as a linear model, it describes poorly the transitions
between widely different steady state speeds. As a non-
linear model it overcomes this failing, but still does
not account for certain pnysical constraints, such as
the limited accelerating capability of a car.

In still another approach, Lighthill and Whitham
(LWl) modeled traffic flow as a continuous process.

They theorized on the existence of shock waves created,
for example, at bottlenecks and signalized intersections
but failed to get good correspondence with real data
since the theory was based on an assumed flow-density
function and neglected the detailed maneuvers of the cars

in changing speed.

 

rt

Others have viewed the traffic problem as a
stochastic one and have attacked it with the tools of the
statistician. Of particular interest is the work of
Beckmann, Tanner, Herman 9E al., Haight and others, in
which the problem of queuing at signalized (BEl, TAl)
and non-signalized (HRl, HA1) intersections is investi-
gated. These intersection models, however, are very
limited in scope since most often only a single inter—
section can be effectively modeled.

None of these modeling approaches are addressed
to the specific problem of modeling a complete signal-
ized traffic system. They are inadequate for describing
the peculiar platooning effects of the traffic signals
(although there is some attempt to simulate the behavior
of the platoon after it is formed). They do not simulate
the traffic routing of an actual arterial system (i.e.,
turning movements). Finally most of them satisfactorily
describe steady state behavior but fail to accurately
describe the acceleration and deceleration transients
occurring at intersections.

Goodnuff (G01) has investigated traffic systems
and established a model which simulates the peculiar com-
ponents (e.g., multi—laned arteries and intersections)
and behavior (e.g., turning movements) usually encountered
in traffic systems. For heavy density operating con-

ditions, he successfully formulated an optimization

 

10
algorithm which clears a grid of queued vehicles in
minimum time. The most important variables in Goodnuff's
system are those describing the queues formed at each
intersection. Since vehicles must stop for each signal,
it is unnecessary to track them as they proceed through
an intersection toward the tail of the next queue--
their position as they traverse the space between queues
provides no useful information. Only when the system is
successfully reduced to a lower density mode do these
movements become important, but at lower densities the
model assumptions are no longer valid.

In the course of solving the control problem for
medium to light density conditions, Chang (CH1) has
developed a traffic model which suitably describes some
of the phenomena of arterial traffic. Chang's model
takes account of the queues and the vehicular flow be-
tween queues. This movement he considers as a continu-
ous flow. Because his ultimate concern is the optimal
setting of traffic signals, the approximations he makes
for velocity (it is always constant), acceleration (he
neglects it), etc., are justifiable. However, without
this ulterior motive the model in its present form in-
adequately simulates vehicular flow and, even further,
has no provisions for describing phenomena such as

turning movements.

 

11

2.2 Present Model

 

The lack of a versatile model for simulating
traffic flow within a system of signalized arteries led
to this investigation. The goal, from a qualitative
standpoint, is to develop a model which describes traf—
fic flow for medium heavy to medium light density con-
ditions such as might exist in morning and evening rush
hours. It should be noted that in this mode the behav—
ior of vehicles in transit is of equal importance with
the behavior of those queued at the intersections.

Before pursuing details of the model, it is
necessary to establish its nature. The model can be
neither too elaborate nor simple An elaborate model
could achieve the stated goal by tracing the path of
each vehicle through the system while maintaining a con-
tinuous surveillance of surrounding vehicles. Predict-
ably, however, it becomes too complex and the computa-
tions inefficient as the system approaches any meaning-
ful size. On the other hand, a continuous model is too
simple since it does not depict all the phenomena that
are important to the control problem.

Between these extremes there exists a suitable
approach to modeling an urban system. Introducing
platoons and queues as the smallest vehicular units allows

studies to be made of large systems without becoming

12

cumbersome. Equally important, it preserves enough of
the identity of the vehicles that acceleration phenomena,
turning movements and vehicle counts can be incorporated.

It has been observed that vehicles in close
proximity to each other behave similarly in many re-
spects, and many mathematical theories rely on this fact
to describe average vehicular behavior and relative
motion between vehicles. For the assumed densities,
then, it is reasonable to model vehicles as platoons and
queues and to describe the platoons and queues by average
vehicular values.

The model can be either deterministic or stochas—
tic in nature. It is assumed that the environment in
which the vehicles move (the medium vehicular density
and the relatively short distances between signals) and
the platooning effects of the traffic signals constrain
the individual's movements so that they are realistically
described in a deterministic way. Some blending with
statistical ideas is achieved in the model via the de-
scription of the generation of input vehicles and of the
vehicle behavior at the intersections where turning is

allowed.

2.3 Basic Component Introduced

A large system is often considered as an inter-

connection of primitive elements or components. The

 

’ “TI

13

properties of such a component can be defined without
reference to any other components. It is not necessarily
the simplest such part since it may be possible to re-
solve it into a set of even simpler pieces.

Between every successive pair of traffic signals
there lies a section of pavement which carries traffic
in one direction. This length of pavement is an arter-
ial section. It serves well as a base for a traffic
system component since all the phenomena of a complete
system can be observed on it. The complete component
consists of the arterial section, the upstream signal,
the upstream queue, and the platoons in transit on the
section. Figure 2.1 illustrates such a component. All
positions are measured positively with respect to the

signalized end of the component.

 

We /,/,/I//1 "T/AI
IL___ DI ____.I

Figure 2.1. Traffic model component.

 

Associated with this component is a set of equa-
tions describing the vehicular units. The platoon de-
scription is complete if its length, its position, and

its number of vehicles are known for every instant of

 

14

time. A queue is described by its length and its number
of vehicles. The set of equations given here are com-
plicated since the platoons and queues are functions

of many primary- and secondary—level factors.

31—: = f(z, Aj,D, t) (Z-D
where
k k k T
Z = (Pld’ Ptr' np, Q, nq) I k = 1,009, n (2.2)

The elements of the 2 vector are the position of the
leading edge of each platoon, PId; the position of the

trailing edge of each platoon, P: the number of vehicles

r'
in each platoon, n2; the queue length, Q; and the number
of vehicles belonging to the queue, nq. The jth phase
of the traffic signal is denoted by Aj, the length

of the arterial section is given by D, and the
independent variable time is represented by t. The
function f (-) is also implicitly a function of street
conditions, prevailing weather, time of day, and other,
more subtle, secondary—level factors.

Since more than one platoon may exist on an
arterial section at a time, the index k is used to dis-
tinguish them. As each platoon is formed it is given a
new index; thus the latest platoon has the highest index
n. The dimension of the vector 2 is 3n + 2.

The relation expressed in equation (2.1) can be

made more tractable if

 

15

(l) The number of elements in 2 can be reduced.

(2) Velocity and vehicular density on an arterial
section are strongly correlated.

(3) The theory of queuing is applicable.

(4) Arterial streets have no inclines, banking
or curves.

When these assumptions are incorporated into the
model, it is possible to spell out the equations explic—
itly and yet not sacrifice accuracy.

The first statement suggests that either some of
the elements of z are redundant or that a better set can
be found.

The second statement, supported by theoretical
and experimental studies, suggests that vehicular dens-
ity is a first—order effect in the determination of
platoon and queue positions. Conversely, the number of
vehicles (or density) in the platoons and queues are
determined almost wholly by the average velocity of
vehicles. The second-order effects (arterial geometry,
weather conditions, etc.) are in comparison negligible
but accounted for implicitly in the velocity-density
relation.

The third merely states that the description of
the phenomena observed at the signalized intersections
can be couched in the terminology of queuing theory.

The last statement disallows peculiar arterial geometries

 

16

and suggests that the effects of geometry in the deter-
mination of densities and velocities be relegated to a
secondary role.

By confronting each of these assumptions in
depth, the simplifications can be achieved. Before pro-
ceeding, however, it is helpful to note that vehicular
density and lane occupancy are alternative measures for
the number of vehicles. Vehicular density is the number
of vehicles per unit pavement length. Lane occupancy is

a normalized density defined in the following manner:

total vehicular length
total pavement length

 

LANE OCCUPANCY =

2.4 Selection of State Variables

In a queue the vehicular density is at a maximum
and the velocity is essentially zero. This value for
density, the ratio of the number of vehicles to the
queue length, is generally assumed to be a constant.
Because the length is proportional to the number of
vehicles, either is sufficient to serve as the state
variable describing the queue. For computational ease,
the length is selected.

The state description of a platoon must include
any set of independent variables from which its length,
position, and number of vehicles can be determined. The

following list suggests most candidates.

 

17

1. Position of leading edge of platoon, (ft).

2. Average vehicular density of platoon, ¥%E),

vehicle ft ) _ (:3
ft '

or average lane occupancy (EEVEEEEE_IE
3. Number of vehicles in the platoon, (veh).
4. Pavement length of the platoon, (ft).
5. Total length of vehicles in the platoon,
(ft).

6. Mean headway between vehicles, (ft or sec).

Variables (2, 3, 5), for example, are not independent.

An arterial section may hold several platoons at
a time. A simplification results if one always assumes
that all the space behind a platoon is occupied by other
platoons, considering a free space as a platoon having
a vehicular density of zero. Consequently, the platoon
state at any time can be provided by any two variables
listed above except for the pairs (1,4), (3,5), (1.6),
and (2,6). The same two must be used to describe each
platoon. For computational reasons the position of the
leading edge and the vehicular length are selected to
describe each platoon.

Since the states of the platoons and queues are
derived from the density and velocity, the relations
between these variables are established by experimental

and theoretical investigations.

 

r---

18

2.5 Velocity-Density Relationg for an

 

Arterial Section

Drivers in a traffic stream, aware or not, react
to increasing density by lowering their speed. This
natural control mechanism was studied closely by Green-
shields in 1934 and led him to conclude a linear relation
between the speed of vehicles in the traffic stream and
the stream density. Subsequent experiments have sub-
stantiated that for many purposes Greenshields' linear

model is realistic. Thus the speed is given by

_ x
3
where vf is the free speed, a mathematical value for

speed as density approaches zero. The jam density, the
density at which the speed goes to zero, is denoted by
xj. A typical value for xj is 40 per cent of the bumper-
to—bumper density. This relation, established under very
restrictive conditions, applies to steady state condi—
tions for vehicles moving on a highway (i.e., an uncon-
trolled artery) and it applies to average values of the
variables.

Although the equation does not reflect it, Green-
shields introduced a kink at the top of the graph to

describeIin a realistic way, the region where the speed

is unaffected by density below a certain limiting value.

 

for

19

This truncation is observed for any speed-density relation.

On a controlled artery when the densities are light
and the speeds are normally higher, the conditions are not
too unlike the steady state stream. With xj about forty
per cent equation (2.3) predicts velocity quite well.

As the density increases, however, the velocity
does not approach zero as quickly as equation (2.3) pre-
dicts. The average velocity is zero on an artery when it
is filled from intersection to intersection with a queue.
Thus under these conditions xj should equal xq.

Stated mathematically, the equations describing

velocity and density for arterial traffic are given as

x
f1 x.l 1

v = v (1 —— —£—), x < x < x. = x (2.5)
f2 sz l - - 32 q

where x. is approximately 0.4. The constants v , v ,
31 fl f2
and x1 are selected to match observed traffic behavior on

particular arteries. Figure 2.2 depicts a typical velocity
- density characteristic.

The relations given in equations (2.4) and (2.5)
are used to determine the average speed for all platoons
on a particular arterial section. Speeds on other sec-

tions are determined similarly. Individual vehicle

 

20

speeds, it must be emphasized, may be somewhat different
from this average. Within the platoons, particularly
when the density is light, accelerating and decelerating

vehicles and passing phenomena may be present.

 

 

Figure 2.2. Velocity—density characteristic.

2.6 Queues Examined

 

In the terminology of queue theory an intersec-
tion is regarded as a rate-limited server which is sub-
ject to breakdowns. However, despite the impressive
amount of literature available on queue theory in gen-
eral and on traffic congestion in particular, most work
has centered on the problem of gap acceptance (i.e.,
vehicle crossing or merging) (TA 1, MR1) or on the rela-
tively simple problem of a single traffic signal on a
two-lane artery (HA1, NEl). The problems encountered

when queue theory is used as a primary analytic tool in

 

21

the study of a complete traffic system are far too diffi—
cult (WEl).

The main difficulties result from the "artificial"
behavior that the signals impose on the traffic. No longer
can the distribution of arrivals at a signal be considered
Poisson or exponential. Instead it is intimately related
to the parameters of the signals (red and green times,
relative phasing). For the same reasons the distribution
of service times at each intersection involves intractable
mathematics.

Nevertheless it is possible to utilize some queuing
concepts to describe the events at the intersections. At
any given instant there are n vehicles in a queue. The
first vehicle in line enters the intersection. The time
elapsing between this first entry and the entry of the
vehicle next in line is the service time ts for the first
vehicle. During this time interval a vehicles arrive at
the queue's end. The number of vehicles in the queue at

the end of the service time is given by
n' = n - l + a (2.6)

It may happen that the original queue has zero length. If
so, it is necessary to await a vehicle's arrival so that

n=1, and consider the service time for it. Then,

n' = a (2.7)

 

22

The two above equations can be combined into the single

equation

n' = max (n-l, 0) + a (2.8)
or even more simply as

n' = n - l + d + a (2.9)
if d is defined as follows

dl if
0

n = 0
n > 0 (2.10)

Alternatively, during a time interval At a total
vehicle length p leaves the queue and a length a? arrives
at its end so that equation (2.9) can be written

(n'-n) I = -p pd a7
——ZE—— XE + XE + XE (2.11)

Average vehicle length is given by I and is used here to
convert the number of vehicles to an equivalent vehicle
length.

In the limit as At approaches zero
Q = -Vq + qu + v (2.12)

where Q is the net rate of queue length change, vq is
the rate of outward flow, and y is the rate of increase
in queue length, all measured in feet/sec or some other
equivalent units.

The effect of the signal can be introduced as a

second, but imaginary, queue served by the intersection

23

which has a "head of the line" priority. It has an
arrival rate which is a constant, one per signal cycle.
Its service rate equals a red phase. Viewed from the
real queue the effect of the imaginary one is to cause
the intersection to switch continuously between opera-
tion and breakdown.

Equation (2.12) describes the observed behavior
at intersections. However, the rate vq at which the
vehicles are served must still be determined. As noted
previously this rate is governed to a great extent by
the signal timing since the timing determines the arter-
ial velocity.

At the beginning of the green phase of a signal
the queue, assuming that its length is not zero, injects
the first vehicle into the next arterial section. After
a moment the queue sends another vehicle into the section
and continues to do so until the queue is dissipated or
the signal changes phase, at which time the next platoon
begins to form. The spacing between vehicles determines
the vehicle density within the platoon. However, this
density differs significantly from the average density
observed on an arterial component.

Platoon density is closely tied to the velocity
Emevailing during its formation. A rule of thumb sug—
sgested by safety advertisements, etc., advises that a

(driver allow a vehicle length between vehicles for each

r,

24

ten miles per hour of speed. This relation stated math—

ematically gives the lane occupancy of a platoon as

x = ————-— , v in mph (2.13)
P l + IO
or
x = _—_l_5__ , v in fps (2.14)
p 1+""147

The "ten" figure is not rigid, it could be some more
accurately determined value. (One suspects, however,
that this advice is not the result of idle daydreaming
but corresponds closely to the natural tendencies of
the average safe driver.) In order to have equation
(2.13) consistent with the requirement that vehicles
at reSt have a density xq the following modified equa-

tion is used instead.

1 (2.15)
+

salt.
Sl“

 

x
P

 

Figure 2.3. Velocity-platoon density characteristic

25

It is now possible to demonstrate the close corre—
lation of the rate at which vehicles are discharged from a
queue and the speed of the platoon on the arterial. This
results from the fundamental requirement that the flow into
an intersection must equal the outward flow. Equating the

inward flows at an intersection during T seconds,
x v T = x v T (2.16)

where x is average density and v is average velocity.

Obviously, then

x
v =52 v (2.17)
q

This equation can be interpreted in two ways. As noted
earlier the queue can be regarded as standing still but
becoming shorter as a "shock wave" moves backward through
it at a velocity vq. The shock wave is the discontinuity
resulting from the difference between the queue and platoon
densities. The second interpretation assumes that the
queue moves forward at a velocity vq and the so—called
shock wave remains stationary at the foot of the intersec-
tion. In either case the rate of change of queue length

is —vq. For the simulation model the first interpretation
is less desirable since it introduces a platoon between the

queue and the entrance to the intersection.

“"1

26

2.7 Component Equations

Some vehicles which leave the queue turn rather

than continue straight. The fraction that continues

straight is given by the constant a.

Using the state variables selected earlier and
the constraint conditions imposed by queue theory, the
component equations are as follows for the ith arterial

section.

={o, if jth phase of ith signal is red
11

1, otherwise (2.18)
P(i,k) = vi, k=l,..., n; 0§P(i,k)<Di-Q(i+l) (2.19)
fi (i'k) = _ng.,k.) 6 (t- Ti, k) , k=l,u ..,n-l k

aivi_§:EAij — Px(i, n) 6 (t— Ti n)' - n
Q 3 o (2.20)
Q(i) = _Vj(x p)iAij + Pac (i- -1 k)<5(t- ‘Ti- 1 k),
q Q(i) : 0 (2.21)

qQ(i+l) +k21 Pr (1, k)
xi = Di (2.22)

 

P(i,k) is the position of the front edge of the kth
platoon; Px(i,k) is the vehicular length of the kth
platoon. Q(i) is the queue length for the ith section.
The characteristic velocity, vi = f(xi), is determined
according to Figure 2.2 once the average density xi is
known. Di is the pavement length of the ith section.

The platoon density mp is determined from equation (2.15).

27

The impulse function is denoted by 5(t); Ti,k is the
time of occurrence for P(i,k) = Di - Q(i+l).

The use of the indices i—l and i+l implies that
the ith queue extends into the (i-l)st arterial section
and the (i+l)st queue extends into the ith arterial
section.

Equations (2.19 - 2.22) are less formidable if
it is kept in mind that vehicles released from the ith
queue during a green phase (2.21) appear as part of the
nth platoon (2.20). The platoon travels the length of
the arterial section (2.19) until it joins the upstream
queue where it ceases to exist as a platoon (2.20). In
the meantime new platoons are added to the ith queue
(2.21). It is this cyclic conversion of the vehicles
from a queue to a platoon to a queue which characterizes
the model.

It is precisely this cycling of vehicles which
allows a traffic system to be represented as an inter-
connection of similar components. Also it permits the
use of iterative techniques to obtain solutions. Be-
cause of the size of the problem, the non-linearities
involved, and the randomness of certain of the vari-
ables, these solutions are best obtained using a digital

computer.

 

a,

CHAPTER III
SIMULATION

Between the phases of modeling a component and
programming a computer there lies the important process
of simulation. This simulation is a demonstration of
how a complete traffic system can be interpreted as an
interconnection of basic components. In this chapter
the simulation models for several traffic structures are
developed. With the addition of acceleration phenomena,

random input generation, and a varying turning pattern

.the simulation can be made adaptable to most situations

encountered in real systems.
3.1 Acceleration Phenomena

With the inclusion of acceleration effects the
model has a greater potential for simulating traffic
behavior. The model presently employs a constant ac-
celeration function which could be generalized to any
function if future experiments dictate a change.

The acceleration process is simple to describe.
When the signal phase becomes green the first vehicle

in the queue crosses the intersection to begin a platoon.

28

 

.—

r*'“‘

29

The velocity of this platoon is initially zero, in-
creasing to the prevailing average velocity vi corre-
spondixjto the average density on the section.

In both real systems and models the process of
decelerating is more complicated. When the signal turns
green, the vehicles accelerate or remain still if their
path is blocked. On the other hand, the driver of a
vehicle approaching an intersection at a constant velo-
city must decide whether to begin braking. He must con-
sider the phase of the approaching signal, the behavior
of the traffic ahead, the distance to the next intersec-
tion, and his own speed. In a simulation model, these
same factors must be weighed.

In the present model, a driver makes the decision
to brake or continue at the same speed through an inter-
section when he is at a critical distance from the
intersection. This critical distance Dcr is a function
of the vehicle's velocity v and the number of vehicles
before it. It is braked for any one of the following
reasons.

1. A queue lies ahead.

2. The signal ahead is red and will remain red for
a time T greater than Dcr/v'

3. The signal ahead is green but will change phase
in time T less than Dcr/V.

4. The vehicles ahead are decelerating.

 

30

In the model, after this decision is made, the
vehicles involved are transferred from their own platoon
to one of two transition platoons which carries them from
the critical point to the intersection at either (1) a

constant velocity or (2) a decreasing velocity.

3.2 Turning Movements

 

The turning movements at an intersection observed
over a long period of time can be used to establish a
value p which is the ratio of the number of vehicles that
turn to the total number that enter the intersection per
unit time interval (a green phase). This average value
can be interpreted as a probability estimate that a ve-
hicle turns. Alternatively, p is the expected value of
the fraction of a queue that turns. Although this frac-
tion can assume any value in the closed interval [0.0,
1.0], for computation it is easier to discretize the set
i - n = o, 1,...,100}. The problem

100 '
Of arriving at the fraction

to the finite set I
"a" of turning vehicles for a.
particular green phase can be handled as follows.

The queue in question is normalized to one hundred

vehicles. A binomial probability function is used to es-

tablish the probability that k vehicles turn

P(x = k) = (1E0) pk (1—p)1°°'k (3.1)

where the random variable X has possible values 0,1,...,100.

 

31

A complete table of these random variables and their cor-
responding probabilities is generated. In the simulation
a random variable r which is uniformly distributed on the
. . . _ k

interval [0.0, 1.0] 15 generated. The turn ratio a — IOO

is determined by searching the table for k such that
Fx(k—1) < r i Fx(k) (3.2)

Here F£-) is the distribution function defined as Fx(k) =

P(x 1 k).

3.3 Input and Output Elements

 

The portion of the traffic network being simulated
may have several sections connecting it to other streets
or parking facilities. The sections by which vehicles
enter the simulated traffic network are the inputs, and
those by which the vehicles leave are the outputs.

The queues of the input sections are formed in a
way uniquely different from the queues of the other sec—
tions. Vehicles are assumed to arrive at these queues in
a random manner. The traffic signals and traffic behavior
outside the system, location of parking lots, time of day,
etc., play significant parts in establishing the distri-
bution of arrivals. However, as a simplified representa-
tion a Poisson probability function is‘used in this model
to describe the arrivals.

An expected value It describes the average demand

 

Observed. The Poisson probability function given by

-}.t k
P(X = k) = 2%?— (3.3)

is used to determine the number of arrivals in the time

interval t. A random number I having a uniform distribu-
I tion on the interval [0.0, 1.0] is generated every t sec-

onds. The number of arrivals k is then established every

t seconds by
; FX(k-l) < r 1 Ex (k) (3.4)

In this latter relation, the distribution function is
denoted by FX(-). For an input section it is easy to con—
tinuously adjust the probability parameters to simulate a
changing arrival pattern.

The output sections are hardly different from the
internal sections. Instead of allowing the vehicles to
form into platoons after crossing the intersection, they
are accumulated in a counter or sink. This implies that
the sinks are of great enough capacity (infinite) that
there is no cumulative congestion which reflects back into
the system. If this is not true, it is necessary to extend
the portion of the network being simulated until such a
section is reached or to reduce the flow rate into the out-

put section as congestion builds up.

   

.n... .-r. a

33
3.4 System Simulation

In the simulation of traffic systems, components
like the one modeled in the previous chapter are connected
together according to the geometric pattern of the system.
This model has associated with it a connection matrix which
details the manner in which these connections are made and
the direction of traffic flow. In all respects it is analo-
gous to the incidence matrix used in graph theory (KTl).
The matrix has dimensions 2 x m, a column for each of the
m components. The first entry of the ith column is the
index of the component following the ith component. The
second entry of the ith column is the index of the compo—
nent preceding the ith. An input or output element has a
zero for its second or first entry, respectively.

In the following examples the components are rep-

resented by the simplified symbol shown in Figure 3.1.

IO I

 

Figure 3.1. Traffic component symbol.

One-way artery

A one-way artery with m signals is the simplest of
all traffic structures to simulate. The complete model
Consists of m traffic components joined end to end as in

Figure 3.2. The first one is an input component and the

 

 

34

 

 

 

 

 

 

 

 

 

(b)
Figure 3.2. (a) Artery with m signalized intersections.

(b) Model of artery (a).

last an output. No indexing scheme is implied by the
formulation, so for simplicity the components are numbered
successively beginning with the input element. For the

system of Figure 3.2 the connection matrix is
_ 2 3 4 0
KM‘I:0123J
Two-way artery

A two-way artery can be considered to be two one-
way arteries side by side as shown in Figure 3.3. It is
important to remember, however, that at each intersection
the signal phase, red or green, is the same for both the

inbound and outbound directions.

Intersection of two one-way arteries 2

The model for a pair of intersecting one-way

streets can be considered as two arterial components at

35

 

 

 

 

 

 

 

OII. O‘Ir or s <—
—>I® JED IE) 1K4) I

(b)

 

 

 

Figure 3.3. (a) Two-way artery.
(b) Model of artery (a).

right angles to each other as shown in Figure 3.4. Each
has a turning coefficient defined for it. Since there is

only one signal for the two components, they must share it.

 

l 1

 

 

 

 

 

Figure 3.4. Basic component for grid structures.
The component equations for this basic grid element are
as follows. In these equations 1 = l, 2.

_ 0, if kth phase is red (3.5)
lk _ {1, otherwise

Elk = logical complement of Alk (3.6)

36

P(i,j) = vi, j = l, n; O : P(i,j) < Di-Q(i+l) (3.7)
—P . . 6 — . . . = —
15 (1.3) s ”0””) (t Tm): 3 1' n l
x
aiVi(93p)iA + (1-a-i) Vi(x i (1-A) (3.8)
x xq

-Px(i,n)<5(t-Ti’n), j = n

Q(i): 0
0(1) -vi(a: )i A + Px(i-l, j) 6 (t-Ti_l’j); Q(i) 3 o
”q (3.9)
A if i =
A = 1k
Kik if i = 2 (3°10)
where i = 1 if i =
2 if i = (3°11)

The turning split factor is ai.

Grid of one-way arteries

Within a grid of one-way arteries the signal at
each intersection is shared by the two competing directions
of traffic. In the model, therefore, the grid components
described above is the basic building element. For con-
venience, especially in a computer simulation, the following
numbering scheme is suggested. Label the signals in any
order from 1 to m. The horizontal component associated
with signal i is labelled i and the vertical component is

labeled m + i, as shown in Figure 3.5. The actual values

37

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

—9:;H _J l_

62.. IL I IE
9 I C...

54 . C

WTI II (a) IT] jTl—

 

 

 

 

 

 

 

 

 

 

 

 

 

 

22 24 25 E;]
[21 27' 26 'Igl
_ 4I

T

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-—9|7 7 6 6 5 5 | 4
1‘ 20 19
(b)
o o 12 o 5 6 o 10 11 2 l3 8

14 ll 0 5 6 7 O 13 O 9 10 3 12 0

0 15 l6 17 o o 22 23 0 27 26 19 20 25

16 17 18 0 26 27 O 21 22 0 28 25 24 0
(C)

Figure 3.5. (a) Traffic grid of one-way arteries.

(b) Model for grid (a).
(c) Connection matrix.

38

for ai (i = 1,...,2m) must be determined through observa-
tion. Note that there are several input components in a
grid. The connection matrix for the system is also given

in Figure 3.5.

Intersection of a two-way artery and a one-way artery

 

A vehicle on a one-way artery which crosses a two-
way artery can turn either right or left while a vehicle
on the two-way artery can turn only right or left depending
on which direction it is traveling. To simulate these
turning Options in a model,a dummy is introduced at the
intersection on the one-way artery as shown in Figure 3.6.
This dummy (labeled 1) has a length D1 = 0, and the right
turning coefficient for the one-way artery is defined on
it. Signals l and 2 may work in unison or the red phase
on the one-way artery of signal 1 may be delayed slightly
in order to simulate the amber phase during which vehicles

from the two—way artery making left turns have the

‘EL_ If

-—> ; —> 1 1 r 2
I I I 3 I
I (a) (b)

Figure 3.6. (a) Intersection of a one-way artery and a
two-way artery.

vb

 

 

 

 

N

 

 

 

 

(b) Model of intersection (a).

39

opportunity to complete their turn. These vehicles turn-
ing left, therefore, do not interfere with the oncoming

platoons.

Intersection of two two-way arteries

 

The model for a pair of intersecting two-way ar-
teries uses four dummy components to simulate the traffic
flow. As shown in Figure 3.7 the dummies are numbered
1, 4, 6, and 7; the right turning coefficients are defined
on them. All four signals must Operate in unison. Un-
fortunately there is no simple way of simulating an amber

phase for all four directions simultaneously.

I

 

 

 

 

 

 

 

I

 

 

 

 

 

 

 

 

 

 

 

 

 

I

 

Figure 3.7. (a) Intersection of two two-way arteries.
(b) Model of intersection (a).

Multiple lane arteries

 

If an artery has more than one lane then some

40

modifications are necessary in the component equations.
If m1 is the number of lanes in the ith arterial section,
then equations (2.20-2.22) are modified in the follow-

ing manner, respectively.

Px(i'j) 6 (t‘T- 0)! j = 1,...,1'1-1

 

1,]
Px(ipj) = , .
miviIxE)i Alk - Px(1'n) 6 ‘t'T1,n" 3 = n
x
‘1 Q(i) 1 o (3.12)
6(1) = -mj_vi(acfi)i Aik + Px(i-l.j) 6 (t-Ti-l,j);
”q of. 0 (3.13)
’2‘
xi = qu(l + l) + j=lPx(l,j) (3.14)
miDi

The simulation ideas presented in this chapter
have been incorporated into a digital computer program
which is capable of handling a significant traffic
area. In the Appendix a flow chart of this program is
given along with a list of the symbols used to represent

the variables in these chapters.

3.5 Example

Under the best of conditions the worth of a simu-
lation model can be demonstrated by comparing data from
a real system and from the model. Evaluating the model

developed in these chapters would be easy if adequate

traffic data were available. Unfortunately, much of the

41

data from large traffic systems consist merely of hourly
vehicle counts. Usually these are for widely separated
points and are acquired over a period of several months.
Consequently, one cannot know accurately how events at
one point affect the vehicular movements at other points
in the system. Little of the data is concerned with
velocity and density variations over short time inter-
vals. Thus, evaluating the wealth of data generated by
the model is difficult.

The system used in the following simulation is
the CBD (Central Business District) of Lansing,
Michigan. The arterial component network is developed
from a street map supplied by the Traffic Division of
the City of Lansing (GE 1). These maps are shown in
Figures 3.9 and 3.8, respectively. In Figure 3.8 there
are 72 signalized intersections. Due to the additional
dummy elements, the number of traffic signals in Figure
3.9 is increased to 100 and the number of arterial com-
ponents is 200.

The number of lanes and arterial lengths are
accurately depicted in the model. The other necessary
data are estimated as accurately as possible. Input
rates, based on typical hourly counts supplied by the
traffic department, range from 0.1 to 0.2 vehicles/sec/
lane. The expected values for the turning split factors

are based on these counts and on the system geometry.

J I

 

42

 

 

 

 

 

 

 

 

 

 

#JI JI__J[_ Logan

I I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I I

II

I V

 

 

I
Oakland :jjéjj \
Saginaw
IJI
IFI ”—7 H
c
O
p .4 u
m s O O) o H .c
c c -u c c m O
H H H ----I (U 'U H
m c Q .2 H o m
. 3 m m 0 L) A
Shlawassee 0 g
I I I
I FTT—
Ionia 44.
Ottawa #44.
. L :3
Michigan .
,Allegan 17* I
I
Washtenaw 'l \
J :I
Kalamazoo 1
N
St Joseph T
;:
Main
L_____ __
Figure 3.8. Traffic grid of Lansing, Michigan's

central business district.

 

 

 

 

 

43

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

£2 :3
o o
.—I 4.) 4.)
4.) o tn 0')
c c :3 .u c: c 'U H .r:
:6 r6 0) c. "-1 --i «4 c: (6 O
0') CO C. .—I o. .c: .c: rd rd :4
O O «4 (U m m m H <1) rd
A .4 n. 3 0 g g 0 O .4
.1 _. _ _
I I I 1 1 Oakland
I::F“'I::u__:I:: 1::r:I;L_:l::1 , <—
__ JDHJH __ __) _| Saginaw
—>:—II __—I I I I I I I9:
_ I T _ Li __J [— L E Shiawassee
CE I?“ 12:: r—“w I I r—‘r—I «—
E ._ E “—1 ,_ I Ionia
9:211: _C:I _:3_:II:::I_::1 “—1
.. .1 .s.
— fl L- -4 F Ottawa
I | I I I I I *-
_ t. __
F L" _: _ y _: Michigan
| L..- I:_ I'_—'1 L___.L__ *-
I. I.— i-I _ E Michigan
+H__1 ___J 1 I": ::|
.1 _.
I: L ._ ‘- L C Allegan
—> "'I____1 _:I,.._J,.1_I ____I _:1
_. _ H r
L— ed _l _—
I I I Lg |_" f""" E—‘—'(._ Washtenaw
_ I— __ — :36 H — F _( Kalamazoo
—> I J.__i.__l 1____1 1“") “—IEEZI
_ T s. _ I. 7’ I H I
l——_I::.r L___I: —I 'I i<—- St Joseph
..._I i I— ._ I—
'9 I I-fi—j I J I I Main
_ I _ I I. L I
Figure 3.9. Model of grid in Figure 3.8.

44

For example, all vehicles traveling west on Michigan Avenue
must turn left at Capitol Avenue, and thus the turning
split factor is zero. It is assumed that all the traffic
signals operate with a 60 second cycle and an equal red-
green split. The relative timing of these signals is more
or less random.

A constant acceleration of 5 ft/sec2 and a free
speed of 60 ft/sec is used throughout the model. Referring
to Figure 2.2, v0, v1, x1 and xq are 57 ft/sec, 30 ft/sec,
0.5 and 0.85, respectively. The vehicles in the system
are assumed to have an average length of 20 feet.

This hypothetical study of Lansing's traffic has
two main objectives. The first is to demonstrate the
effectiveness of the model in simulating a large, realistic
system and to establish a measure for the ratio of computer
time to real time. The second objective is to illustrate
the variety of investigations which are possible with the
model. These investigations may be either macrosc0pic--
dealing with such variables as vehicle counts and velocities

for an extensive area of the system--or microscopic--dealing

with the detailed behavior on a small portion of the system.

Simulation

The simulation was performed on a CDC 6500

45

computer. Using a time increment of 1.0 second, 2000
seconds were simulated in 252 seconds. Based on these
figures, approximately 0.63 millisecond is required to
simulate the traffic behavior on a single component for
one increment of time. As indicated in the following

studies the data available as output varies widely.

Microscopic Study

Between Saginaw and Shiawassee streets on Logan,
the southbound platoon and queue behavior were studied as
a function of time. Data were printed every second.

The leading and trailing edges of the platoons and queues
are plotted on space-time coordinates in Figure 3.10.

It is easy to follow the cyclic behavior of the vehicles:
their accumulation at Saginaw, their acceleration, their
transit to Shiawassee and their deceleration. In the
figure one also notes that during the red time vehicles
are appearing on the street due to the turning movements
from Saginaw.

A cross-section of Figure 3.10 taken at a par-
ticular time produces a picture of the platoon and queue
states like Figure 2.1. As more vehicles are added to
the street the average vehicular velocity decreases, and
this is reflected in the decreasing slopes of the lead-
ing and trailing edges of the platoonszat t = 0 the

velocity is 57 ft/sec and at t = 140 it is 42 ft/sec.

H0H>mnmn msmsv paw cooumam oa.m whomflm

 

46

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

msmsv #4
IN d\\\. moosv.. \ \\\\ msmsv an o
3mcfimmm I I I

i.

M

%
gr pm com

.1.

\\ _

a

_

F

msmsw msmsv
mommmsmwam

I ~ ~ I~ WI “.5 coma

 

I 14

Avon omv own Aomm omv smmum

47

Note that the ends of the queues, having zero velocity
during the red times, are represented by lines with
zero slope.

Since it is assumed that the whole platoon be-
comes part of a queue once its leading edge has reached
the end of the queue, an impulse appears in the position
of the platoon's trailing edge. Although this descripe
tion is unrealistic, it is felt that no serious conse-
quences result. First, the important variables are the
average transit time per vehicle, which is determined
for the leading edge, and the time headway between the
two edges. If it were necessary to approximate the true
behavior of the trailing edge, using the above data
this would be easy. Secondly, since the queue serves
the vehicles on a first come-first served basis, the
vehicles from the end of the platoon (even though it is
assumed that they arrive early) will not be served until

their turn.

Macroscopic Study

The input rate to Capitol Avenue was assumed to
have a normal value of 0.15 veh/sec/lane. After an
initial period of 300 seconds, in which the system
reached a more or less steady state, this input rate

was increased by a factor of 3 for an interval of 100

seconds and then returned to normal. The average

48

velocity for several sections of Capitol Avenue are plotted.
The velocity on the input section (between Oakland and
Saginaw) drops considerably due to the increased load. On
the following section (between Saginaw and Shiawassee) the
effect is less evident. This is probably due to the greater
distance between intersections (1200 versus 800 feet).
Between Shiawassee and Ionia the velocity demonstrates the
same drastic response to the increased load as the input
section.

Between Ottawa and Allegan the disturbance is still
strongly felt, but the velocity in this region is also
influenced considerably by the turning movements onto
Capitol from Michigan. Note, for example, that the veloc-
ities in this region, even for normal operation, tend to be
lower than observed elsewhere. Finally between Washtenaw
and Kalamazoo the effect of the disturbance has been greatly
diminished--only a slight depression is noted. The effect
of the signal timing is one factor which influences the

results--e.g., the rate of the disturbance propagation.

49

OOON

 

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was»
cord cows com

masoH can mommmsmasm cmmsumn uuuuu I

mommmsmflSm paw Bacfimmm cmmsumn ll.l||

BmsHmmm pom Usmaxmo cmmzumn

 

 

mm

(ass/4;) KirooIeA

50

Umdcwusoo mscm>< Houfimmu mcoam >ua00am> .NH.m onsmflm

Aowmv mafia

ooom coma coca com
_ _ _ _

oonEMHMM can swampsmmz cmmzuon lllll I

ammoaad paw cmmflsoflz smmzumn II..II|

cmmasoflz can mzmupo cmmsumn

 

mN

(ass/4;) KurooIeA

CHAPTER IV
CONTROL OF URBAN TRAFFIC SYSTEMS

Within a system of signalized intersections traf-
fic flow is governed almost entirely by the traffic sig-
nals. By a judicious choice of signal variables one can
minimize total vehicle travel time and time spent waiting
in queues while maximizing vehicle counts. For medium
density traffic conditions an effective control strategy
based on minimal queue build-up is the establishment of
progressions on the arteries. A progression can be de-
fined as a steady state mode of Operation which allows
vehicles to travel at a specified velocity (the design
speed) from one end of an artery to the other without
stopping. The portion of the signal cycle for which this
is possible is called the bandwidth.

A method developed by Morgan and Little (MLl) is
a useful basis for determining a particular progression
design on a two-way artery. With the introduction of two
important theorems, it is possible to inspect a wide
range of designs with a minimum of calculations.

While a prerequisite for smooth, efficient flow

on an artery is the establishment of a progression, the

51

52

possibility that queues develoP always exists. Since the
signal parameters which satisfy the design specifications
of bandwidth and velocity are not unique, adjustments
within the framework of a particular design can discourage
the build-up of queues. In situations where queues have
developed, it may be necessary to perturb the progression
settings to eliminate them.

A discussion of traffic control would be incomplete
if it did not touch on the special problems encountered
on traffic grids. The extension to a grid of the control

. methods used on arteries is possible if some preliminary
ground rules are established.

This chapter presents some of the important ideas
pertaining to progressions. It also presents methods for
establishing progressions on one-way and two-way streets
as well as grids and, finally, some auxillary techniques

for maintaining progressions in the face of disturbances.
4.1 Space-Time Diagrams and Traffic Signals

In studying the motion of a body in a one—dimen-
saional space, a plot of its displacement from some refer-
ence point as a function of time is often helpful. On
Slush a graph the velocity at any instant is given by the
Slope of the plot.

Engineers, studying the behavior of vehicle pla-

'UMDIis on an artery, have long used space-time graphs as a

53

visual aid. These graphs display the locations of the
leading and trailing edges of the platoon as functions of
time. The spatial length of the platoon is measured as
1p; its length in time (headway) is measured as tp. The
ratio of these variables is the platoon velocity v.

When the behavior of vehicular platoons in the two
directions of a street are displayed on a space-time dia-
gram at the same time, a complete picture of traffic flow
on the street is obtained (GAZ). However, the diagram's
usefulness is limited to illustrating the flow for a given
set of traffic signal parameters and design velocities.
Under limited circumstances it may be possible to use the

diagram for noting how a change in a parameter affects

the flow. For example, in Figure 4.1 it can be seen that

 

 

 

 

 

 

 

Intersection
1 A—\— 7—- ~ - --— —L—--1~ --
2 \. // 1....

o

o

5 3 — A A — —

p

U)

-a

'0
4 _ u .
5 I 7/ Y _ \ time

Figure 4.1. Typical space-time diagram.

54

if the timing of signal 5 is advanced slightly the size of
the platoons which can be accommodated increases in both
directions. Such observations produce only limited quali-
tative information for improving the system's flow.

It is appr0priate to introduce a list of some
basic traffic terms and the symbols which represent them.
These terms occur frequently enough in what follows to

warrant their inclusion.

CYC - common cycle of signals (sec); a cycle con-
sists of successive red and green phases, the amber phase
being relegated to the red or green.

Gi - green time of the ith signal (sec).

Dij - distance between intersections i and j (ft).

vij - design velocity for vehicles traveling from

intersection i to intersection j (§§%)

Tij - transit time for vehicles traveling from
D

intersection i to intersection j (sec); Tij = ij.

V"
13
BW - bandwidth, the measure of the band for which

vehicles can travel the length of the artery without stop-

ping (sec); the bandwidth-cycle ratio is B = gga .
Bij - offset of signal j measured with respect to

signal 1 (sec); Bij is measured from the center of a green
of signal i to the center of the first green of signal j

such that (0 5 Bij < CYC).

55

These parameters are illustrated on the space-time diagram

of Figure 4.2. Often it is convenient to normalize the

 

 

 

 

time parameters by dividing by "CYC." Thus, gi = SIC ,
0': 91 < 1; etc.
F———— CYC R. G.
9f ‘1'“19’ l
» ' —_1 signal i
D. .
13
I- signal j
‘é—Bij A

 

Figure 4.2. Traffic parameters defined.
4.2 Steady State Queuing

The problem of attaining an efficient traffic sys-
tem and of maximizing flow is closely related to the prob-
lem of queuing. Therefore, a useful (though incomplete)
measure of a control system's effectiveness is the total
time that the vehicles spend waiting in queues.

As previously indicated determining vehicle behav-
ior at the intersections of a large system using mainly
statistical methods is nearly impossible. A study of
steady state queue behavior, however, is a reasonable ob-

jective. For the purposes of the following discussion steady

56

state implies that the vehicles are flowing continually
through the intersections during the green phases and that
the velocity is always assumed to be the same. A constraint
implied by these assumptions is that the green phases of
all signals are equal and the cycle length is the same for
all signals.

It will be shown that the formation of a queue at
an intersection is a function of the offset 3 of that inter-
section's signal measured with respect to the previous
signal, the velocity v of the vehicles and the green time
to cycle length ratio, g.

Used as an aid in the discussion of the queuing at
a single intersection, Figure 4.3 is a Space-time diagram
for two intersections illustrating the vehicle flow between
them. From the figure, it is evident that four cases exist
depending on the values of g and B. The first two cases
(a and b) correspond to situations where the trailing por-
tion or the leading portion, respectively, of platoons
leaving the first intersection encounters the red phase at
the second intersection. In the third case (c) the entire
platoon encounters the red phase, and in the last case (d)
the center portion of a platoon arrives at the second inter-
section during the red phase.

For each case three new variables are defined.
Vehicles arriving at the second intersection when the sig-

nal is red form a queue. The queue continues to grow until

57

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(C) (d)

Figure 4.3. Steady state queuing at an intersection.

58

the signal changes to green (Figures 4.3 a, d) or until
vehicles cease to arrive at its end (Figures4.3 b, CL In
either event, the variable n represents the time interval
of queue formation.

If the signal turns green and if vehicles are still
arriving at its end (Figures 4.3 a, d), the queue length
remains constant until vehicles cease to arrive at its
end. Alternatively, if vehicles cease to arrive at its
end but the signal remains red (Figures 4.3 b, c), the
queue length remains constant until the signal becomes
green. In either event, the time interval that the queue
exists with a constant (nonzero) length is T.

Finally, the queue begins to shorten at the first
instant when both the signal is green and no vehicles are
arriving at its end. The time required to dissipate the
queue is assumed to be the same as that required to form
it, n_

The third variable defined is 8'. Measured at the
second intersection, 8' is the time between the arrival of
the first vehicle in the platoon and the start of the next
green phase.

These variables are defined mathematically as

n = min [8" 1 - B', 9: l - g] (4-1)
T = '8' — 9' (4.2)
s' = B —’T + nCYC (4.3)

where T is the transit time between the signals and n is

59

an integer selected so that 0 5,8' < CYC. These relations
are more apparent after examining Figure 4.3

In Figure 4.3 there is presented also a graph de-
picting the queue length behavior during a cycle. The
average vehicle time spent waiting in a queue per cycle is
proportional to the integral of queue length with respect

to time over a cycle.

1 . CYC2

0-3
ll

[(nY) g + (nY) r + (nY) g

= [n2 + nT] Y°CYC2 (4.4)

Substituting (4.1) and (4.2) into (4.4) results in

 

 

Tq = min [8'2 + B'~ 8'-gl. <1—B'>2 + (1-8')- B'-g|.
92 + 9-IB'-g|, (1-9)2 + (l-g)-|B'-g|]y-CYC2
(4.5)

For a fixed value of g the minimum total wait per

cycle of all vehicles is given by

 

Tq(min) = mi?'IminlB'2 + B'-|s'-g|. (l-B')2
I I 2 I 2
+ (1‘8 )' B '9! . g + gole -g|, (l-g)
+ (l-g)-|s'-g|1} y - cyc2 (4,6,

The value for B' which achieves this minimum is 0 so that

T = 0. This result is readily seen from equation

q(min)

60

(4.6) or from Figure 4.4 which illustrates equation (4.5).
The conclusion that can be gleaned from the fore-

going discussion is that the Optimal selection of 8 occurs

2
Q Tq (max) (1-9) M CYC

 

 

Figure 4.4. Queue integral as a function of offset 8'.

when 8' = 0, (i.e., B = T—nCYC). At this value no queues
are formed and the flow y is maximum. For all other choices
queues are present and they have the detrimental effect of
reducing Y. For the assumptions in this example it is

clear that the best way to control an artery is to set the
signal variables so that vehicles can travel from one end

to the other without stOpping somewhere in between to form

a queue. For a real artery the same conclusion applies:

it is desirable to set the signal variables in conjunction
with the prevailing vehicle speed so the vehicles do not

need to stop at intermediate intersections.
4.3 Progressions

A method attributed to Morgan and Little can be
used to determine the signal settings for progressions on a

two-way artery when the velocities are specified everywhere.

61

Briefly summarized, they have shown that with each
offset value of either zero or one half of a cycle (that
is, half cycle synchronization), there results inbound
and outbound bandwidths which are equal. Among all the
possible half cycle synchronizations there exists a com-
bination which maximizes this equal bandwidth value.

The procedure suggested by them to achieve maximum
equal bandwidths is basically simple: A bandwidth is es-
tablished between the first and second signals. This band-
width can be maximized by selecting the proper half cycle
offset value (either 0 or %). The third signal can then
be selected with either 0 or % cycle offset so that the
bandwidth is reduced as little as possible. The procedure
is continued until all signals are considered. The pro-
cedure is then repeated for every pair of initial signals
and from the resulting bandwidthsthe combination of off-
sets producing the maximum bandwidth is selected.

The method has not been fully exploited for design
purposes. For example, they have shown in a corollary to
the main presentation how a design having equal bandwidths
for the two directions can be modified by reapportioning
the total available bandwidth between the two directions.
However, no sound criterion is given for this redistribu-
tion.

Another shortcoming is the lack of information re-

garding bandwidth as a function of velocity (or of any

62

variable, for that matter). If the progression velocities
are fixed precisely in advance, this information is not
needed to determine a progression design. However, this
is generally not true in a genuine design situation. The
better approach is to consider all designs for a range of
acceptable velocities and to select the one which pro-
vides the most bandwidth. Since the bandwidth is measur-
ably affected by even small changes in velocity, it is
worthwhile to have this information.

The simplest, yet most common, problem encountered
on a two-way street is to establish a proqression in each
direction when only two velocities are specified, one for
each direction of flow. A progression design exists for

each point of the subset defined by

2 - .
V - {(v1, v2). 0 < vi 5 Vmax}

For this important case, the following theorem demon-
strates how the bandwidth can be depicted as a function of

a single variable, thereby simplifying the design problem.

Theorem 1. Between a pair of intersections i and
j, if the bandwidths BW1 and BW2 are realizable for a

), then these same band-

design velocity pair (vi ,

. v..
3 31
widths are realizable with the design velocities (Vij,

!
vji) where

63

Vl‘ + v.1._.= ;%—. + v$—- (4.7)
ij ji ij ji

Proof: Assume that the progression bands 1 and 2 are as-
sociated,respectively, with the design velocities vij and
v between the intersections i and j. The distance be—
tween the intersections is designated Dij (or Dji). Bij
is the offset value for which the original design is

ji

realized.

Transit times for vehicles in the two bands are

D..
T.. = —£1 (4.8)
13 v..
1]
Dji
.. = v.. (4.9)
31 31
If the offset Bij is altered so that
I .—
Bij — Bij + AT (4.10)
new transit times can be defined
01'
I ._ _
Tij - Tij + AT — ‘ng- (4.11)
D'i
' = —- =
Tji Tji AT §+ (4.12)
31
Thus
+T.. =T! +T! (4.13)

Tij :1 1j 31

64

and

91.4.; —I—+-11— (4.14)
ij ji ij ji

The change in transit time does not affect the
relative time spacing of the leading and trailing edges
of the progression bands, thus the bandwidths BWi remain
fixed.

With this result it is possible to define a new
velocity ve such that

l l 2

5;; + ngr— V;_ (4.15)
where V6 is the value when the two velocities are equal.
For the two-way street example cited above, the design
problem is reduced to examining the equal bandwidth pos-

sibilities corresponding to the points of V where

V = {v : 0 < v < v I
e e - max

This information can be set forth in a graph. The
ordinate, bandwidth-cycle ratio, is the total available
bandwidth which can be freely apportioned between the two
directions, subject only to the constraint, that the band-
width in either direction must not exceed the minimum

green time, g Similarly, each abscissa point (velocity-

min'
Cycle product) represents a set of inbound and outbound

65

velocities constrained only by equation (4.14). Once a
point on the graph is selected and the individual veloci-
ties are fixed, one needs only to determine the cycle
length to complete the design of the progression.

As an example of such a plot consider the artery
used by Morgan and Little in their presentation: a two-

way street having ten intersections with specified green

phases. An immediate observation is that the graph of

025‘

B/cyc

 

 

l
5000 V-CYC (ft)

Figure 4.5. Bandwidth/cycle versus velocity-cycle.

Figure 4.5 is very erratic having no truly periodic com-
(ponents. Despite this lack of mathematical periodicity,
however, most values of the function are repeated many
izimes suggesting that the total bandwidth range may be
realized over a relatively narrow velocity domain.

This fact is especially fortunate since it is

66

likely that the choices of velocities and cycle length
come from relatively narrow ranges. That is, the veloci—
ties may range from 20 to 80 ft/sec and the cycle length
from 30 tc>100 sec; thus the product may range from 600
to 8000 ft. However, low velocities are usually associ-
ated with long cycles and vice-versa. Under these cir-
cumstances, the abscissa interval of interest is more
likely to be 2000 to 4000 ft. Over this interval all
values from the bandwidth range are realized and only
this part of the graph needs to be determined.

In the general problem the desired velocity along
the street may not be constant. In this case, it is use-
ful to divide the street into n segments, each segment
having constant inbound and outbound velocities. For
each point in the set v?“ there exists an equal bandwidth

progression design where

2n _ .
V — { (V1, 0 o o ,Vzn) o 0<ViSVmax }

where

i = 1,..., Zn

The previous theorem permits reducing the dimension of
this set by a factor of two, thereby cutting the search
thne. A greater simplification for this n-segment street

can be achieved with the result of the following theorem.

67

Theorem 2. If a street for which a progression
is being designed is partitioned into n segments such that
a different design velocity vek prevails over each seg-
ment, the band is equivalent to one having a universal
velocity ve obtained by defining an equivalent length for

each section such that

135.: D.. £- (4.16)

/

I

Proof: The transit time for vehicles in any seg-

ment is given by

D..

T.. = —1—1 (4.17)
13 v k
e

If the transit time is kept constant, then

e
OLD.. Di'
T.. = ——1% = —l (4.18)
13 av ve
e
Thus
k-
and
v
e _ e
Dij — __F’Dij (4.20)
ve

TTHe bandwidth remains unchanged with these transformations

0f velocity and length.
The preceding result is useful in determining the

Progression design for a street which may be partitioned

68

into segments, each having distinct design velocities.
Although the result does not reduce the dimension of the
space to be searched, it allows a more systematic search
to be made. It is possible to make a plot of available
bandwidth as a function of the single velocity ve. This,

however, does not constitute the complete picture.
4.4 One-way Streets

Although the problems encountered when establish-
ing a progression on a one-way street can be readily
solved using the methods for two-way arteries, the wide
usage of one-way streets, particularly in central business
areas, justifies a separate discussion.

The specifications for a one-way street include
the velocity over each arterial segment and the bandwidth.
The bandwidth can be set equal to the minimum green time
on the artery no matter what the specified velocities are.
Morgan and Little's method could be used to determine the
offset values which accommodate this specified design.
However, these offsets can be determined more simply by

the following relation.

3.. = iii-acre (4.21)

l] Vij

‘where n is selected so that 0 s Bij < CYC. Such a set of

69

offsets locates the progression band centrally within each
signal's green time.

Consider the one-way street progression of the
space-time diagram of Figure 4.6 For simplicity a constant
velocity is assumed along the street so that v12 = v23 =
... = v. Intersection l is the primary input for vehicles.
However, at each intersection vehicles are injected onto

the artery from cross-streets by turning movements (second-

ary inputs). A vehicle leaving intersection l at the

9‘ % r—‘Gl-T-BW

 

 

 

 

g

BW

 

 

 

Figure 4.6. Progression bandwidth.

progression velocity v during the interval (T 5.t 5_T + BW)
can travel the artery without stopping. The group of
vehicles leaving during this opportune time is a progres-

sion platoon.

70

Vehicles within the system which are not part of
a progression platoon result from one of the following
phenomena.

1. Vehicles which enter the artery at the primary

input at time t where (0 fi_t < T or T + BW < t E'TG ).
l

2. Vehicles which enter the artery at one of the
secondary inputs. (These vehicles necessarily enter the
artery during a red phase.)

3. Vehicles, which fail to maintain progression

speed, falling away from the platoons.

These vehicles are (probably) stOpped at one of
the succeeding intersections (especially intersection i)
forming queues. Unless these queues are dissipated before
the platoons arrive, they interfere with the movement of
the platoons. Under severe conditions, the queues cause
the breakdown of the progression, and, for this reason,
their effects must be minimized. Several steps can be

taken to this end.

Light Density Conditions

The first intersection of an artery is the primary
.input for vehicles. Forcing vehicles from this intersection
into progression platoons tends to minimize the number of
'vehicles which form queues. Therefore, by reducing the
green time of this first signal so that it has the smallest

green time of all signals, any vehicle entering the artery

71

at this point is in a progression platoon and can travel
to the other end without stopping.

This corrective action may be less desirable if
there are a significant number of vehicles which could
use the non-progression band and leave the artery before
being queued. Figure 4.7 shows some of these non-platoon
vehicles which leave the artery before reaching the criti-

cal signal i.

Q ueue __

V —
Non-platoon “
vehicles ‘

Progression
band

 

 

Figure 4.7. Non-platoon vehicles and queue on one-way
artery.

The band occupies the entire green phase of the
Ininimum green signal. For the other signals it is wise to
distribute the excess green time so it occurs to the left
Of the band; that is, so it occurs earlier in time. (See

Figure 4.8.) This shift in offset provides time for any

 

_
—
I
O
.—

: “‘\_
__\__

 

 

Figure 4.8. Progression with excess green distributed to
left of band.

queue formed at the intersection to start moving before

the arrival of the scheduled platoon, thereby minimizing

the interference between queue and platoon. The new off-

set value is given by

D.. G. —G.
8.. = _il + —$————1-—— n CYC (4.22)
13 vij 2

where n is selected so that 0 $.Bij< CYC. Figure 4.9

illustrates how this relation is obtained.

At troublesome intersections (e.g., intersection
i in Figure 4.7) the above measures are not sufficient to
completely dissipate the queues. In such cases a transient
control can achieve the desired result. If all the off-

sets are changed by the same amount the steady state set—

tings of the signals remain unchanged. During the transient

73

time, however, the queues are provided an opportunity to

clear.

--——~——‘I‘. .—-————>
Gi/ I" 13
signal

Si
2

(— B..+nCYC —J
1]

 

 

 

 

 

Figure 4.9. Illustration for Bij determination.

The phasing of the signals should be performed in
the following manner.

1. Increase, successively, the red times of sig-
nals k (k = l, 2,..., i - l) by 0 seconds.

2. Then increase successively the green times of
signals k (k = i, i + l,..., n) by a seconds.

This procedure increases temporarily the time for
which the queue at intersection i can move while maintain-
ing the usual number of vehicles entering at the primary
input. Repeated intermittenly, this procedure helps to
clear queues. Figure 4.10 illustrates this transient con-

trol.

Heavy Density Conditions

In a well-designed progression the timing of the

signals is such that as a platoon approaches an intersection

74

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Transient Period

Figure 4.10. Clearing of queues by transient effects.

75

the signal turns green. Under heavy density conditions in
which vehicles are queuing at the intersections a good de-
sign is difficult to maintain. When a platoon reaches

the end of a queue it has for all purposes reached the
intersection, and the signal should be turned green at
this time. As a general policy, therefore, as the queue
at an intersection increases, the offset of that inter-
section should decrease with respect to the preceding
intersection. Equation (4.21) becomes

Di' - aQi.
8.. = 3 3 -- n cyc, 0.31.0 (4.23)
13 vij

 

If a is selected greater than one, the queued vehicles
have a chance to accelerate before the platoon arrives.
Under extreme conditions where a queue is formed at every
intersection and extends over the entire artery, the off-
sets should be reduced to zero so that all signals turn
green simultaneously. Goodnuff discusses this problem in
detail (601).

By increasing the green phase of all signals but
the first, the vehicles on the artery have more time to
pass through each intersection. At the same time, all the
inputs, primary and secondary, are regulated (i.e., the
number of vehicles permitted onto the artery is decreased).

Lengthening the cycle of all signals results in a
lower progression speed, hOpefully coinciding with the

lower natural speed dictated by the heavier density

76

conditions. (See Figure 2.2.) This policy is very effec-
tive and easy to implement on an artery which has pro-
gression settings. Figure 4.11 illustrates a progression

for which the cycle is increased at t = T.

 

 

i-l

 

 

 

'i+1‘

 

 

Figure 4.11. Effect of increasing cycle length.

4.5 Grids

The study of arterial traffic leads naturally to
the study of arterial networks having at least one complete
circuit. A grid, as such a system of intersecting arteries
isicalled, operates most effectively when a progression is
established on each of its arteries. The same techniques
for'establishing progressions on isolated arteries can be
applied to the arteries of a grid, but there are constraints
for the signal parameters which cannot be violated.

A grid can be visualized as a mesh of arteries

77

having one or more internal circuits or loops. A simple
grid loop may consist of four arterial sections formed
into a closed path as in Figure 4.12. The following
definitions are useful in establishing the important re-
lation for the signal offsets around such a loOp.

Let the underlined index refer to the green phase
for north-south flows and the other index to the green
phase for east-west flows. Thus Bil is the relative off—
set of signal j with respect to signal i, measured from
the center of the east-west green for the ith signal to
the center of the north-south green of the jth signal.

The following relations are obvious from any space-

time diagram.

Bii = 959" (4-24)
Bij = QCYC _ Bjil a = 0'1 (4.26)

Since offsets are positive fractions of a cycle, the a's
are necessary to maintain this status.

The offsets around a closed 100p sum. to an integer
number of cycles. To show this, the following relations

can be used. (See Figure 4.12.)

+ 323 + 533 + alcyc, a1 =-1,-2

813 = 812 + 523
(4.27)

78

 

 

 

 

 

 

 

 

 

 

60 C3)

 

 

Figure 4.12. A grid loop model.
and

813: 811+ 814+Bi4+ B43+a2CYC, a = -1,-2 (4.28)

2

Using the relations (4.24-4.26) above and equating (4.27)

and (4.28) the following can be stated.

812+832 = 812 + 843 + (a2 -al)CYC (4.29)
312+3£2+623+83§fcyc = B£1+Bl4+84§CYC

+343+(a2-dl)CYC (4. 30)

a3 = 0,1,2 (4.31)

Since the 8's are positive, (a3 + a2 - a1) must equal
(OI 1, 2, Or 3).

By using Morgan and Little's method iteratively

79

it may be possible to establish progressions on a grid,
which satisfy the geometrical constraints of the arteries
and equation (4.31) for every closed grid 100p.

It was noted earlier that on an arterial more than
one design could be realized. The designs that are pos-
sible for a grid are even more varied. One may seek a
design which establishes a progression on every part of
the grid. Theoretically this is possible but it may re-
quire an enormous amount of computation time and ultimate—
ly result in very narrow bandwidths. It is reasonable,
therefore, to establish less restrictive objectives for
a grid and to devise methods to achieve them.

One effective technique is to divide the procedure
into two stages. In the first, the grid is broken into
simple subsystems of single grid 100ps and arteries. Com-
plete progressions can be established on these pieces
using material presented previously. In the second stage,
when the system is re-joined, one may not be able to
maintain the progressions established; however, one may be
able to minimize the delay on each artery by a compromise
shift of offsets at the tie points.

Consider the hypothetical grid consisting of three
north-south arteries crossing three east-west ones. As-
sume that in-order of demand priority, the highest is
labeled A and so on to the lowest which is labeled F.

This grid is broken into two subsystems as indicated in

80

Figure 4.13. For the first stage of the design the pro-
cedures presented previously produce progressions on each
of the arteries. In the second stage the system is re-
assembled an artery at a time beginning with A. The off-

sets on A, B, and C remain unchanged after assembly. An

B-—ll In _—

WV [I

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4.13. Subdivided grid.

interruption in the progression on D may result where D

crosses C. This is due to the fact that the progression
on C has already fixed the offset of the signal at the

intersection. Most likely, the progression established
on D fixed this offset at some other value. Since only
one offset is possible the progression on either artery
(or both) must be disrupted at this intersection. Simi-

larly one disruption results in the progression on E

81

where E crosses D and on F where it crosses B. If any of
these arteries are one-way arteries or two-way arteries
with traffic flow predominantly in one direction, it may
be desirable to have these disruptions occur either up-
stream or downstream depending on the particular arterial
traffic conditions. This would dictate to some extent
how the grid is sub-divided in the first stage.

Fortuitous geometries of certain grids having a
large number of one-way streets make the problem of es-
tablishing progressions less formidable. For very regular
geometries it is always possible to satisfy equation
(4.31) using zero, half-cycle or quarter cycle signal
offsets. However, even with a good choice of cycle length,
the service on the established progressions may be low in

quality.

4.6 Example

 

As a further illustration of the diverse applica-
tions of the simulation model developed in the preceding
chapters a simple control problem is considered. Within
one section of the computer program it is possible to
adjust the timing of the traffic signals on a one-way
street to obtain a progression. The basis for these set-
tings is equation 4.22

Along Walnut Street the signal offsets were ad-

justed initially for a progression velocity of 40 ft/sec,

82

and these were not changed during the run. On Pine Street,
which is parallel to Walnut, the offsets were set arbitrar-
ily initially but during the run were intermittently (every
60 seconds) adjusted by the program so that a progression
was obtained. The input rate for each street is 0.25
vehicles per second. Figure 4.14 depicts the input compo-
nent velocities of these two streets. Succeeding figures
(4.15-4.19) illustrate successive downstream component
velocities. The average steady state velocity is 9 per
cent higher when the signal settings are adjusted (36 vs

33 ft/sec).

One notes, however, that the velocities observed
along Pine are more susceptible to oscillations. This
illustrates a serious problem which must be dealt with in
the future: In an attempt to attain a certain steady state
control strategy the signals have to be changed. During
the transient period resulting from this change the state
of the system may change enough that a different steady
state strategy is called for. This stability problem has
not been adequately considered in the program.

In Figure 4.20 the output as a function of time is
plotted. The steady state outputs (determined from the
slope of the linear region) are 650 and 595 veh/hr/lane
for Pine and Walnut, respectively. The output is improved
by 9.2 per cent when the adjustments are made. In Figure

4.21 the accumulated time spent by all vehicles in a queue

83

is shown. More time (7 per cent) is spent waiting in
queues when no adjustments are made. The steady state
rates are 41.4 and 44.3 veh-sec/sec for Pine and Walnut,
respectively. (It seems reasonable to expect that a more
sophisticated adjustment of the settings could result in
a reduction of the "41.4" figure.)

For an individual vehicle the improvement is more
pronounced. In the steady state the average vehicle must
spend a total of 115 and 135 sec in queues while traveling
the lengths of Pine and Walnut, respectively--a 17.4 per
cent improvement with the adjustment. The average total
trip time on Walnut is 12 per cent greater.

The conclusion that can be drawn from this example
is obvious. The better the progression matches the condi-
tions existing on the artery the smoother will be the
traffic flow. The velocity will be increased, the wait
time decreased and the output increased. Although one
cannot determine a synchronization which will be good for
all conditions, any synchronization is better than a random
setting of signals. On the other hand, stability becomes

a serious problem when corrections are attempted.

84

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CHAPTER V

OPTIMAL PROGRESSION DESIGN

Although the theory Of Optimal control is a fairly
well—established one, its application to real systems is
often severely limited. Many times the equations for the
system are not known with a great degree of accuracy, and
without an accurate description nothing like an Optimal
control is possible. On the other hand, if all the facts
were accounted for, by sheer magnitude the problem of Op-
timality for certain (large, nonlinear) systems would
overwhelm even the most ambitious engineer.

It would be unfair to generalize by concluding
that an Optimal control for a traffic system cannot be
devised for (l) a lack of knowledge and (2) an overabund-
ance and complexity of variables. Sufficient evidence
indicates that a traffic system can be efficiently con-
trolled using a progression. A reasonable next step is
to select the progression design in a manner which best
satisfies the demands of the artery. This chapter sets
forth a practical method for arriving at this Optimal

design.

92

93

5.1 Preliminary Remarks

The specifications for a progression design include
the bandwidths and velocities over every segment of the
artery. If this design is achieved without regard to the
prevailing traffic conditions, it is likely to be inade—
quate. For example, the velocities cannot be arbitrarily
specified since they are closely correlated to the number
Of vehicles demanding use Of the artery. It is necessary
and reasonable, therefore, to include this demand some-
where in the procedure.

In the discussion to follow a demand, speaking
quantitatively, is a measure of the number of vehicles
per unit of time desiring the use of an artery. Similarly,
a flow rate is a measure of the number of vehicles per
unit time actually using an artery.

For a one-way street it may be possible to find a
progression design which handles a maximum demand at a
good average velocity. For a two-way artery on which
neither flow demand is negligible, selecting a good de-
sign is more difficult. There are many combinations of
design parameters (bandwidths and velocities) to try and
many that satisfy the given traffic conditions. The prob-
lem of selecting a best design is further complicated when
the demand does not remain constant along the artery.

A space-time diagram is useless in this instance.

94

While it illustrates the structure Of a particular design,
it fails to show if the system is effectively meeting the
flow demands. Likewise, while the methods of Morgan and
Little are useful for determining the signal settings for
a two-way street progression, they too fail to give an
indication to the user as to which sets Of parameters
satisfy the flow demands on the artery.

In fact present methods for establishing a pro-
gression on two-way streets avoid this question almost
completely. They (l) a priori specify the velocities based
on gross Observations Of the traffic, and then (2) deter-
mine the bandwidths that these velocities produce. For
the two-way street either the bandwidths in the two direc-
tions are made equal and maximum or one is maximized
subject to the restriction that the other does not drOp
below a pre-assigned minimum. (This latter alternative
hinges on the well-established fact that an increase in
the bandwidth Of one direction is usually accompanied by
a decrease in the bandwidth Of the other direction.) In
no way, however, do these techniques provide an answer
to the Optimal progression design problem. Yet very
little, if any, work on progression design uses more.

A progression that is optimally adjusted to the
prevailing traffic demands must be implemented with
traffic signals which can be automatically changed by a

central digital controller. Initial steps toward a

95

central traffic control have already been taken in Toronto,
San Jose, and Witchita Falls (HUI, SJl, CA1) with signifi-

cant gains reported.

5.2 The Mathematical Model

 

For a progression to Operate effectively no con-
gestion can occur and the demand everywhere must be
matched exactly by the flow rate. To achieve this when
the demand does not remain constant complicates the prob-
lem.

Often the ends of an artery do not comprise the
only vehicular entry and exit points on the artery. A
large number of vehicles may enter at one or more internal
intersections, and possibly many vehicles leave the artery
before reaching its end. As a result of these turning
movements both onto and Off the artery or due to a change
in the number Of traffic lanes, the demand along each
direction of flow may not remain constant. An excellent
approximation is that this demand varies in a stepwise
fashion along the artery's length. It is natural, there-
fore, to assume each point (intersection) where the de-
mand changes significantly represents a boundary between
segments having unequal demands for one or both directions.
Thus an artery is, for the present purposes, divided into
n segments if at n-l internal intersections there is a

change in either the inbound or outbound demand.

96

The first point Of concern is to determine the
flow rate that a given progression can sustain. The
number of vehicles per unit of time that can be served
by the street depends not only on the bandwidth of the
progression but also on the density of the vehicles in
the band. The relationship between average vehicular
velocity and density has already been presented in Chap-
ter II. Greenshields and others have established by the-
oretical and experimental studies that for a single lane
of traffic there is an almost parabolic relationship be-
tween the flow rate and the lane occupancy as shown in

Figure 5.1 (GRl). The corresponding speed of the traffic

 

 

Y
ys y = G(x)
Y1 = 313'”)
7 x
0 xj

Figure 5.1. Flow-density characteristic based on Green-
shields' linear model.

stream varies with occupancy as shown in Figure 5.2.

97

 

Figure 5.2. Velocity-density characteristic.

Inasmuch as the same basic phenomena governing speed and
density are effective in a platoon as in the continuous
stream of vehicles, the flow-occupancy characteristic for
the progression system is given by the product of the
continuous stream's characteristic and the bandwidth to
cycle length ratio. Figure 5.1 illustrates both charac-
teristics. The continuous stream flow rate is given by
y8 = G(x), the lane occupancy by x and the ratio of band-
width and cycle length by B. There is a unique graph for
each distinct bandwidth: yi = Bi G(x).

In a continuous steady state traffic stream the
vehicles adjust their speed so that the flow rate matches

the demand exactly, provided the capacity Of the traffic

98

lane is not exceeded. If the demand exceeds the capacity
Of the street, congestion quickly occurs upstream. Below
the saturation level, an increase in demand is accommo-
dated by a lower'individual vehicle speed and higher lane
occupancy, as demonstrated in Figures 5.1 and 5.2.

Analogously, within a progression system the speed
should be selected so that the service flow rate always
matches the demand and so that the platoons within the
band are filled with vehicles at the density dictated by
the speed—occupancy characteristic. Setting the progres-
sion Speed higher than the value for which demand and ser-
vice rate are balanced invites queuing and congestion,
while setting it lower implies a needlessly lower quality
Of service.

If the demands on each part of the partitioned two—-
way street do not exceed the capacity then many combina-
tions of speeds and bandwidths for a progression design
are possible. The problem is tO determine the unique
combination which Offers the highest quality of service.

In the analysis which follows, Greenshields' traf-

fic model

y = 4ysm (cf—I) (l - f—j (5.1)
3 J
v = vf (l — jig (5,2)

3'

99

is used where ysm is the peak flow rate, 33 is the jam
lane occupancy and vf is the free speed. Comparable re-
sults may be obtained using other models which have been
prOposed.

If'x is eliminated from (5.1) and (5.2) the result
is

_. _LJL
Y - 4ysm (1 Vf) (Vf) (5.3)

Applying the principle that bandwidths and progression
speeds should be selected such that service rate equals

demand on each arterial segment, requires

= 4

 

 

1 V2k-l (Vzk-l)
" I

y2k-l B2k-l ysm Vf vf

k: 1,000,!) (504)

and

V V
_ _ 2k 2k _
sz ' 4B2k Ysm (' VE—I (GE-Io k - l.--..n (5.5)

where y2k-l and y2k are the inbound and outbound demands

and v2k-l and v2k are the progression speeds on the kth

segment.

5.3 The Optimization

 

A reasonable expectation for the progression is
that it satisfy the demands of the artery while minimiz-

ing the total vehicle-hours of travel time in the system.

100

A measure Of total vehicle-hours of travel time under

steady state Operation is

y2k-1 sz

V D+——D
1 2k-1

(5.6)
k v2k

 

'11
II
II Mb

k k

Here y is a measure of the number of vehicles served in
an interval Of time. The individual trip time over each
segment of the artery is given by D/v, the ratio of the
segment length and steady state velocity. The function
F must be minimized while taking into account the con-
straints Of realizability Of the bandwidths and progres-
sion speeds selected.

Before proceding with the Optimization process,
the relation established in Chapter IV, equation (4.16)

is recalled.

k=lp000pn (507)

 

Within each segment the progression speeds can vary with-
in this constraint without changing the bandwidths of
that segment. Note that vek is the progression speed
which would prevail if v2k-l and v2k are equal. Inspec-
tion Of equation (5.7) indicates that allowing vek to
increase without disturbing bandwidths results in higher

permiSSible values of VZk-l and V2k .

101

If equations (5.4) and (5.5) are substituted into

(5.6), the result is

 

 

 

3.: % 4sz-l ysm Dk 1 _ v2k-1 +
k=l Vf Vf
4B2k ysm Dk 1 _ Vzk (5 3)
Vf Vf

If F is minimized by the method of Lagrange mul-
tipliers subject to the constraint of (5.7), the Optimum

occurs when

 

 

3 n
-5—v-—-——- F+ Eu.[ 1 +3—1—-""2_] =0k=l’000,n
2k-1 j=1 3 V2j-1 2j ej
(5.9)
n
33 F+ 211 [V1 +—-V1 __V2] -0 k=1,...,n
2k ._13 2j-l 2j ej
3‘ (5.10)

where u is a Lagrange multiplier.
The simultaneous solution of (5.9) and (5.10)

after substitution Of (5.8) results in

 

 

k = 1,...,n (5011)

2
v2k-1 B2k
Vzk

To be sure that the extremum Obtained is a minimum and

not a maximum, note that a maximum for F requires all the

102

velocities to be zero--clearly this is not the case. The
relationship given in equation (5.11) is for a minimiza-
tion of F.

This latter relation must be satisfied within each
segment on the artery. Taking the ratio of (5.5) and (5.4)

results in

 

 

 

 

Y2k B2k Vzk (Vf ' Vzk' _
u-y— —B V TV -V ) k-lp000'n
2k-l 2k-1 2k-l f 2k-1
(5.12)
Substituting (5.11) into (5.12) yields
sz = v2k-1 (Vf'vzkI = Vzk _
y2k-1 V2k Wf'vzk-l,’ Vf _ 1
v2k-1
k = 1,...,n (5013)

Simultaneous solution Of (5.7) and (5.13) results in

 

 

 

 

y
v (1 + y 2k )
2k-1 = 2k-1 k = 1’...’n
Vf sz + zvf 1
Y2k-1 Vek (5'14)

and

103

 

 

 

Y _
v 1 + 2k 1
_2i<_ = y2k
Vf 2v k = 1,...,n (5015)
y2k-1 + f _ 1
y2k vek

A realizable design, Obtained using the Morgan and Little
procedure and the theorems introduced in Chapter IV, ac-
counts for the physical dimensions of the artery, the
signal green times and the maximum allowable speeds.

In applying these methods it is implied that the total

bandwidth along the artery is constant,

sz-1 + 32k = c k = 1,...,n (5.16)

The Optimal design results from an iterative
search employing the exact equations (5.14) and (5.15)
and the design method of Morgan and Little, subject to

the minimization of the cost function of equation (5.8).

5.4 The Search

 

The search is initiated by selecting a point Vo
having n coordinates, each coordinate representing the
equal progression velocity on a segment of the artery.

Each coordinate is restricted to the interval

0.5 v 5 vf k = 1,...,n (5.17)

f 5 Vko

104

A design with one or more velocity coordinates in the
neglected interval, 0 S'Vko < 0.5 vf, always provides
lower quality service than one with all coordinates in
the interval (5.17). This means that the flows are al—
ways achieved on the left half Of the flow-density curve
of Figure 5.1.

The second theorem of Chapter IV permits trans-
forming the point VO associated with the given artery A to
the equivalent point V5 associated with the equivalent

artery A'. The point V5 is given by

V; = (l, 1,...,1)VO

Theorem 1 permits the determination Of the maximum band-
width which is realizable On the artery A' for the vel-
ocities V6, or equivalently, on A for V0. Equations
(5.14) and (5.15) determine the inbound and outbound
progression velocities on each segment. Finally, these
values are substituted into the cost function expression
and stored as F.

The coordinates of V0 are successively incremented
and in each instance the above procedure is repeated to
determine Fi' This Fi is compared with F0 to determine
if better service is achieved. The procedure is repeated
until no improvement is Obtained. The incremental step

size is then reduced for finer resolution Of the function

105

space. A steepest descent approach is used tO speed the
rate of convergence.

The combination of the arterial geometry and the
signal variables produces a bandwidth-velocity function
which is not convex as noted in Chapter IV, see Figure
4.5. As a result the function F(-) to be minimized is
not convex. Therefore, one cannot achieve a global mini-
mum for F(-) by a direct application of the above search
procedure, although a local one is assured. The global
minimum is Obtained by determining the minimum for the
function resulting from a number of different starting

pOints V0.

5.5 Example

 

Present attempts to establish progressions on
two-way streets are mostly concerned with the heavier
demand directions. TO adjust to a change in the demand
once the system is in Operation, the signal cycle is
modified. Such action decreases or increases the pro-
gression velocities in both directions by the same factor.
This unduly penalizes the traffic in the low demand
direction.

Euclid Avenue in Cleveland, Ohio, is a typical
street on which progressions might be established. It
has ten intersections, unequally spaced over its 6050

feet, and traffic signals having a variety Of red-green

106

splits. Initially it has been assumed that the two
demands are constant along the street length. The results
in Figure 5.3 illustrate that it is possible to give
better service to traffic in both directions in terms of
a higher progression Speed by taking account of the light
demand as well as the heavy.

When the street is partitioned into two or three
sections corresponding to lengths of constant demands,
similar results are obtained and examples are given in

Figure 5.3

107

 

l 0 Bi BO Vi 52'
ysm ysm f f
0.23 0.23 0.237 0.237 0.589 0.589

0.20 0.23 0.226 0.249 0.669 0.638
0.15 0.23 0.210 0.265 0.767 0.682
0.10 0.23 0.196 0.280 0.850 0.711
0.07 0.23 0.188 0.288 0.896 0.724
2-sections
section 1 0.23 0.20 0.257 0.235 0.663 0.693

2 0.15 0.10 0.259 0.229 0.824 0.875

section 1 0.15 0.10 .266 0.237 0.831 0.8801

2 0.15 0.18 0.238 0.257 0.804 .773J
3-sections

section 1 0.23 0.20 0.259 0.237 0.668 0.698

2 0.15 0.20 0.233 0.265 0.798 0.747

3 0.18 0.10 0.270 0.221 0.788 0.870

Figure 5.3. Optimal progression characteristics.

CHAPTER VI

CONCLUDING REMARKS

The primary objectives at the outset of this re-
search were tO develop a simulation model for arterial
traffic systems and to establish a steady state control
strategy for the system for medium to light density traf-
fic conditions. During the research many other facets
of the problem were recognized. The intent of this con—
cluding chapter is to summarize the material presented
in this dissertation and to indicate potential areas for

future research.

6.1 Traffic Model

 

Crucial to the development of the traffic model
were the identification of certain entities and the selec-
tion Of variables to describe them. The features which
distinguish the arterial system are the platoons and queues
formed as a consequence of the traffic signal controls.
Describing these phenomena in the most direct manner dic-
tated to a great extent the choice Of position and density
as the basic variables for the system. Thus the descrip-

tion of each platoon and queue in the system requires a

108

109

pair of these variables. The differential equations for
these variables were based on published experimental and
theoretical investigations and other approximations due
to the author. These equations given for the platoons
and queue Of an arterial section comprise a component
state model. When a number of these component models are
combined according to the procedures of system theory,
they form a state model (KTl). With such combinations
the simulation Of all traffic structures (e.g., two-way
streets, grids) is possible.

Despite the relative simplicity of the simulation
model several factors make analysis difficult. (l) The
simulation model for ordinary traffic systems quickly
becomes large. Its order at any time is given by 2p + q,
where p is the number of platoons and q the number of
queues. The number of platoons is not constant--it is
affected by the density conditions and the signal con-
trols. (2) The velocity-density relationship is a non-
linear one. (3) Certain random phenomena used to give
more realistic results make the system only quasi-deter—
ministic.

The present model has several advantages over other
models making it a desirable and effective simulation
tool.

1. Variety of Applications. A large number of

traffic situations which might require investigation by

110

simulation have densities ranging from light to medium.
The model is develOped especially for these conditions.
Without requiring a prohibitively large amount of data,
it functions equally well in the study of single arteries
or complete grids.

2. Design and Analysis Capabilities. Since the
effects of some changes in a traffic system cannot be
predicted, the capability to evaluate these changes be-
fore they are instituted proves to be an economic, time-
saving, safety, and even political advantage. The simu-
lation model can be used to determine the best locations
and settings of traffic signals, to specify speed limits,
to evaluate the desirability Of one-way streets and other
prOposed changes in the traffic system, and to predict
effects, for example, Of a parking lot on traffic pat—
terns. Further, it may prove useful in the investigation
of disturbance propagation along an artery and through-
out a grid.

3. Ease of Computer Programming. To simulate a
traffic system requires a minimum of data preparation,
and the variables (e.g., vehicular velocity, position,
and counts) which are time and Space dependent, are
readily available as outputs. Although velocity, accel-
eration, and vehicle input behavior have been simulated

in the model, it is possible to rapidly change these

111

descriptions if Special studies are required or if fur-

ther research dictates better ones.

6.2 Control

 

In Chapter IV it was shown how the procedure
developed by Morgan and Little can be used as a basis
for designing progressions. The question Of selecting
design parameters is easier to answer with the inclusion
of two results which provide a means for displaying the
velocity and bandwidth dependence and the effects of a
nonconstant velocity along an arterial, respectively.
These results find the greatest application in the Opti-
mal design procedure presented in Chapter V.

The Optimal design is the steady state control
of an artery which minimizes the total travel time on
the artery. The iterative search procedure used to ac-
complish this considers not only the Optimality conditions
relating velocities and bandwidths but also the constraints
imposed by the geometry of the artery and the traffic

Signal variables.

6.3 Future Investigations

 

Although many aspects of the arterial problem
have been considered, several studies are suggested by
this dissertation.

Most of the control strategies presented apply to

112

steady state operation. It is possible to use the model

to investigate stability when the input demands change

or the signal parameters are varied. Such a study would

reveal how quickly the system controls should be adjusted
to meet new demands while minimizing transients.

The biggest menace to progression systems is the
formation of queues. Although some strategies were sug-
gested in Chapter IV for avoiding queue development, in-
vestigations for the future could place greater emphasis
on the queue phenomenon and how it contributes to conges-
tion. These studies may suggest the best control strat-
egy to return a congested system to a normal one having
progression. Transient stability would be a significant
factor in this analysis.

In the event that it is not possible to establish
progressions throughout a traffic system, a simulation
test can be made to determine alternative traffic settings
which will minimize travel and wait time. Although this
has been done before, vehicle acceleration and decelera-
tion have always been neglected. It is likely that tak—
ing these into account will reveal significant differences
between the results Obtained when acceleration is con-

sidered and when it is ignored.

APPENDIX A

The primary goal achieved in Chapters II and III
is the development of an efficient Simulation model for
urban traffic systems. The equations which comprise
this model are best solved using a digital computer. A
Fortran program for this purpose is available. The pro-
gram consists of a main program called TRAFIK and three
subroutines: BINOM, POISSON, and GENERAT.

Within the program the following scalars (con-

stant) are used.

ACL average car length

B vl (see Figure 2.2)

BRAKE deceleration constant

D1 x1 (see Figure 2.2)

F lane occupancy for queues

H time increment

KN length of table for Poisson distributed
inputs

MN artery (1) or grid (2) indicator

MPN number of outputs

MSF number of expected values of turn
coefficients

NEX ramp (0) or exponential (l) accelera-

tion indicator

113

114

N00 number of inputs

NP number Of arterial components
NS number of traffic Signals

TF total simulation time

TQUEUE accumulated queue waiting time
ZOOM acceleration constant

The following vectors (constant) are used. Unless

otherwise specified, the vectors have NP coordinates.

CM free velocity

DT length of arterial section

EXPVAL expected inputs per unit time (NOO
coordinates)

KM3 integer denoting index of appropriate

expected value Of turn coefficient
(1 f. KM3 (I) _<_ MSF)

M number of traffic lanes
TURN expected value of turn coefficient (MSF
coordinates)

The matrix KM is used to specify how the system is
connected. Its dimensions are NPxZ.

KM(I,l) index of the component succeeding the
Ith (equals 0 if Ith is an output)

KM(I,2) index Of the component preceding the Ith
(equals 0 if Ith is an input)

The variable T is time.

The following vectors (variable) each have NP
coordinates.

D output counter (if KM(I,1) = 0), distance

from lead vehicle of lead platoon to end
of queue (otherwise)

Of rows

in each

DCR

DENMAT

115

critical distance at which vehicles

either begin braking or continue at

constant velocity

average lane occupancy for component
queue length

current turn coefficient

position Of leading edge Of lead pla-
toon

In the following matrices (variable) the number

is always NP. The number of columns is indicated

instance.

ACCEL

CRIT

DECEL

PD

accelerated velocity, (2 columns)
(col 1) for platoon; (col 2) for queue

steady state velocity, (2 columns)
(col 1) for platoons; (col 2) for
queue

description of transition platoon ex-
isting between platoons and queue and
having a constant velocity, (4 columns)
(col 1) length of platoon

(col 2) vehicular length of platoon
(col 3) distance to end of queue

(col 4) velocity of platoon

description of transition platoon ex—
isting between platoons and queue and
having a decelerated velocity, (4
columns)

(col 1) length of platoon

(col 2) vehicular length of platoon
(col 3) distance to end Of queue

(col 4) velocity of platoon

length of platoon, (KP columns, see
below)

vehicular length of platoon, (KP col-
umns, see below)

116

The vector NDECL having dimension NP is an indi—
cator for the presence of CRIT and DECEL platoons

(a) neither type platoon present, indicated by 0

(b) only a CRIT platoon present, indicated by 1

(c) both type platoons present, indicated by 2

(d) only a DECEL platoon present, indicated by 3.

9%
Equations (2.17 - 2.20) for the states of the ‘I

platoons and queues are deceptively simple, but require

 

meticulous care in programming. AS a result several L ‘
variables, not explicit in the equations, have to be ii
introduced.

KQ is a vector with NP coordinates which describes
the nature of the queue.
(a) the absence of a queue, indicated by 0.
(b) a queue whose vehicles are accelerated across
the intersection, indicated by +1.
(c) a queue whose vehicles move at a constant {1

velocity across the intersection, indicated

e. .' J...‘.-' o. -

by -10

Since the number of platoons on the arterial'sec-

 

‘§ ‘."L..' 2‘! .:

tion is variable, the vector KP of dimension NP denotes
this number.
The variables of the traffic signals are contained

in the matrix MT having dimensions NS x 5.

ll7

(col 1) the current time such that 0 :_MT(I,1)
<MT(I,2)

(col 2) the signal cycle length

(col 3) the green time

(col 4) indicator for current phase
(a) signal is green, indicated by 0.
(b) signal is red, indicated by 1.

(col 5) indicator for phase of previous time
increment.

(Note: If MT(I,4) = MT(I,5), then signal has not changed

phase during the time increment.)

Three subroutines are used in the program TRAFIK.
Two of them, BINOM and POISSON, determine tables of dis-
tribution functions, and the third, GENERAT, using these
tables selects random variables which are Poisson or bi-
nomially distributed.

BINOM: For each expected value, TURN, for the

turn ratio, a row of the matrix ATPOI is determined such

that
I-l
ATPOI (',I) = Z P(X = J)
J=0
where P(X = J) = (130) TURNJ (l—TURN)100"J

POISSON: For each expected value, EXPVAL, for the
input arrival rate, a row of the matrix THRESH is determined

such that

,‘F'fll-nlé-u .I!" - . ..--
n.

 

k- T:
. I"

'I

IL

 

118

 

I-l
THRESH (HI) = Z P(X = J)
J=0
where
J
p(x = J) = exp (EXPng)(EXPVAL)
GENERAT: A uniformly distributed variable R is

generated (R = RANF(-l)) and is compared with a table to
determine another random variable NGEN which is either

Poisson or binomially distributed, depending on the table

used.

THRESH b, NGEN-l) < R g THRESH (-, NGEN)

An abbreviated flow diagram of the program is pre-
sented in Figure A.l. Since the program is long (over
500 statements) and uses over 150 "IF" statements, only
the major logical branches are included. Most of the
blocks in the diagram represent a "DO" loop with "J =
l, NP" or "J = 1, NS." It is clear in each instance which

one is intended.

119

 

 

 

[tREAD input data

 

V

 

{-Determine MPN,NOO

 

 

 

 

PRINT input data

 

 

 

 

 

Set initial values for
DECEL, CRIT

KQ
X
MT

DENMAT
C

ACCEL

TQUEUE

 

 

 

 

 

CALL POISSON
CALL BINOM

 

 

 

 

 

‘ Increment time
T = T + H

 

 

 

 

 

IF T = nMT(I,2)
CALL GENERAT to
determine binomially
distributed turn ratios

 

 

 

db

120

93

 

 

CALL GENERAT
generate vehicles at inputs
Q = Q + n veh

 

 

 

 

110
T :_TF? r————~‘:l’I
1 yes

Update MT

 

 

 

 

 

 

 

 

 

no
MT(I,4) = MT(I,5)?

 

 

 

 

yes

 

 

Determine DENMAT(I),
C(I,J)

 

 

 

 

 

 

 

KP(I) = KP(I) + l

 

 

 

 

Update DECEL

 

 

 

 

I

 

Update ACCEL

 

 

 

 

I

 

X = 2P

O

 

 

 

121

 

 

MT(I,4)

 

 

 

 

I: 0

Decrement Q

5}

Increment PD, P
(For output component
increment counter.)

I

Determine distance
to end of queue
D = DT -X “Q

I

Update CRIT

I
Form DECEL, CRIT
platoons if possible

I

Transfer vehicles from
CRIT, DECEL to Q if necessary

I

Update KQ

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I
Determine TQUEUE

 

 

 

 

 

 

I

PRINT output

 

 

 

 

Figure A.l Flow diagram for TRAFIK

APPENDIX B

To find the optimal set of velocities and band-

widths for progressions on a two-way street when the

demand is not

written. The

constant the Fortran program SEARCH was

program uses two subroutines: INITIAL and

MORLIT. The variables used in SEARCH are listed below.

The dimensions of the matrix variables are clearly in-

dicated.

BW

BMAX

DELTA

DIST

TWOB

VA

XE

bandwidth requirement for inbound (row 1)
and outbound (row 2) directions, (2 x M)
maximum available bandwidth

length of section, (M)

step increment (3)

pavement length between successive traffic
Signals, (N-l)

number of sections

number of traffic signals

total required bandwidth

normalized velocity of inbound (row 1) and
outbound (row 2) traffic flow, 0.5 < VA 3
1.0, (2 x M)

reciprocal equal velocity, (M)

122

123

XET trial reciprocal equal velocity, (M)
Y flow demand of inbound (row 1) and outbound
(row 2) traffic, (2 x M)

The purpose of the subroutine INITIAL is to deter-
mine a set of velocities VA (and XE) and bandwidths BW
(which meet the realizability requirements of a solution)
to serve as a starting point in the search.

The subroutine MORLIT is based on the equations
developed by Morgan and Little. It has been written so
that for a range of equal velocities the maximum avail-
able bandwidth BMAX is determined. This subroutine con-
siders the constraints imposed by the green-red splits

of the traffic signals and the distances between them.

The following flow diagram, based on the ideas
developed in Chapter V, outlines the main features of

the program SEARCH.

124

 

READ input data:
N,M
RED
DIST
DELTA
D
Y

 

 

 

 

II

 

Estimate XE

 

 

 

 

 

CALL INITIAL
determine VA, XE, BW

 

 

 

 

I

 

 

Determine FO:

 

 

 

 

 

 

 

F0 = z D(J)v(I,J)/VA(I.J)
I=l,2
J=1,M
XET = XE
I
J = 1

 

 

 

 

 

125

 

 

 

 

 

no

 

 

 

XET(J) = XE(J) -DELTA 1

 

 

 

 

Determine VA
corresponding to XET

 

 

 

 

 

 

 

 

J=J+l .4 no 0.5 5 VA 5 1.0?

 

 

 

 

 

 

yes

 

 

TWOB = max{BW(l,J)Y(1,J) + BW(2,J)Y(2,J)} I
J

 

 

I

 

 

CALL MORL IT
determine BMAX

 

 

 

 

I10

 

BMAX Z. TWOB?

 

 

 

 

yes

 

 

PRINT,
END

126

 

2 DIJ)Y(I,J)/VA(I,J)

FJH

CIH
n n
‘2‘»

 

 

yes

 

 

yes

 

 

 

 

 

 

 

 

 

Decrease Step size.

Third decrease?

 

 

no

 

 

 

PC = F, XE = XET
Increase step size

in direction of success

 

 

Figure B.l Flow diagram for SEARCH

(BEl)

(CA1)

(CH1)

(GAl)

(GA2)

(GEl)

(G01)

(GR1)

(HA1)

BIBLIOGRAPHY

Beckmann, M. J., gt 31. Studies in the Economics
of Transportation. New Haven: Yale University
Press, 1956.

 

 

Casciato, L. and Cass, S. "Pilot Study of the
Automatic Control Of Traffic Signals by a General
Purpose Electronic Computer," Highway Research
Bulletin, 338 (1962), 28-39.

 

Chang, A., "Synchronization Of Traffic Signals in
Grid Networks," IBM Journal, 11 (July, 1967),
436-441.

 

Gazis, D. C., Herman, R., and Rothery, R. "Mathe-
matical Car-Following Theory of Traffic Flow,"
Proceedings American Society of Civil Engineers
(Engineering Mechanics Division), 89 (December,
1963), 29-46.

 

Gazis, D. C. "Traffic Control, Time-Space Dia-
grams and Networks" in Horton, T. R. (ed.), Traffic
Control-Theory and Instrumentation. New York:
Plenum Press, 1965.

 

Gervais, E. F., and Associates; and Wiley, T. T.,
and Associates, Traffic Study--Central Business
District Lansing, Michigan, Report, February, 1969.

Goodnuff, J. L. An Application of System Theory
to the Optimal Control of Vehicular Traffic Net-
works, Ph.D. Thesis, Michigan State University,
1967.

 

 

Greenshields, B. O. "A Study of Traffic Capacity,"
Proceedings of the Highway Research Board, 14,
(1934), 448-477.

 

Haight, F. Mathematical Theories of Traffic Flow.
New York: Academic Press, 1963.

 

127

128

(HRl) Herman, R., Montroll, E. W. "Traffic Dynamics:
Analysis Of Stability in Car-Following," Opera-
tions Research, 7 (1959), 96-106.

 

(HUl) Heumann, G. H. "Digital Computers Rev Up Traffic
Control," Control Engineering, 14 (August, 1967),
95-100.

 

(KTl) Koenig, H. E., Tokad, Y. and Kesavan, H. K.
Analysis of Discrete Physical Systems. New York:
MCGraW—Hill' 19670

 

(LWl) Lighthill, M. J. and Whitham, G. B. "On Kinematic
Waves: II. A Theory of Traffic Flow on Long
Crowded Roads," Proceedings of the Royal Society,
London, A 229 (1955), 317-345.

 

(MLl) Little, J. D. C., Martin, B. V., and Morgan, J. T.
"Synchronizing Traffic Signals for Maximal Band-
widths," Highway Research Board Record, 118 (1966),
21-470

 

(MR1) Miller, A. J. "Analysis of Bunching in Rural Two—
Lane Traffic," Operations Research, 11 (March-
April’ 1963), 236-2470

 

(NEl) Newell, G. F. "Statistical Analysis of the Flow
Of Highway Traffic through a Signalized Intersec-
tion," Quarterly Applied Mathematics, 13 (1956),
353-369.

 

(SJl) "San Jose Traffic Control Project, Final Report,"
Data Processing Division, IBM, Kingston, New
York.

(TAl) Tanner, J. C. "A Theoretical Analysis Of Delays
at an Uncontrolled Intersection," Biometrika,
49: l and 2 (June, 1952), 163-170.

 

(WEl) Weiss, G. H. "A Survey of Some Recent Research
in Road Traffic," Proceedings of the Symposium
on Congestion Theory, Smith, W. L. and Wilkinson,
W. E. (ed.). Chapel Hill: University Of North
Carolina Press, 1965.

(WLl) Wilde, D. J. Optimum Seeking Methods. Englewood
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