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Traffic-responsive strategies for personal-computer-based traffic signal control systems

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Traffic-responsive strategies for personal-computer-based traffic signal control systems
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Luh, John Zenyoung, 1952-
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English
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x, 137 leaves : ill. ; 28 cm.

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Subjects / Keywords:
Approximation ( jstor )
Highway operations ( jstor )
Intersections ( jstor )
Modeling ( jstor )
Motor vehicle traffic ( jstor )
Propagation delay ( jstor )
Signals ( jstor )
Simulations ( jstor )
Terminal operations ( jstor )
Traffic delay ( jstor )
Electronic traffic controls ( lcsh )
Traffic engineering -- Mathematical models ( lcsh )
Traffic signs and signals -- Automation ( lcsh )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Thesis:
Thesis (Ph. D.)--University of Florida, 1989.
Bibliography:
Includes bibliographical references (leaves 131-135).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by John Zenyoung Luh.

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University of Florida
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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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21060272 ( OCLC )

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TRAFFIC-RESPONSIVE STRATEGIES FOR
PERSONAL-COMPUTER-BASED TRAFFIC SIGNAL CONTROL SYSTEMS














By

JOHN ZENYOUNG LUH


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1989














ACKNOWLEDGMENTS


One of the pleasures of finishing the dissertation is thanking the

people who contributed to it. My gratitude cannot be adequately

expressed to the Florida Department of Transportation, which sponsored

this research as part of the traffic signal retiming project, nor to the

University of Florida, which provided financial support during the

entire four years of my study.

I have profited greatly from the suggestions and assistance given

by the members of my supervisory committee. I am eternally grateful to

them.

Professor Kenneth G. Courage has been my advisor since I first

arrived in Gainesville. I have obtained tremendous benefits from his

professional competence and vast practical experience in traffic signal

control. His honoring me by chairing my supervisory committee and his

enthusiastic support have given me opportunities that otherwise would

have been impossible.

Dr. Charles E. Wallace, Director of the Transportation Research

Center and McTrans Center, has been my employer for over three years.

My primary duties under him have been associated with the technical and

software maintenance for the FHWA's Highway Capacity Software. From

him, I learned what high professionalism is. His commitment to

excellence is evident in every piece of my work.








Dr. Chung-Yee Lee has helped me far beyond simply being a member of

the committee. He assisted me during the entire course of this study.

He asked inspirational questions, offered helpful suggestions, and

criticized the draft of the dissertation. Without his invaluable time

and effort in assisting me, this study would have been impossible.

Dr. Joseph A. Wattleworth has encouraged me like a good friend over

the past four years. He has also been a source of inspiration and

motivation. I enjoyed his classes and obtained most of my knowledge of

freeway operation modeling and highway safety analysis from him.

Dr. Mark Chao-Keun Yang served as the outside member of the

committee. He assisted me in the development of mathematical models in

this research. His knowledge of probability and his sometimes

perplexing questions have allowed me to grow intellectually.

Discussions with my colleagues, Lawrence T. Hagen and Charles D.

Jacks, always helped me get around problems encountered with this

research.

Special appreciation is due to Mr. Thomas R. Sawallis, an

instructor at the University of Florida Writing Center, for his

professional work in editing this dissertation.

I am indebted to my wife, Tracy, for her advice, her support, and

her understanding. Finally, I would like to dedicate this dissertation

to my dear parents.















TABLE OF CONTENTS


Page

ACKNOWLEDGMENTS ................................................... ii

LIST OF TABLES ................................................... vi

LIST OF FIGURES .................................................... vii

ABSTRACT ........................................................ ix

CHAPTERS

ONE INTRODUCTION ........................................... 1

Background ............................................. 1
Objectives and Scope .................................... 5
Organization ............................................ 7

TWO LITERATURE REVIEW ....................................... 9

Signal Performance Evaluation ........................... 9
Decision Between Coordination and Free Operations ....... 19
Exploration of Personal-Computer-Based Signal Systems ... 21

THREE ANALYTICAL METHOD FOR PERFORMANCE EVALUATION
OF SEMI-ACTUATED SIGNALS AT LOW VOLUMES .............. 27

Introduction ............................................ 27
Assumptions and Notations ............................... 27
Method Development ...................................... 32
Isolated Control ........................................ 39
Coordinated Control ..................................... 50
Test of the Analytical Method ........................... 57
Comparison with a Pretimed Model ........................ 60
Summary and Discussions ................................. 65
Equations Development ................................... 66

FOUR APPROXIMATION METHOD FOR PERFORMANCE EVALUATION
OF SEMI-ACTUATED SIGNALS AT LOW VOLUMES ............... 79

Introduction .......................................... 79
Concept of the Method ................................... 80
Method Development ...................................... 81
Tests of the Approximation Method ....................... 88
Summary and Discussions ................................. 92
iv














Page

FIVE DECISION BETWEEN COORDINATION AND FREE OPERATION ........ 95

Introduction .......................................... 95
Decision Making Process ................................. 96
The DELVACS Program ..................................... 100
Case Studies .......................................... 104
Application on PC-Based Signal Control Systems .......... 117

SIX CONCLUSIONS AND RECOMMENDATIONS ......................... 122

Conclusions .......................................... 123
Recommendations ......................................... 126

BIBLIOGRAPHY .................................................... 131

BIOGRAPHICAL SKETCH ............................................ 136














LIST OF TABLES


TABLE Page

3.1 Patterns (in Terms of Red and Green Times) and Their
Probabilities at a Three-Phase Semi-Actuated Signal
Under Isolated Control .................................... 45

3.2 Patterns (in Terms of Red and Green Times) and Their
Probabilities at a Three-Phase Semi-Actuated Signal
Under Coordinated Control ................................. 55

3.3 A Further Comparison Between the Analytical Method
and the Pretimed Model Method ............................ 64

5.1 Measures of Effectiveness and Performance Indices
Under Coordinated Control in Case 1 ...................... 108

5.2 Measures of Effectiveness and Performance Indices
Under Isolated Control in Case 1 ........................... 109

5.3 Measures of Effectiveness and Performance Indices
Under Coordinated Control in Case 2 ...................... 110

5.4 Measures of Effectiveness and Performance Indices
Under isolated Control in Case 2 .......................... 111

5.5 Desirable Operations in the Two cases of the Example ....... 112














LIST OF FIGURES


FIGURE Page

2.1 Basic Configuration of Urban Traffic Control Systems ....... 22

2.2 Basic Configuration of PC-Based Traffic Signal
Control Systems ......................................... 24

3.1 Block Diagram of the Analytical Method ................... 34

3.2 Stop Probability and Average Delay Equations and the
Conditions for Their Use Under the Analytical Method ....... 37

3.3 Transitions, Red Times, and Transition Probabilities
at a Three-Phase Semi-Actuated Signal Under
Isolated Control ......................................... 41

3.4 Example of the Determination of Pattern Probability ........ 49

3.5 Transitions, Red Times, and Transition Probabilities
at a Three-Phase Semi-Actuated Signal Under
Coordinated Control ......................................... 52

3.6 Stop Probability and Average Delay Comparisons,
the Analytical Method vs. Simulation ...................... 61

3.7 Stop Probability and Average Delay Comparisons,
the Analytical Method vs. the Pretimed Model Method ........ 63

3.8 Actuated Signal Operation Examples of Single Ring
without Overlap and Dual Ring with Overlap ................ 67

3.9 Queue Length Example at a Signalized Intersection .......... 76

4.1 Determining the Total Effective Red Time and the
Number of Green Occurrences ............................... 84

4.2 Stop Probability and Average Delay Equations and the
Conditions for Their Use Under the Approximation Method .... 89

4.3 Stop Probability and Average Delay Comparisons,
the Approximation Method vs. Simulation ................... 91

4.4 Stop Probability and Average Delay Comparisons,
the Approximation Method vs. the Analytical Method ......... 93













FIGURE Page

5.1 Block Diagram of the Decision Making Process ............... 98

5.2 Flow Chart of the DELVACS Program .......................... 101

5.3 Artery Layout, Phase Sequences, and Timing Settings
of the Example .......................................... 105

5.4 Traffic Flow Rates in Cases 1 and 2 of the Example ......... 107

5.5 Intersection Layout, Phase Sequence, and Timing Setting
of the Switching Condition Example ......................... 114

5.6 The Switching Thresholds of Total Intersection Volume
for the Example Intersection with a Two-Phase
Semi-Actuated Signal ....................................... 116


viii














Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

TRAFFIC-RESPONSIVE STRATEGIES FOR
PERSONAL-COMPUTER-BASED TRAFFIC SIGNAL SYSTEMS

By

John Zenyoung Luh

May, 1989

Chairman: Kenneth G. Courage
Major Department: Civil Engineering

Traffic on the artery generally moves more smoothly during peak

periods if signals in proximity are coordinated. During off-peak

periods when traffic is light, however, free operations may be more

efficient than coordination from the standpoint of system performance.

The change from coordination to free operation (and vice versa) is a

typical traffic-responsive problem, but the choice between them is often

subjective, since the literature offers very few methods for the

decision.

This dissertation proposes a decision making process whereby

traffic engineers can make more effective choices on light traffic

signal operations, based on system performance in terms of vehicular

delay and number of stops. Two methods are developed to supply models

needed for evaluating semi-actuated signals at low volumes, under both

coordinated and isolated controls.









The decision making process has been implemented as the DELVACS

program. A hypothetical example is given to demonstrate the application

of the process using DELVACS. The possible approaches for implementing

the process on personal-computer-based signal control systems are

explored. The advantages and disadvantages of the approaches are

discussed, and the resolutions to the potential problems are proposed.

The decision making process provides a viable tool in traffic-

responsive controls. The process and the semi-actuated signal

evaluation methods developed in this research can also be applied on

other types of traffic-responsive signal systems, such as Urban Traffic

Control Systems which have been predominant in the United States in the

past two decades.














CHAPTER ONE
INTRODUCTION


Background

Traffic-responsive control of signalized intersections represents a

significant advance in the traffic control field, and is the subject of

extensive research, as well as development efforts. With proper design

and operation, traffic-responsive control, which has been in practical

operation during the past two decades, can effectively reduce stops,

delay, fuel consumption, and emissions. A variety of traffic control

techniques in response to traffic demand has been developed, and some

have been evaluated in several countries [13, 20, 23, 26]. The

underlying philosophy of all the techniques has been to smoothly move

traffic with as few obstacles as possible. Coordinating the timing of

adjacent signals to allow progressive traffic movements has been

recognized as one of the most effective means to achieve this goal.

The need to coordinate adjacent signals varies according to the

geographic and traffic environment. The most important factors have

been identified as intersection spacing and traffic volumes [7, 59].

For widely spaced intersections, signals usually operate individually

because of the small influence on traffic flows from their neighbors.

Where signalized intersections in close proximity influence each other's

traffic flows, traffic volumes become an important factor in their

coordination.





2



Early thinking on the subject of signal control always indicated

the need to interconnect signals into a single system and to work toward

maximizing progressive movement during peak periods, though during off-

peak hours, free running or flashing control may better handle the

traffic [54]. The reason is rather evident. Coordinating the signal

timing to allow traffic on the artery to travel smoothly without being

stopped at intersections (which is called progression) is often

accomplished at the expense of extra delays to the cross street traffic.

When the arterial traffic decreases during off-peak periods, the

benefits gained on the artery may not offset the losses on cross street

traffic. This phenomenon has been observed by Riddle and Hazzard [43].

From a series of simulation and field tests, they concluded that

coordination is superior to free operation under all conditions where

volumes exceeded 350 vehicles per hour.

These findings suggest that it is preferable for signals in

proximity to be coordinated during peak periods. During off-peak

periods when traffic is light, however, free operation may be more

efficient from the standpoint of system performance. The change from

coordination to free operation (and vice versa) is a purely traffic-

responsive problem. But the choice between coordination and free

operations is often subjective, since the literature offers very few

methods for the decision. Developing a method to allow traffic

engineers to make more effective choices on light traffic signal

operations is the subject of this research.

Asking whether to change from coordination to free operation (or

vice versa) is equivalent to asking which operation offers better

performance. In other words, the decision rests mainly on the









performance evaluation of the signals within the system under the two

alternatives.

A variety of models has been offered to evaluate signal

performance. Most of the models available, however, have centered

around pretimed signals mainly because their operation is simple. Under

pretimed control, the sequence of right-of-way assignments (phases) and

the length of time interval for each phase are fixed. Pretimed signals

are well suited to coordination operation since progression can be

easily maintained.

Unfortunately, the literature offers very few models dealing

exclusively with the growing number of actuated signals, since their

operation is more complex. Traffic-actuated control attempts to adjust

the sequence of phases (through skipping of phases with no traffic

demand), and the length of time interval for each phase (through adding

extension intervals to the minimum green interval) in response to

traffic demand. Traffic-actuated signals can be further classified into

two types of operations: (1) full-actuated operation and (2) semi-

actuated operation. In full-actuated operation, since each phase is

controlled by traffic demand, it is difficult to maintain progression on

the artery. Hence, full-actuated signals are seldom used in

coordination control. In semi-actuated operation, the designated main

street is governed by a non-actuated phase and cross streets by actuated

phases. The non-actuated phase has a green light at all times until

cross street has traffic demand. Semi-actuated signals are widely used

on arterial roadways, since they are able to not only maintain

progression on the major street but also provide flexible control to









cross street traffic. In conclusion, semi-actuated signals are the

major concern of this research.

A common approach for semi-actuated signal evaluation is to use

averages, such as average cycle lengths and green times, in a model for

pretimed signals. The use of pretimed models to evaluate semi-actuated

signals is reasonable when traffic volumes are moderate to high. In

this case, each actuated phase appears in almost every cycle, with

phase lengths varying from cycle to cycle. Longer phase length

represents heavier traffic demand and shorter represents lighter. Since

the average phase length reflects the average traffic demand, the use of

its value would represent the average performance.

When traffic is light, however, this method is no longer

applicable. In low volume situations, phase skipping (a phase being

skipped due to lack of traffic demand) and dwelling (the green light

stays on the non-actuated phase waiting for cross street traffic demand)

occur frequently. Phase skipping and dwelling make it very difficult to

determine the average cycle length and green times. Even though the

averages can be determined, as demonstrated in Chapter Three, the models

for pretimed phases are no longer valid for actuated phases due to phase

skipping and dwelling. Therefore, they cannot be used for semi-actuated

signals under low volume conditions.

The lack of a model to evaluate semi-actuated signals under low

volume conditions is a critical problem in the decision between

coordination and free operations with low traffic volumes. The decision

is easier to make when only pretimed signals are involved, since an

ample choice of models is available to evaluate their performance. When

semi-actuated signals are involved, however, the decision is difficult









due to the lack of appropriate models. Hence, one challenging aspect in

considering the choice between coordination and free operations is the

development of a model for evaluating semi-actuated signals at low

volumes, under both coordinated and isolated controls. The development

of such a model is the first task of this research.

The second task of the study is to develop a method to help make

the decision between coordination and free operations more effectively.

An opportunity for this arose recently when the City of Gainesville,

Florida, installed a personal-computer-based signal control system (also

called a closed loop signal system by some manufacturers) to supervise

the area-wide traffic signals. Both the Florida Department of

Transportation and the City of Gainesville have requested the University

of Florida to examine the system's operation to pursue better

performance. This project was originally initiated for the Gainesville

System, and was entitled "Traffic-Responsive Strategies for Personal-

Computer-Based Traffic Signal Control Systems." However, it should be

noted that the method as well as models developed in this research can

also be applied on other types of traffic-responsive signal systems,

such as Urban Traffic Control Systems (UTCS) which have been predominant

in the United States in the past two decades.
Objectives and Scope

This dissertation presents a methodology for making the choice

between coordination and free operations. Two models are developed for

evaluating semi-actuated signals under low volume conditions. The first

model performs a thorough calculation using probability theory and

stochastic processes. The second model uses average values to

approximate measures of effectiveness. The first model is referred to









as the Analytical Method and the second as the Approximation Method.

A method to make the decision between coordination and free operations

is proposed. The application of the method on personal-computer-based

signal systems is discussed.

The specific objectives of the research are as follows:

1. Review the literature with respect to signal performance

evaluation and the contrast between coordination and free

operations.

2. Review the control capabilities of personal-computer-based

signal control systems.

3. Develop models to evaluate semi-actuated signals at low

volumes, under both isolated and coordinated controls.

4. Propose a method for making the decision between coordination

and free operations in response to traffic demand.

5. Discuss the application of the decision making method on

personal-computer-based signal systems.

The primary emphasis of this research deals with semi-actuated

signals under low volume conditions. The signals have a fixed common

cycle length when they are coordinated, but not when run freely. The

system of interest is an arterial roadway controlled by semi-actuated

signals with light traffic on the cross streets.

The two signal evaluation methods developed in this research have

been implemented as two computer programs. The purpose of the two

programs is to ease the tedious calculations while demonstrating the

feasibility of the two methods. But the program implementing the

Approximation Method will be further expanded to the program called

DELVACS (Delay Estimation for Low Volume Arterial Control Systems) under









the sponsored project associated with this research work. The DELVACS

program implements the proposed method to provide a tool for making

choices between coordination and free operations. The DELVACS program

is designated to have interface with the Arterial Analysis Package (AAP)

[3] to allow the user to run the AAP and DELVACS together, using a

common data set.

Since presentation of computer programs is not within the scope of

this dissertation, the two programs developed are not intended to

represent complete software packages. Nevertheless, they are

operational, and can be used immediately within their capabilities.

The two evaluation methods developed in this study are based on

existing theories which have been well accepted by the traffic

engineering community. The feasibility of the two methods is validated

against microscopic simulation. The applicability of the method in

deciding between coordination and free operations is discussed.
Organization

The dissertation is organized according to the objectives stated

previously. The next chapter reviews literature that is pertinent to

this study. The literature search includes traffic signal performance

evaluation and the techniques for making the decision between

coordination and free operations. Along with the methodology search, a

review of the capabilities of personal-computer-based signal systems is

included.

Chapter Three presents the Analytical Method to evaluate semi-

actuated signals under low volume conditions. The method uses

probability theory and stochastic processes to deal with individual

cycles. The method produces good estimates, but is complicated to









apply, particularly when signals have more than three phases or have

special control capabilities (such as permissive periods).

Chapter Four presents the Approximation Method. The method uses

average parameters (such as red and green times). The method offers a

cost-effective alternative to the Analytical Method. This is

particularly important where conditions are unfavorable to the

application of the Analytical Method.

Chapter Five proposes a methodology for making the decision between

coordination and free operations. The application of the methodology

on personal-computer-based traffic signal control systems is discussed.

Finally in Chapter Six are the conclusions and recommendations

emanating from this study. The conclusions include the summary of the

findings, and the recommendations suggest the areas for future study.














CHAPTER TWO
LITERATURE REVIEW


The purpose of the literature review is to collect the earlier

efforts in order to avoid duplication and stimulate new methodology

development. The literature review includes three parts which are

pertinent to this study. The first part focuses on the methodologies of

traffic signal performance evaluation. The second part reviews the

techniques in deciding between coordination and free operations. The

last part explores the capabilities of personal-computer-based signal

control systems.

Signal Performance Evaluation

Before reviewing the various methodologies, it is necessary to

select measures of effectiveness (MOE). The MOEs serve as a basis for

signal evaluation, and a variety of MOEs have been used for this

purpose. The major ones include delay, number of stops, degree of

saturation, excess fuel consumption, and accident rate reduction [54].

Among them, delay and number of stops have been most widely used. The

reasons are evident [24]: they can be measured; they are the motorists'

major concerns; they have economic worth; and they are easily understood

by both technical and non-technical people. The 1985 Highway Capacity

Manual (HCM) [19], for instance, uses delay as the sole indicator of

level of service at signalized intersections. The TRAffic Network StudY

Tool (TRANSYT) [45] and the Traffic SIGnal OPtimization Model (SIGOP)

[51] aim at minimizing the combination of delay and number of stops.









Because of extensive use in the traffic engineering community, delay and

number of stops were selected as the major MOEs in this study.

Delay and number of stops may be measured directly in the field, or

estimated by either simulation or analytical models. A number of

methods are available for making the field measurement, such as test-car

observations, arrival and departure time recording, and stopped-vehicle

count [19, 40]. Field measurement is easy to apply, but is costly and

time consuming. It cannot be applied to non-existent situations, such

as a projected signal control plan.

Simulation provides a convenient tool for studying delay and number

of stops, especially when the real-world system is complex, or when

proposed operations are to be evaluated [29]. Experimental conditions

can be much better controlled in a simulation than would generally be

possible in a field measurement. Simulation also makes it possible to

study a system with a long time frame in compressed time, or

alternatively to study the detailed workings of a system in expanded

time. A number of simulation models are available for signal operation

evaluation [4, 18, 31, 44, 45].

But using simulation has disadvantages. Simulation models are

often expensive and time-consuming to develop and maintain. Each run of

a stochastic simulation model produces only one estimate of a system's

true characteristics, and only for a particular set of input parameters,

so that several independent runs are required for each set of input

parameters, and statistical methods are needed to analyze simulation

output data [14]. For this reason, simulation models are generally not

as good at optimization as they are at comparing a fixed number of

specified alternative system designs [29]. In contrast, an analytical









model, if available, can easily determine the actual characteristics of

the system. Thus, an analytical model is generally preferable to a

simulation model if such a model can be developed. In other words,

analytical models are always of interest.

Pretimed Signal Evaluation

For pretimed signals, a wealth of theoretical analysis models for

delay and stops has been offered in the literature [1, 35, 38, 45, 57,

58]. Webster's delay model provided the foundation for most subsequent

models. Robertson's model (which basically modified Webster's model to

deal with high degrees of saturation) was used in the TRANSYT program.

The 1985 HCM model [19] provides a recent addition to this field.

Because the models have been broadly reviewed and compared from various

aspects [2, 17, 24, 25], they will not be repeated here.

Actuated Signal Evaluation

Delays and stops for pretimed signals have been successfully

analyzed, mainly because the operation of pretimed signals is simple.

For semi-actuated signals, the dynamic characteristics of signal

operation do not lend themselves well to analytical treatment.

Simulation has been the most popular approach to assess delays and stops

for actuated signals [4, 31, 44].

The limited analysis models for semi-actuated signals (or more

broadly, for actuated signals) can be categorized into two approaches:

direct analysis and indirect approximation. In the direct approach, the

operational characteristics of an actuated phase are analyzed, and

delays and stops are derived from the analysis. In indirect

approximation, a substitute model (often borrowed from pretimed signals)

is used to approximate delays and stops for semi-actuated signals.









In the direct analysis approach, Newell [36] developed a

mathematical model for evaluating delays at actuated signals. The model

is complicated even in the case of one-way streets. Following the work,

Newell and Osuna [37] applied the model to two-way streets with

symmetric traffic flows under the control of a two-phase signal. A

purely theoretical model was studied by Lehoczky [30], which relies on

input data that is difficult to obtain. Hoshi [22] developed an even

more complicated model for a three-way intersection.

Papapanou [39] modified Robertson's pretimed model to assess the

saturation delay under actuated control. Papapanou's model was

incorporated into the Signal Operations Analysis Package (SOAP) [52].

Because saturation delay is significant only when traffic volumes are

moderate to high but is negligible when traffic is light, Papapanou's

model is not applicable in handling low volume conditions with phase

skipping.

Lin [32] developed an algorithm to estimate average green times for

full-actuated signals. The algorithm estimates the steady state green

time of each actuated phase with an implicit assumption that the phase

appears in every cycle. The assumption holds true when traffic volumes

are moderate to high. In this situation, the actuated phase appears in

almost every cycle, with phase length varying from cycle to cycle.

However, the phase assumption is not valid under low volume conditions,

since phase skipping occurs frequently due to lack of traffic demand.

Cowan [11] investigated average delays and average lengths of red

and green phases for a two-street intersection operating with a full-

actuated signal. However, his results cannot be applied to semi-









actuated signals, because semi-actuated signals are not a special case

of full-actuated signals.

In conclusion, the existing analysis models for actuated signals

either have limited applications, or rely on information that is

difficult to obtain, or are restricted to moderate to high volumes.

None of them are applicable to general operation of semi-actuated

signals under low volume conditions.

The second approach of indirect approximation is more common and

easier to apply. In the 1985 HCM [19], delay is first computed without

distinguishing among signal control types, then is multiplied by

different factors, according to signal control and traffic arrival type.

(For example, in random arrival condition, the factors are 0.85 for

actuated signals and 1.0 for semi-actuated signals.) In the delay

calculations, average values (such as average cycle length and green

times) of actuated signals are used in the delay equation originally

developed for pretimed signals.

To assess stop probability for actuated signals in SOAP [52],

average cycle length and green times of actuated signals are entered

into the stop probability equation originally developed by Webster and

Cobbe [57] for pretimed signals. TRANSYT Enhanced Version for Actuated

Signals (TRANSYT-7FC) [5], although it does a better job in estimating

timing for actuated signals, adopts this same method to assess delays

and stops for actuated signals.

The underlying assumption of the indirect approximation approach is

that the actuated phase appears in every cycle. As stated above, the

assumption is tenable only when traffic volumes are moderate to high,

and is false under low volume conditions. Therefore, this approach is








not directly applicable when traffic is light, unless phase skipping and

dwelling are taken into account.

So far, the signal evaluation has been based on the implicit

assumption that signals are isolated, so that vehicle arrivals are more

or less random. When signals are coordinated, however, the random

arrival assumption is generally not valid, since the arriving vehicles

are the output stream from the upstream intersection. In this case,

arriving vehicles are usually grouped into platoons by the upstream

signal. The delay and stops at the intersection will be significantly

affected by progression in terms of the relationship between the signal

green time and the platoon arrival. The approaches to account for

progression have evolved broadly in two directions: system approach and

individual intersection approach. In the system approach, more than one

intersection (at least the two adjacent intersections) are handled

simultaneously, and in the individual intersection approach, each

intersection is treated separately.

System Approach for Progression

TRANSYT [45] provides an excellent example of the system approach.

TRANSYT simulates macroscopic traffic flows in the form of platoon

dispersion from the upstream intersection to the downstream one. In

doing so, the signal timing and traffic flows of the adjacent

intersections are considered simultaneously.

A flow profile which combines the vehicle arrival and departure

curves at each intersection is estimated in TRANSYT for the purpose of

estimating delay and stops. Delay is then calculated by integrating the

area under the curve in the red and saturation green times and by

applying Robertson's random delay equation. The number of vehicles









stopped is basically equal to the number of vehicles arriving while a

queue is present.

TRANSYT calculates delays and stops from predicted flow profiles.

The Split, Cycle, and Offset Optimization Technique (SCOOT) [23], which

was developed by the Transport and Road Research Laboratory in Great

Britain, measures the upstream flow profiles by using detectors on the

streets, then projects them downstream using the same model in TRANSYT.

In any event, delays and stops are from detailed analysis of traffic

flow profiles in the two systems.

As pointed out by Hurdle [24], the detailed analysis of flow

profiles is a good approach to account for progression, but just as

obviously it is not always a practical one. It is a computer simulation

oriented approach, so that it is generally not practical for hand

calculation. Its advantages are likely to be realized only when good

data are available, and the approach is highly dependent upon signal

timing. The approach is most useful for a large system, rather than for

the usual situations in which only a few intersections are to be

studied.

Individual Intersection Approach for Progression

The individual intersection approach provides an easier way to

assess the effect of signal progression, especially for hand

calculation.

Staniewicz and Levinson [53] used graphic analysis on time-space

diagrams to develop delay equations for two extreme conditions, namely

the first vehicle in the platoon arrives during a green interval, and

the first vehicle in the platoon arrives during a red interval.

Accompanied with the simplified assumptions, such as all vehicles









traveling in the platoon with no platoon dispersion, the study provided

an analytical tool to estimate delay for special cases.

A more complicated analysis to estimate delay under progression was

proposed by Rouphail [50]. The study considered platoon dispersion by

assuming that traffic flow occurs in two distinct flows, one inside the

progression band and another outside the band. With this assumption and

a regression model of platoon dispersion, which is a function of travel

time, Rouphail provided a method to estimate delay using information

available from a time-space diagram.

The 1985 HCM [19] offers an easier way to account for progression.

In the HCM, a progression adjustment factor (PF) is used to assess the

influence of traffic progression, rather than using complicated

equations. Delay for each intersection is originally computed with no

consideration of progression, then a final progression adjustment is

performed. The values of PF vary primarily based on arrival type and

degree of saturation. In this approach, the classification of arrival

types is very important, since it attempts to describe the nature of the

progression. The dominant arrival flows are classified into five types.

Higher type indicates better progression.

There are two major concerns associated with the HCM method: (1)

arrival type is difficult to assess subjectively, and (2) progression

adjustment factors may not reflect the actual influence of progression

quality. To cope with the difficulty in subjectively assessing arrival

type, the HCM suggests the use of platoon ratio to quantify the arrival

type. The platoon ratio is defined as









PVG
Rp (2.1)
PTG

where R = platoon ratio;

PVG = percentage of all vehicles in the movement arriving during

the green phase; and

PTG = percentage of the cycle that is green for the movement.

PTG can be easily computed from the green and cycle times, but

field observations are needed to determine PVG. The relationship

between platoon ratio and arrival type is provided in Table 9-2 of the

HCM.

To overcome the drawback that platoon ratio must be determined from

field observations, Courage et al. [10] developed an estimator of the

platoon ratio, which is called band ratio. The band ratio is defined as


C B Pa (Gd -B)( Pa)
Rb = [ + ]
Gd Go C Go


wh


di


(2.2)


iere Rb = band ratio;

C = cycle length;

B = band width;

Go = the green time at the upstream signal;

Gd = the green time at the downstream signal;

Pa = the proportion of traffic entering the upstream intersection

from the artery; and

1 Pa = the proportion of traffic entering the upstream intersection

from the cross street.

The band ratio can be derived from traffic volumes and time-space

agrams without field studies. The test of the band ratio indicated


1


n









that the field measurement of platoon ratio provides more reliable

indicator of progression quality, but the band ratio offers acceptable

substitute when field data are not available.

With regard to the second concern, accuracy of adjustment factors,

Prevedouros and Jovanis [42] reported from a limited validation at

actuated signals in Illinois that PF values collected from the field are

lower than those provided in Table 9-13 of the HCM. A paper by Courage

et al. [10] compared the PF values in the HCM with those generated from

TRANSYT-7F [56], and concluded that there was good agreement between the

two models, but that a wider range of PF values exists than the HCM

recognizes. The study suggested that some extrapolation of the HCM PF

values may be warranted to cover exceptionally good and exceptionally

poor progression. Another paper by Courage and Luh [8] attempted to

compute service volumes at signalized intersections, and reported that

due to the threshold PF values in the HCM, the effect of progression on

delay is discontinuous.

Recognizing these problems, the Texas Transportation Institute

initiated a comprehensive study, which was completed recently [6]. The

study examined the PF values in the HCM using extensive simulation and

delay data collected nationwide. The results of this study have not yet

been published.

From the above literature review, it can be concluded that the 1985

HCM provides a practical method for evaluating signal performance under

traffic progression. When applying the method, arrival type can be

assessed either by field observations (if applicable) or by computing

band ratio from time-space diagrams.









Decision Between Coordination and Free Operations

A number of techniques have been found in the literature which are

relevant to making the decision between coordination and free

operations.

Yagoda et al. [59] developed the coupling index, a simple ratio of

link volume and link length, to determine which signals to coordinate.

In its application, the coupling index for each link is computed. A

coupling threshold is defined so that all links with coupling indices

below the threshold are assumed not to require coordination. By

adjusting the coupling threshold, the network (a group of intersections)

can be broken into several parts. The boundaries among these parts are

thereby identified as suitable candidates for subdivision of

coordination. An analysis of benefits is finally conducted by using a

signal optimization program to establish the validity of the

coordination divisions. This method provides a simple and

straightforward technique to determine which signals to coordinate.

Based on this idea, a more complicated interconnection desirability

index was developed by Chang [7]. The index attempts to determine the

need to coordinate a pair of adjacent signals. The index considers many

factors, such as intersection spacing, traffic traveling speed, traffic

volumes, and number of lanes. The index has a range of values from zero

to one. When the index has a value of 0.25 or less, isolated operation

is recommended. Conversely, when the index has a value of 0.50 or

greater, interconnection of the two adjacent signals is recommended.

Other evaluation methods are needed to assist in the interconnection

decision if the calculated value of the index is between 0.25 and 0.50

[7]. Since the index considers only adjacent pairs of signals at any









one time, a piece-wise coordination might result from this method.

Furthermore, though deemed necessary by the author when the index has a

value between 0.25 and 0.50, the supplementary evaluation indicators

were not provided.

Another technique for deciding about coordination is found in the

Second Generation (2-GC) of Urban Traffic Control Systems [13, 20]. The

2-GC of UTCS contains an on-line optimization routine which determines

the control parameters that will minimize the total delay and stops

within the system in response to current traffic conditions. A sub-

network configuration model called RTSND [13] resides in this system to

determine whether to break the coordination among signals. The

determination is primarily based upon cycle lengths. The coordination

of signals is maintained if they have similar cycle lengths; otherwise,

the interconnection is released.

Another similar method was used in the Sydney Coordinated Adaptive

Traffic System (SCATS) [33], which was developed by the Department of

Main Roads, New South Wales in Australia. In SCATS, a coordination

criterion and a coordination vote computational algorithm were developed

to on-line determine whether to coordinate signals or not. A

coordination vote is calculated every cycle. The vote is in favor of

coordination if the difference of cycle lengths is less than 9 seconds.

A counter is increased by one if a vote is in favor of coordination, but

is otherwise decreased by one. The signals are coordinated when the

counter attains a value of four, but the interconnection is broken when

the counter reaches zero.

The techniques used in the above two systems (2-GC of UTCS and

SCATS) use similar cycle lengths as the sole criterion in the









determination of whether or not to coordinate. The reason is primarily

based on the fact that most signal coordination operations require a

common cycle length. In other words, the techniques do not consider any

potential benefits in making the decision.

Exploration of Personal-Computer-Based Signal Systems

A variety of traffic-responsive, computer-controlled systems has

evolved over the past two decades. Most of the systems presently

operating in the United States have been centered around Urban Traffic

Control Systems (UTCS), which were established by the Federal Highway

Administration in the early 1970s. Personal-computer-based (PC-Based)

signal control systems are a new tool to supervise area-wide traffic

signals. The number of PC-Based signal systems in this nation has grown

significantly since the mid-1980's. Since this type of signal control

is newer, its capabilities are explored in this study.

Because the traffic engineering community is familiar with UTCS, it

would serve a good basis for understanding PC-Based systems. In the

following, UTC systems and their major disadvantages are first briefly

reviewed. PC-Based systems are then described by way of their contrasts

with UTCS. Last is the discussion of the PC-Based systems'

capabilities.

Although there exist some variations among UTC systems, the basic

configuration in predominant use in UTCS is illustrated in Figure 2.1

[41, 54]. The dominant characteristic of UTCS is the location of all

decision making capability at one geographic point and on one level.

Data from all detectors in the system are fed into the central computer,

where all decisions and control commands are made and dispatched to

local controllers.



















Central


Communication
Link


Local Controller
and Dectectors

Urban Area Limits
Urban Area Limits


Figure 2.1. Basic Configuration of Urban Traffic Control Systems.









The heavy workload on the central computer and the direct

communication between the central computer and local controllers require

most UTC systems to depend on large computers and complex dedicated

communication systems. Because of this, several disadvantages have been

experienced in the design and maintenance of such large centralized

control systems. UTC systems are costly. Since the exclusive use of

user-owned twisted pair cable has been accepted as standard for traffic

control communication in UTCS for many years [54], the communication may

constitute as much as 50 to 60% of the total project cost [28]. Hence,

a project cost could be tremendously high when control is desired over a

great number of signals, spread over a large geographic area.

User involvement is another concern associated with UTCS in many

communities. Staff in small to medium communities generally lacks

expertise in the area of traffic control systems. The capabilities of

staff in small jurisdictions are limited. The limited personal resource

cannot be allocated to a continuous commitment for operating a complex

control system [27]. Also, a complex system requires longer set-up

time. These disadvantages have prevented many jurisdictions from taking

advantage of traffic-responsive, computer-controlled systems [9].

In contrast to UTC systems, PC-Based traffic signal systems offer a

cost effective alternative for traffic-responsive computer-control. The

basic configuration of PC-Based systems is illustrated in Figure 2.2. A

central personal-computer is interconnected with several on-street

masters or a number of local controllers. Typically, telephone lines

are used between the central personal-computer and on-street masters.

Each on-street master is simultaneously connected to a number of local
































Telephone Line


Local Controller
and Detectors


Figure 2.2.


Basic Configuration of PC-Based
Traffic Signal Control Systems.


- -


(









controllers with more sophisticated, higher-capacity media, such as

twisted pair cable.

There are many features which separate personal-computer-based

systems from UTC systems. The major ones involve the role of the

central personal-computer and the communication system. The central

personal-computer is not a master-control, system-host machine. It

functions as a system data logger and as a central editing and reporting

station for overall monitoring. Close monitoring and interface with the

on-street masters is not necessary, and the central personal-computer

can be turned off or used for other activities. The tasks of selecting

signal plans in response to traffic demand and supervising local

controllers are distributed to on-street masters. The on-street masters

function on a stand-alone basis, and primarily operate in traffic-

responsive mode, based on surveillance data gathered from sub-system

detectors.

The communication system, as illustrated in Figure 2.2, groups

local signals into sub-systems. A sub-system may consist of a single

intersection but normally contains one section of an arterial, a small

grid network, or any other signal group requiring coordination. Sub-

systems are generally independent unless coordination is provided from

the central site. This structure of building blocks permits future

expansion (additional sub-systems) without major modification to

existing hardware.

PC-Based signal systems generally provide three operational modes:

manual, time of day and day of week (TOD/DOW), and traffic responsive

(TRSP). The three modes represent three degrees in response to traffic

demand. The manual mode is often used for handling unusual traffic









conditions, such as traffic congestion resulting from football game or

concert. The TOD/DOW mode changes the timing plans according to a

preset schedule. The TRSP mode selects the plan from the timing plan

library that is best suited to the current traffic conditions. The

match is based on the combination of volume and occupancy data gathered

from system detectors. (The Gainesville, Florida, Signal System

currently operates on the TOD/DOW mode.)

The timing plans used in both the TOD/DOW and TRSP modes are

developed off-line, using programs such as TRANSYT. A day is broken

into several time periods to represent the various traffic

characteristics, and the average parameters (such as volumes) in each

period are used in the timing development.















CHAPTER THREE
ANALYTICAL METHOD FOR PERFORMANCE EVALUATION
OF SEMI-ACTUATED SIGNALS AT LOW VOLUMES


Introduction

This chapter presents a method to directly estimate stop

probabilities and average delays for semi-actuated signals at low

volumes, under both isolated and coordinated controls. The method uses

probability theory and stochastic processes to consider the various

cycle patterns in terms of red and green times. Also, stop probability

and average delay of each cycle pattern are estimated using equations

developed in this chapter. The method will be referred to as the

Analytical Method.

This chapter first addresses the assumptions in modeling delays and

stops. Then the Analytical Method is explained, and applied to both

control types. Next the mathematical model in this method is

validated against simulation. After the validation, results obtained

from the Analytical Method are compared with those from a model for

pretimed signals. The summary and discussions follow. Finally, the

stop probability and average delay equations used in the Analytical

Method are developed.

Assumptions and Notations

This section first addresses the assumptions in modeling delays and

stops. The signal operations under consideration then follow. The

notations used throughout the dissertation are listed finally.









Arrival and Departure Times

As in many delay models, vehicles are regarded as identical in size

and performance. They are assumed to arrive and depart the intersection

at a constant speed, and delay is regarded as time spent at the stop

line. In other words, stopped delays are of concern, and deceleration

and acceleration are not considered. The queue is regarded as a

vertical stack at the stop line. Vehicles are assumed to pass the stop

line at a constant rate so long as there are vehicles waiting at the

stop line. After the waiting queue has fully discharged, vehicles

arriving in the remaining green time are assumed to pass the stop line

without any delay. Yellow and lost times are not considered.

Vehicle Arrivals

Moreover, vehicles of the actuated phases are assumed to have

Poisson arrivals [2, 16], which means that the probability of n arrivals

in a time interval has a Poisson distribution [16]. (Please note that

it is unnecessary to assume Poisson arrivals for the non-actuated phase

in this study since the non-actuated phase is not controlled by vehicle

detections.)

Steady State

It is assumed that the system has operated for a sufficiently long

time with the same average traffic volume and vehicle departure rate to

have settled into a steady state [24].

Signal Operations

The method presented in the dissertation is based on a typical

semi-actuated signal with the following characteristics [54]:

1. Detectors are located only on actuated-phase approaches.

2. Detectors are placed near the stop line (presence detectors).









3. The non-actuated phase receives at least the minimum green

interval each cycle.

4. The actuated phases receive green upon actuation, provided that

the non-actuated phase has completed its minimum green interval.

5. A signal under coordinated control has a background cycle

length, but a signal under isolated control does not.

6. The following are preset: the background cycle length for

coordinated control, the minimum green time for the non-actuated

phase, and initial green intervals for actuated phases.

For simplicity, it is assumed that there is only one non-actuated

phase, and all others are actuated. Actuated phases are assumed to

terminate their green times after the initial green intervals elapse,

since the probability of requiring green extensions after the completion

of the initial green interval is usually very small under low volumes.

The probability of an actuated phase having arrivals more than the

departure capacity of the initial green interval (i.e., requires green

extensions) can be estimated by the following equation:

capacity un e-u
probability = 1 E (3.1)
n=O n!


where u = (red time + green time) volume / 3600

capacity = (green time)/(saturation headway)

saturation headway = 2 seconds


It can be easily verified that in most cases the probabilities are very

small where volumes are less than 300 vehicles per hour per lane

(vphpl). Because of this, 300 vphpl is taken as the upper bound for

this method.









Since the actuated phases of a semi-actuated signal may be skipped,

it is not clear at the first glance in determining a cycle. Under

isolated control, two successive occurrences of the non-actuated phase

will be referred to as a cycle. As will be explained later, a cycle

consists of the non-actuated phase plus at least one actuated phase.

Under coordinated control, a background cycle will be referred to as a

cycle. In this case, a cycle generally consists of the non-actuated

phase plus actuated phases, but may solely consist of the non-actuated

phase.

Traffic Progression

Stop probabilities and average delays at individual intersections

are influenced by the quality of traffic progression from their

neighbors. When the signal progression is favorable to the subject

traffic movement, stop probabilities and average delays will be

considerably less than those for random arrivals. Conversely, when

signal progression is unfavorable, stop probabilities and average delays

can be considerably higher than those for random arrivals [19]. When

signals are under isolated control, no progression is considered. When

signals are under coordinated control, the non-actuated phase (which

controls the major street traffic) is considered under the influence of

progression, but actuated phases (which control the minor street

traffic) are not.

Notations

The following notations are used throughout the dissertation.

Where applicable, a subscript i indicates the ith phase.

C = cycle length in seconds;

CB = background cycle length in seconds of a coordinated signal;













































N




P(

Pk(


P


d = average delay in seconds per vehicle;

D = total delay in seconds in a cycle;

dint = average delay in seconds per vehicle for an intersection as a

whole;

f(t) = probability density function that the headway of traffic is t;

F(t) = probability that the headway of traffic is less than t;

g = green time in seconds;

G = average green time in seconds;

Gdwl = expected dwell time in seconds of the non-actuated phase;

Gmin = minimum green interval in seconds of the non-actuated phase;

gs = saturation green time in seconds;

Gs = average saturation green time in seconds;

Gsla = slack time in seconds of the non-actuated phase;

N = number of vehicles in a cycle;

Nc = number of cycles;

Nd = number of green occurrences with a zero effective red time;

Ng = number of green occurrences;

Nr = number of green occurrences with a positive effective red time

Ns = number of phase skip occurrences;

stop = number of vehicles stopped in a cycle;
P = transition function, which is a square matrix of P(i,j);

P(i) = stationary probability that cycle 1 is in state (i);

i,j) = transition probability from state (i) to state (j);

i,j) = sub-transition probability of sub-transition k from state (i)

to state (j);

Ps = stop probability;

'sint = stop probability for an intersection as a whole;


;










r = red time in seconds;

R = average red time in seconds;

r(n,t) = probability of having n arrivals in a time interval t;

s = saturation flow rate in vehicles per second;

v = volume in vehicles per second;

x = volume to capacity ratio;

T = stationary distribution, which is a horizontal vector in the

form of (P(1),P(2),...,P(n)); and
S= transposed vector of 7 ;

Method Development

An actuated phase will be either skipped or present in a cycle. If

the phase is skipped, it has no traffic demand up to the yield point,

which is defined as the moment when the controller is ready to serve the

phase. If the phase is present, all its traffic demand is assumed to be

serviced in the green time (as previously stated). Therefore, the

presence of the actuated phase in the next cycle depends simply on

whether, if skipped in the current cycle, it receives actuations after

the yield point, or, if present in the current cycle, it receives

actuations after the end of the green time. In other words, two

consecutive cycles are involved in determining the presence of an

actuated phase. Because of this property, the operation of a semi-

actuated signal can be described by two consecutive cycles, and the

system can be treated as a Markov chain [21]. For notation convenience,

the previous cycle is denoted as cycle 1 and the current cycle as cycle

2. A phase sequence in a cycle will be treated as a "state" in terms of

Markov chains, and two consecutive cycles will constitute a









"transition", again, in Markov terms. It is clear that there are total

of n*n transitions if there are n states in one cycle.

Method Description

The method is conducted on a phase by phase basis for individual

intersections. There are two basic inputs: traffic volumes and signal

settings, such as the background cycle length and the minimum green

intervals. The complete analysis procedure is illustrated in Figure

3.1, which shows the five computational steps: determine transitions,

compute transition and stationary probabilities, determine patterns,

compute pattern probabilities, and compute stop probabilities and

average delays. The first four steps are to consider the various cycle

patterns and their probabilities. A cycle pattern is defined as a

specific combination of a red time and a green time. The last step is

to estimate stop probability and average delay for each cycle pattern

and for the intersection as a whole.

In Step 1, transitions are determined from all the combinations of

possible states in two consecutive cycles. Step 2 computes transition

and stationary probabilities. Transition probability, which is a

conditional probability of cycle 2 being in one state given that cycle 1

is in a known state [21], is determined from the Poisson distribution in

the following way. Assuming that the volume of an actuated phase is v,

the probability that this phase is not activated during a time interval

of r is e-vr, while the probability of it being activated in r is

1 e-vr

Since the system is a Markov chain, stationary probability, which

is the probability that cycle 1 is in a given state, is computed from

the following equation [21]:























































Figure 3.1. Block Diagram of the Analytical Method.









7T p = T t (3.2)

Step 3 is to identify all the possible cycle patterns in terms of

red and green times. Red and green times can be directly measured on

transitions. Identical red and green times may appear in differing

transitions. For the non-actuated phase, both the red and green times

may vary from pattern to pattern. For actuated phases, only red times

may vary since the phases are assumed to terminate the green times after

the initial green intervals complete.

In Step 4, pattern probability is then computed from summing up the

product of stationary and transition probabilities of the transitions

that produce the pattern.

Finally in Step 5, stop probability and average delay for each

cycle pattern are computed using the following equations which are

developed in the last section of this chapter.


Ps = r + gs (3.3)
r + g
r+g

r
d =- Ps (3.4)
2.6


r + g
Ps = e-vr + (1 e-vr) (3.5)
r+g


r2 g(g + 2r) gs2
d = r + e-vr g) + (1 e-vr) g(3.6)
2(r + g) 2(r + g) 2(r + g)


1
Ps = (3.7)
vg + 1


d=O 0


(3.8)









vr
where gs = (3.9)
s v

For clarity, the above equations and the conditions for their use

are summarized in Figure 3.2. Equations (3.3) and (3.4) are for the

patterns of the non-actuated phase. Equations (3.5) and (3.6) are for

those of the actuated phases with positive effective red times.

Otherwise, for actuated phases with a zero effective red time, Equations

(3.7) and (3.8) should be used. The difference between a positive

effective red time and a zero effective red time will be explained later

in Step 3 of the section of Isolated Control.

Once the stop probability and average delay of each cycle pattern

is computed, the next task is to compute the phase stop probability and

average delay. The phase average delay is treated first.

The total vehicle arrivals (N) in one cycle of a pattern can be

estimated by multiplying the average arrival rate and the sum of the red

and green times of the pattern, i.e.,

N of pattern j = v(r + g) (3.10)

The total delay in one cycle of the pattern can also be estimated by

multiplying the average delay and the total arrivals, i.e.,

D of pattern j = v(r + g)d (3.11)

The expected arrivals across the patterns is the sum of the products,

for all patterns, of the total arrivals of each pattern multiplied by

the corresponding pattern probability, i.e.,


N of phase i

= E [(N of pattern j) (probability of pattern j)] (3.12)
all j

The expected delay across the patterns is the sum of the products, for

























































vr

s v


r = effective red time


Figure 3.2. Stop Probability and Average Delay Equations
and the Conditions for Their Use Under
the Analytical Method.








all patterns, of the total delay of each pattern multiplied by the

corresponding pattern probability, i.e.,

D of phase i
= E [(D of pattern j) (probability of pattern j)] (3.13)
all j

Combining (3.12) and (3.13), the average delay for phase i is the

expected delay divided by the expected arrivals, i.e.,

D of phase i
di =
N of phase i

Z [(D of pattern j) (probability of pattern j)]
all j
(3.14)
E [(N of pattern j) (probability of pattern j)]
all j

The average delay for the intersection as a whole, therefore, is

# of phases
E (divi)
i = 1
dint = (3.15)
# of phases
vi
i = 1

The computations of stop probability follow the same logic. The

number of vehicles stopped (Nstop) in one cycle of a pattern can be

estimated by multiplying the stop probability and the total arrivals,

i.e.,

Nstop of pattern j = v(r + g)Ps (3.16)
The expected Nstop across the patterns is the sum of the products, for

all patterns, of the Nstop of each pattern multiplied by the

corresponding pattern probability, i.e.,

Nstop of phase i
= E [(Nstop of pattern j) (probability of pattern j)] (3.17)
all j










Combining (3.12) and (3.17), the stop probability for phase i is the

expected Nstop divided by the expected arrivals, i.e.,


Nstop of phase i
N of phase i

E [(Nstop of pattern j) (probability of pattern j)]
all j
(3.18)
Z [(N of pattern j) (probability of pattern j)]
all j

The stop probability for the intersection as a whole is

# of phases
S (Psivi)
i = 1
Psint = (3.19)
t # of phases
E vi
S vi
i = 1

The applications of the method to both isolated and coordinated

controls are presented below.

Isolated Control

A three-phase signal under isolated control is illustrated, but the

logic can be systematically expanded to signals with more than three

phases.

Step 1: Determine Transitions. There are three possible states in any

one cycle, based on whether phase 2 or 3 is skipped or not. The three

states are (1) phase 1, then phase 2, and finally phase 3; (2) phase 1

followed by phase 2 only; and (3) phase 1 followed by phase 3 only.

State 1 is the normal phase sequence, in which no actuated phase is

skipped. State 2 means that phase 2 is followed by phase 1 of the next

cycle. Namely, phase 3 is skipped. This state occurs when phase 3 is

not activated before the end of phase 2. Similarly, phase 2 is skipped








in state 3. This state occurs when phase 2 is not activated before the
end of phase 1, but phase 3 is. The nine transitions constituted by the
three states are shown in Figure 3.3.
It can be seen in this figure, each transition is further split
into two sub-transitions based on phase 1's green length in cycle 2. In
sub-transition 1, phase 1's green time (gl) in cycle 2 is the minimum
green time (Gmin). In sub-transition 2, gl is greater than Gmin. Sub-
transition 1 occurs when at least one of phases 2 and 3 is activated
before the end of Gmin. In this case, the green light will switch from
phase 1 to the activated phase(s) after the completion of Gmin. Sub-
transition 2 occurs when both phases 2 and 3 are not activated before
the end of Gmin. In this case, the green light will dwell on phase 1
until any one of them is activated. Therefore, phase 1's green time in
sub-transition 2 is always longer than that in sub-transition 1. The
green time of phase 1 extended beyond Gmin is called dwell time. The
length of dwell time can be estimated in the follow way:
Let f2(t) be the probability density function that the headway of
phase 2 is t, and F3(t) be the probability that the headway of phase 3
is less than t, the expected dwell time of the case that phase 2
terminates the dwell is

Gdwl2 = t f2(t) [1 F3(t)] dt

t v2 exp(-v2t) exp(-v3t) dt

= v2 ft exp[-(v2 + v3)t] dt


(3.20)
(v2 + v3)2








Cycle 1


State 1




g 92 93 State 2
I I --

State 1

State 3
L I .


Cycle 2


State 1
I


State 3


-- phase 2's red time
==== phase 3's red time


Transition No.


1.1


1.2

2.1


2.2

3.1


3.2


4.1


4.2

5.1


5.2

6.1


6.2


7.1


7.2

8.1


8.2

9.1


9.2


Figure 3.3.


Transitions, Red Times, and Transition Probabilities at a
Three-Phase Semi-Actuated Signal Under Isolated Control.








Transition No. Transition probability

1.1 Pi(1,1)={1-exp[-v2(93+Gmin)]}{l-exp[-v3(Gmin+g2)

1.2 P2(1,1)=exp[-v2(g3+Gmin)]exp(-v3Gmin)2/(v2+v3)[1-exp(-v3g2)]
2.1 Pl(1,2)=({-exp[-v2(g3+Gmin)]}exp[-v3(Gmin+g2)

2.2 P2(1,2)=exp[-v2(g3+Gmin)]exp(-v3Gmin)v2/(v2+v3)exp(-32)
3.1 Pl(1,3)=exp[-v2(93+Gmin)][1-exp(-v3Gmin)

3.2 P2(1,3)=exp(-v2Gmin)exp(-v3Gmin)v3/(v2+v3)

4.1 Pl(2,1)=[1-exp(-v2Gmin)]{1-exp[-v3(Gmin+g2)]

4.2 P2(2,1)=exp(-v2Gmin)exp(-v3Gmin)v2/(v2+v3)[1-exp(-v3g2)
5.1 Pl(2,2)=[l-exp(-v2Gmin)]exp[-v3(Gmin+92)

5.2 P2(2,2)=exp(-v2Gmin)exp(-v3Gmin)V2/(v2+v3)exp(-v32)
6.1 P1(2,3)=exp(-v2Gmin)[1-exp(-v3Gmin)]

6.2 P2(2,3)=exp(-v2Gmin)exp(-v3Gmin)v3/(v2+v3)

7.1 Pl(3,1)={1-exp[-v2(g3+Gmin)]}{1-exp[-v3(Gin+92)]

7.2 P2(3,1)=exp[-v2(g3+Gmin)]exp(-v3Gmin)v2/(v2+v3)[-exp(-v3g2)
8.1 Pl(3,2)={1-exp[-v2(g3+Gmin)]}exp[-v3(Gmin+2)

8.2 P2(3,2)=exp[-v2(g3+Gmin)]exp(-v3Gmin)v2/(v2+v3)exp(-v3g2)
9.1 Pl(3,3)=exp[-v2(93+Gmin)][-exp(-v3Gmin)]

9.2 P2(3,3)=exp(-v2Gmin)exp(-v3Gmin)v3/(v2+v3)

Figure 3.3. continued.









Similarly, the expected dwell time of the case that phase 3 terminates

the dwell is


Gdwl3 = 3 (3.21)
(v2 + v3)

Please note that it is unnecessary to split the parallel difference

in cycle 1, since whether phase 1 dwells or not in cycle 1 will not

affect the presence of the actuated phases in cycle 2.

Step 2: Compute Transition and Stationary Probabilities. The transition

probabilities are also shown in Figure 3.3. Although they look

complicated at the first glance, they can be easily established from

basic Poisson distribution function. The following example illustrates

how they are obtained.

Let P2(1,1) be the probability of sub-transition 2 from state 1 to

state 1. It is shown in Figure 3.3 that in this sub-transition, phase

2's red time is g3+Gmin+(dwell time), and phase 3's is Gmin+(dwell

time)+g2. This sub-transition occurs when the following conditions are

satisfied: (1) both phases 2 and 3 are not activated before Gmin

expires; (2) phase 2 is first activated when the green light dwells on

phase 1; and (3) phase 3 is activated during phase 2's green time.

These conditions are formulated as follows:

P2(1,1)
= P(phase 2 is not activated before Gmin expires)

P(Phase 3 is not activated before Gmin expires)
P(phase 2 is first activated when the green light dwells on phase 1)

P(Phase 3 is activated when phase 2 is green)








= r(O, g3+Gmin) r(0, Gmin) [v2/(v2+v3)] [l-r(O, g2)]
= exp[-v2(g3+Gmin)]*exp(-v3Gmin)*[v2/(v2+v3)]*[-exp(-v3g2)] (3.22)

Once Pk(i,j) is determined, k=sub-transition 1 or 2, the next task

is to compute stationary probabilities. Let P(i,j)=P1(i,j)+P2(i,j) for

all i and j, and let P(i) be the stationary probability of state i,

where i=1 to 3, P(i) can be obtained from solving the following

equations:
P(1,1) P(1,2) P(1,3) P(1)

[ P(1),P(2),P(3) ] P(2,1) P(2,2) P(2,3) P(2) (3.23)
P(3,1) P(3,2) P(3,3)_ P(3)
with P(1)+P(2)+P(3)=1.

Step 3: Determine Patterns. All the possible patterns in terms of red

and green times for the three phases are summarized in Table 3.1. For

phase 1, the dwell time is included in the green time, since dwell
extends the green time. For phases 2 and 3, the green times are fixed,

but the determination of red times needs explanations.

When dwell occurs, the actuated phase following phase 1 terminates

the dwell. In this case, the actuated phase has no arrival during the

red time until it is activated by a vehicle (i.e., until the end of the

dwell time). Similarly, when an actuated phase is skipped, there is no

vehicle arrival during the corresponding red time. In order to

distinguish from the actual red time, effective red time is used for the
actuated phases, which subtracts the red time without arrivals from the

actual red time. The effective red time is always positive in Sub-
transition 1, but is zero for the actuated phase immediately following

phase 1 in Sub-transition 2. In Table 3.1, Patterns 1 and 2 of









Table 3.1.


Pat.


Patterns (in Terms of Red and Green Times) and Their
Probabilities at a Three-Phase Semi-Actuated Signal
Under Isolated Control.


Type of pattern


- I I


r=g2+g3


Pattern probability


P(1)[P(1,1P(1,)+P(1,2)+P(1,3)]


r=g2 9+g3
2 P(1)[P2(1,1)+P2(1,2)]
g=Gmin+Gdwl2

r=g2+93
3 P(1)P2(1,3)
g=Gmin+Gdwl3

r=g2
4 P(2)[P1(2,1)+P1(2,2)+P1(2,3)]
g=Gmin

r=g2
5 P(2)[P2(2,1)+P2(2,2)]
g=Gmin+Gdwl2

r=g2
6 P(2)P2(2,3)
g=Gmin+Gdwl3

r=g3
7 P(3)[P1(3,1)+P1(3,2)+P1(3,3)]
g=Gmin

8 3P(3)[P2(3,1)+P2(3,2)]
g=Gmin+Gdwl2


r=g3

g=Gmin+Gdwl3


where Gdwl2


P(3)P2(3,3)


Ps and d
equations


(3.3)

(3.4)


V2

(v2 + V3)2


V3
dwl3 2
(v2 + v3)

r =red time
g = green time


Phase












Table 3.1. continued.


Phase Pat. Type of pattern


Pattern probability


I -i I


r=Gmi n
rGmin

9=92

r=g3+Gmin

9=92


r=O

g=92

r=Gmin+92

9=93
r=g2

9=93


r=Gmin

9=93


r=O

9=93


P(2)[P1(2,1)+P1(2,2)]


I


P(1)[P1(1,1)+P1(1,2)]+[P(1)P(1,3)+P(2)

P(2,3)+P(3)][P1(3,1)+P1(3,2)]/[1-P(3,3)]


P(1)[P?(1,1)+P (1,2)]+P(2)[P2(2,1)
+P2(2,2)]+P(1)P(1,3)+P(2)P(2,3)+P(3)]
[P2(3,1)+P2(3,2)]/[1-P(3,3)]


P(1)P1(1,1)+P(3)P1(3,1)]+[P(1)P(1,2)

+P(2)+P(3)P(3,2)]P1(2,1)/[1-P(2,2)]


P(1)P2(1,1)+P(3)P2(3,1)]+[P(1)P(1,2)

+P(2)+P(3)P(3,2)]P2(2,1)/[1-P(2,2)]


P(1)Pl(1,3)+P(3)P1(3,3)]+[P(1)P(1,2)

+P(2)+P(3)P(3,2)]P1(2,3)/[1-P(2,2)]


P(1)P2(1,3)+P(3)P2(3,3)]+[P(1)P(1,2)

+P(2)+P(3)P(3,2)]P2(2,3)/[1-P(2,2)]


where r = effective red time
g = green time


Ps and d
equations


(3.5)

(3.6)


(3.7)

(3.8)


(3.5)

(3.6)


(3.7)

(3.8)


I


I I


- I








phase 2 have positive effective red times, while Pattern 3 has a zero

effective red time. Similarly, Patterns 1, 2, and 3 of phase 3 have

positive effective red times, and Pattern 4 has a zero effective red

time.

Step 4: Determine Pattern Probabilities. Pattern probability is

computed from summing up the product of stationary and transition

probabilities of the transitions that produce the concerned pattern.

Table 3.1 shows the pattern probabilities of the three phases. The

first two patterns of phase 2 are explained below to illustrate how they

are obtained.

Pattern 1 has the green time of g2 and the effective red time of

Gmin. In Figure 3.3, it can be observed that this pattern appears in

the following two sub-transitions:

State 2 to state 1, sub-transition 1; and

State 2 to state 2, sub-transition 1.

Since their transition probabilities are Pl(2,1) and Pl(2,2),

respectively, and the stationary probability of state 2 is P(2), the

pattern probability is P(2) Pl(2,1) + P(2) PI(2,2), i.e.,

P(2)[PI(2,1) + P(1(2,2)].

Pattern 2 is more complicated. Its green time is g2, and its

effective red time is g3+Gmin. As observed in Figure 3.3 again, this

pattern appears in the following (sub-)transitions:

State 1 to state 1, sub-transition 1;

State 1 to state 2, sub-transition 1;

State 1 to state 3;


State 2 to state 3;









State 3 to state 1, sub-transition 1;

State 3 to state 2, sub-transition 1; and

State 3 to state 3.


This pattern is depicted in Figure 3.4 as separated into three

components according to the status in cycle 1: state 1, state 2, and

state 3. The pattern probability contributed by each component is shown

in the figure, and their sum is the pattern probability.

The third component shows that the pattern appears in the following

three transitions:

State 3 to state 1, sub-transition 1;

State 3 to state 2, sub-transition 1; and

State 3 to state 3.

For easy reference, the three transitions are referred to as

transitions a, b, and c here. The pattern probability contributed by

transitions a and b, as explained above, is

P(3)[PI(3,1) + Pl(3,2)]. (3.24)

However, the probability contributed by transition c involves an

infinitive series. When this transition occurs, the green and effective

red times will not appear in the current transition due to phase 2 being

skipped. But they may appear in the next transition if the next

transition is either transition a or b. The probability of this

situation is

P(3)P(3,3)[P1(3,1) + Pl(3,2)]. (3.25)

Similarly, if the next transition is c again, then the above

situation may repeat and therefore the probability is








Pl(1,1)


P(1) I 91 gl
State 1 P(1,3)


P1(3,1)

,2)


9 gl 93 Gmi 2 i P(3,1)
P(3,3) Pl(3,2)

91 93
P(3,3)
Probability=P(1)[P1(1,1)+P1(1,2)]+P(1)P(1,3)[P1(3,1)+P1(3,2)]/[l-P(3,3)]

Gmin 92 93
91 92 91 93 I P1(3,1)
P(2) P(2,3) min- 92 P (3,2)
State 2 9 min 93 P1(3,1)

P(3,3) PI(3,2)

91 93
P(3,3)
Probability=P(2)P(2,3)[PI(3,1)+PI(3,2)]/[l-P(3,3)]


P1(3,1)


State 3


P(3,3)


effective red time
'== green time


Pl(3,1)

,2)


gl 93
P(3,3)
P(3,3)


Pattern probability=P(1)[P (1,1)+PI(1,2)]+[P(1)P(1,3)+P(2)P(2,3)
+P(3)][P1(3,1) Pl(3,2)]/[C-P(3,3)]
Figure 3.4. Example of the Determination of Pattern Probability.


Probability::P(3)[P,(3,1)+Pl(3,2)]/[1-P(3,3)]








P(3)P(3,3)[PI(3,1) + P1(3,2)]
+ P(3)P(3,3)2[Pl(3,1) + Pl(3,2)]

+ P(3)P(3,3)3[PI(3,1) + Pl(3,2)]+...

= P(3)P(3,3)[PI(3,1) + Pl(3,2)]/[1 P(3,3)] (3.26)

Combining (3.24) and (3.26), the total probability contributed by this

component is

P(3)[P1(3,1) + PI(3,2)] + P(3)P(3,3)[P1(3,1) + PI(3,2)]/[1 P(3,3)]
= P(3)[PI(3,1) + P1(3,2)](1 + P(3,3)/[1 P(3.3)]}

= P(3)[P1(3,1) + PI(3,2)]/[1 P(3,3)]. (3.27)

The other two components will not be explained since they follow

the same logic.

Step 5: Compute Stop Probabilities and Average Delays. By entering r

and g of each pattern along with the phase traffic volume to the

equations numbered in the last column of Table 3.1, the pattern stop

probability and average delay can be computed. Then by applying

Equations (3.14) and (3.15), the average delays for the three phases and

for the intersection as a whole can be estimated. Similarly, by

applying Equations (3.18) and (3.19), the stop probabilities for the

three phases and for the intersection as a whole can also be estimated.

Coordinated Control

Under coordinated control, a background cycle length is imposed on

each individual signal in order to achieve green bands for progressive

movements on the major street. A three-phase signal under coordinated

control is presented in this section.

Step 1: Determine Transitions. There are four states in one cycle,

based on whether phase 2 or 3 is skipped or not. They are (1) phase 1,

then phase 2, and finally phase 3; (2) phase 1 followed by phase 2, then









returns to phase 1; (3) phase 1 followed by phase 3, then returns to

phase 1; and (4) phase 1 only. State 1 is the normal sequence, in which

no actuated phase is skipped. State 2 is the case that phase 3 is

skipped due to lack of demand before phase 2 completes. State 3 means

that phase 2 is skipped, but phase 3 is not. State 4 indicates that

both phases 2 and 3 are skipped, so that the green light stays on phase

1 to complete the whole cycle. The sixteen transitions constituted by

the four states are shown in Figure 3.5.

It can be seen in states 2 and 3 that the green light returns to

phase 1 to complete the cycle after the activated phase (either phase 2

or phase 3) has been serviced. The time interval beyond the end of the

last activated phase to the end of one cycle is called slack time. The

maximum slack time, as can be observed in state 4 in which both phases 2

and 3 are skipped, is the cycle length minus Gmin-

Moreover, as shown in transitions 4, 8, 12, and 16, both phases 2

and 3 are skipped in cycle 2, since they are not activated before Gmin

completes. In this case, if a vehicle arrives for phase 3 right after

the end of Gmin, the vehicle has to wait till the next cycle to be

serviced. This is certainly not efficient. Some commercial controllers

use permissive periods to improve this situation. A permissive period

in this example would allow phase 3 to receive the green time in cycle

2, if activated after the end of Gmin, but before the end of the

permissive period. To accomplish this, transitions 4, 8, 12, and 16

would have to be split into several further transitions, to represent

all the possible outcomes. Since permissive periods greatly increase

the complexity of the analysis, they will not be considered in this

example.







Transition No.


State
r-
I
State
Gmin 92 93
1 Gmn I I_


State 1


Gmin
I I L


St


State


State


State
r--

State
g2 slack -

State
rate 2

State


State
r

State
Gmin 93 slack -

State
State 3

State
L -


I Gmin ,


slack


State

State
IState
I...


State
State 4

Phase 2's red time State
Phase 3's red timeL -


1 GmIn


2 Gmin 92 slack


3 Gmin 93 slack


4 Gin slack
min

1 Gmin 92 93
S I I

2 Gmin 92 slack


3 Gmin 93 slack

4 G slack
min


1 Gmin 92 93
-I

2 Gmin 92 slack


3 Gmin 93 slack
... I

4 Gin slack


1 Gmin 92 93


2 Gmin 92 slack


3 Gmin 93 slack


4 Gmin slack
-I


Figure 3.5. Transitions, Red Times, and Transition
Three-Phase Semi-Actuated Signal Under


Probabilities at a
Coordinated Control.


Cycle 1


Cycle 2


9g 9g








Transition No. Transition probability

1 P(1,1)={l-exp[-v2(g3+Gmin)]}{l-exp[-v3(G +g2)]}

2 P(1,2)={l-exp[-v2(g3+Gmin)]}exp[-v3(Gmin+g2)

3 P(1,3)=exp[-v2(g3+Gmin)][l-exp(-v3Gmin)]

4 P(1,4)=exp[-v2(g3+Gmin)]exp(-v3Gmin)

5 P(2,1)={1-exp[-v2(g3+Gmin)]}{1-exp[-v3(g3+Gmin+g2)]

6 P(2,2)={1-exp[-v2(g3+Gmin)]}exp[-v3(g3+Gmin+92)

7 P(2,3)=exp[-v2(g3+Gmin)]{1-exp[-v3(g3+Gmn)]}

8 P(2,4)=exp[-v2(g3+Gmin)]exp[-v3(g3+Gin)]

9 P(3,1)={1-exp[-v2(g3+g2+Gmin)]}{1-exp[-v3(g2+Gmin+2)

10 P(3,2)={l-exp[-v2(g3+g2+Gmin)]}exp[-v3(g2+Gmin+g2)

11 P(3,3)=exp[-v2(g3+g2+Gmin)]}{1-exp[-v3(g2+Gmin)]

12 P(3,4)=exp[-v2(93+92+Gmin)]}exp[-v3(g2+Gmin

13 P(4,1)={1-exp[-v2(g2+g3+Gmin)]}{-exp[-v3(92+3+Gmin+92)]

14 P(4,2)={1-exp[-v2(g2+g3+Gmin)]}exp[-v3(92+3+Gmin+92)

15 P(4,3)=exp[-v2(g2+g3+Gmin)]}{1-exp[-v3(g2+g3+Gmin)]}

16 P(4,4)=exp[-v2(g2+g3+Gmin)]}exp[-v3(92+3+Gmin)]

Figure 3.5. continued.








Step 2: Compute Transition and Stationary Probabilities. The transition

probabilities are shown in Figure 3.5. Similarly to isolated control,

stationary probabilities, P(i), i=1 to 4, can be obtained from solving

the following equations:

P(1,1) P(1,2) P(1,3) P(1,4) P(1)

P(2,1) P(2,2) P(2,3) P(2,4) P(2)
[ P(1),P(2),P(3),P(4) ] = (3.28)
P(3,1) P(3,2) P(3,3) P(3,4) P(3)

P(4,1) P(4,2) P(4,3) P(4,4) P(4)

with P(1)+P(2)+P(3)+P(4)=1.

Steps 3 and 4.

Since the processes of Steps 3 (Determine patterns) and 4

(Determine pattern probabilities) are similar to those under isolated

control, only their results are shown in Table 3.2.

Step 5: Compute Stop Probabilities and Average Delays. The processes of

this step to arrive at phase and intersection stop probabilities and

average delays are similar to those under isolated control. But the

effect of progression on phase 1 should be further considered.

Up to this point, the stop probabilities and average delays

obtained assume Poisson arrival conditions. As stated in the section of

Assumptions and Notations, the non-actuated phase (i.e., phase 1, which

controls the major street traffic) is assumed to be under the influence

of traffic progression from the upstream intersection. Hence, the stop

probability and average delay of phase 1 should be further adjusted to

take this factor into account. As reviewed in Chapter Two, the

influence of traffic progression can be accessed by using the

progression adjustment factors (PF) provided in the 1985 Highway

Capacity Manual (HCM) [19]. The stop probability and average delay for















Table 3.2. Patterns (in Terms of Red and Green Times) and Their
Probabilities at a Three-Phase Semi-Actuated Signal
Under Coordinated Control.


Type of pattern


r=g2+g3


Pattern probability


P(1)[P(1,1)+P(1,2)+P(1,3)]


g=Gmin
r=g2 P(2)[P(2,1)+P(2,2)+P(2,3)]

g=Gsla2+Gmin


r=g3

g=Gsla2+Gmin


r=g2+g3, n=2,3,...
4 P(1)[P(4)+P(1,4)]P(4,4)n-2
g=(n-1)CB+Gmin, n=2,3,... [P(4,1)+P(4,2)+P(4,3)]


r=g2

g=nCB-g2, n=2,3,...

r=g3

g=nCg-g3, n=2,3,...


P(3)[P(3,1)+P(3,2)+P(3,3)]


P(2)[P(4)+P(2,4)]P(4,4)n-2
[P(4,1)+P(4,2)+P(4,3)]


P(3)[P(4)+P(3,4)]P(4,4)n-2
[P(4,1)+P(4,2)+P(4,3)]


Ps and d
Equations


(3.3)

(3.4)


where Gsla2
Gsla3
r
g


92
93
=red time
=green time


Phase


Pat.


1









Table 3.2. continued.


Phase Pat. Type of pattern

r=g3+Gmin


19=92 1


r=Gsla3+Gmin

g=g2
r=g3+Gsla2+Gmin

9=92
r=Gsla+Gmin

9=92


Pattern probability


P(2)[P(2,1)+P(2,2)]


P(1)P(1,3)+P(2)P(2,3)+P(4)P(4,3)

+P(3)[P(3,1)+P(3,2)+P(3,3)]

P(1)P(1,4)+P(2)P(2,4)+P(3)P(3,4)

+P(4)[P(4,1)+P(4,2)+P(4,4)]


- -I I-- I


r=Gmin+92

9=93
r=Gmin

9=93 Gmi

r=Gsla2+Gmin+ 92


9=93


P(1)P(1,1)


P(1)P(1,3)


P(3)P(3,1)+{P(1)[P(1,2)+P(1,4)P(5,2)]+
P(2)+P(3)[P(3,2)+P(3,4)P(5,2)+P(4)
P(4,2)}P(2,1)/{1-[P(2,2)+P(2,4)P(4,2)]


r=Gsla2+Gmin P(3)P(3,3)+{P(1)[P(1,2)+P(1,4)P(5,2)]+
4 P(2)+P(3)[P(3,2)+P(3,4)P(5,2)+P(4)
9=93 P(4,2)}P(2,3)/{1-[P(2,2)+P(2,4)P(4,2)]}

r=Gsla+Gmin+92 (P(1)[P(1,4)+P(1,2)P(5,4)]+P(2)[P(2,4)+
5 P(2,2)P(5,4)+P(3)[P(3,4)+P(3,2)P(5,4)+
9=93 +P(4)}P(4,1)/{(-[P(4,4)+P(4,2)P(2,4)]}


r=Gsla+Gmin

9=93


Ps and d
equations


(3.5)

(3.6)


(3.5)

(3.6)


{P(1)[P(1,4)+P(1,2)P(5,4)]+P(2)[P(2,4)+
P(2,2)P(5,4)+P(3)[P(3,4)+P(3,2)P(5,4)+I
+P(4)}P(4,3)/(1-[P(4,4)+P(4,2)P(2,4)]}I


where Gsla2 = 92
G~la3 g93
G = g+g3

P(5 = P 4,2)/[1-P(4,4)]
P(5,4) = P(2,4)/[1-P(2,2)]
r = effective red time
g = green time


I


P(1)[P(1,1)+P(1,2)]










phase 1, computed by this method, are multiplied by PF to account for

the influence of traffic progression. Since the determination of PF

values has been reviewed in Chapter Two, it will not be repeated here.

The PF values are simply regarded as input in this method.

Test of the Analytical Method

Since the quantitative analysis of average delays and stop

probabilities is very complicated, some approximations are used in the

development of this method. Justification is needed for these

approximations. A simulation which deals with individual vehicles in

determining delays and stops was employed for this justification. The

purpose of this simulation is to test, under the same assumptions,

whether the Analytical Method will produce the same results as does the

simulation. If the results are close, it would be reasonable to

conclude that the mathematics in this method is correct.

It should be emphasized, however, that this simulation is simply to

test the mathematical model in this method under the previously stated

assumptions. Field data would be required to further validate whether

the assumptions are reasonable.

It would be best to test the mathematical model if existing

simulation programs can be used. NETSIM [31] could be a reasonable

choice for this purpose, since it is the most widely used simulation

program in the traffic engineering community. However, NETSIM assumes

uniform arrivals on entry links, but the Analytical Method assumes

Poisson arrivals. Therefore, they are not under the same assumptions.

Because of the difference in the underlying assumptions, NETSIM could

not be used and a new simulation program was developed.









Since the Analytical Method is conducted on an intersection basis,

the simulation deals with only individual intersections. Also, since

the Analytical Method originally computes average delays and stop

probabilities with no consideration of progression (though it then uses

the HCM's progression adjustment factors to account for the effect of

progression on the non-actuated phase), the simulation does not simulate

platoon arrivals either. Only random flows are simulated, regardless of

whether the signal is under coordinated control or isolated control.

The results from the two methods, before the adjustments for

progression, are compared in this section.

In the simulation program, a sequence of vehicle arrival times for

each actuated phase is first generated, with exponentially distributed

inter-arrival times. These arrival times represent the times of vehicle

arrivals at the stop line (which also represent the detection times,

since a presence detector is presumed to be located near the stop line).

The mechanics of the semi-actuated signal operations are built into the

program to determine the start times of the green phase of each actuated

phase. For example, for coordinated operation, the program first checks

if the first actuated phase has any arrivals up to the yield point. If

it does, the actuated phase is given green time right after the yield

point, and the program checks if the second actuated phase registers any

arrivals up to the end of the green time of the first actuated phase.

On the other hand, if the first actuated phase does not have any

arrivals up to the yield point, it is skipped and the program checks if

the second actuated phase has any arrivals up to the yield point. If

the second actuated phase also has no arrivals, the next actuated phase









is checked. After all the actuated phases have been checked, one cycle

is completed, and the program starts the next cycle.

The queued vehicles arriving in the red time of an actuated phase,

and those (if any) arriving before the queue has fully discharged, are

released at the saturation flow rate. The moment a vehicle is released

represents the departure time of the vehicle leaving the stop line. The

delay of each stopped vehicle was calculated as the departure time minus

the arrival time. The average delay per vehicle was calculated as the

total delay divided by the total number of vehicles, with total delay as

the sum of the delay of all stopped vehicles. The stop probability was

obtained by dividing the total number of vehicles stopped by the total

number of vehicles. Isolated and coordinated controls were treated

separately.

From the above description, it can be seen that vehicle arrival

times of actuated phases are the only random variables generated in the

simulation process. After they are generated, the remaining processes

are deterministic based on these arrival times, the signal operation,

and the saturation departure rate.

A three-phase signal was simulated under both isolated and

coordinated controls. The minimum green time of the non-actuated phase

and the green times of the actuated phases were taken as 20 seconds.

Under each kind of control, 216 sets of volumes were tested-the three

phases were given volumes from 50 vehicles per hour (vph) to 300 vph,

with 50 vph increments. Each set of volumes was simulated for ten

replications, and each replication was simulated for 6,000 seconds.

The Analytical Method was coded into a computer program to compute

stop probabilities and average delays for the 216 sets of volumes. Part









of the results are shown in Figure 3.6, which presents four comparisons:

stop probabilities and average delays of phase 2 under both isolated and

coordinated controls, when phase 3's volume is 50 vph. The average

delays obtained from simulation runs are plotted to illustrate both the

central tendency and the dispersion, but only the central tendency of

stop probabilities are plotted, since their standard deviations are very

small (less than 0.3%). These comparisons show that the estimations

from the method are visually fitted to the data points obtained from

simulation runs. Since other sets of results show the same pattern,

they are omitted. This finding indicates that the approximations in

this method are acceptable.

Comparison with a Pretimed Model

According to the review in Chapter Two, very few models exist which

deal exclusively with semi-actuated signal evaluation. A common

evaluation practice, given this lack of appropriate models, is the use

of a model for pretimed signals, with average green and red times as

input. The purpose of this section is to investigate the difference

between using a pretimed model and using the Analytical Method. The

method using a pretimed model will be referred to as the Pretimed Model

Method, in contrast to the Analytical Method.

The three-phase signal is used again in this comparison, as it was

used in the previous section for validation against simulation. Recall

that the minimum green time of the non-actuated phase and the green

times of the two actuated phases are taken as 20 seconds. Because an

actuated phase has 20 seconds of green time when present, but has zero

seconds of green time when skipped, and because the phase's red time

depends on the presence of the other actuated phase, it is very





61

Under isolated control, phase 3'
Average delay of phase 2 (sec/veh) Stop


1.0

0.9


0.7-

0.6-


s volume=50 vph

probability of phase 2


15-

13-

11-

9-

7-


I I I I I I 1
300 50 100 150 200 250 300

2's volume in vph


Under coordinated control, phase
Average delay of phase 2 (sec/veh) Stop


3's volume=50 vph
probability of phase 2


+- 1
devia


1.0

0.9

U 0.8

8 0.7:

B-
S0.5-

0.4-
I I I I 1 I
50 100 150 200 250 300
Phase 2's volume

results froi
std. fo_ mean delay values
ition from simulation ru


I I I I I I
50 100 150 200 250 300
e in vph

n the Analytical Method

o mean stop probabilities
ns from simulation runs


Figure 3.6. Stop Probability and Average Delay Comparisons,
the Analytical Method vs. Simulation.


I I I 1 1
50 100 150 200 250

Phase


I s


8 8


8 8


11 8









difficult to estimate the phase's average red and green times without

conducting a field observation or performing a simulation. For

simplicity, it is assumed that the average green time of each phase is

20 seconds. In other words, the three phases are assumed to appear in

every cycle with 20 seconds of green time. Under this assumption, each

phase has a red time of 40 seconds and a green time of 20 seconds in

each cycle.

Since Equations (3.3) and (3.4) are for pretimed phases, they are

used in the Pretimed Model Method to compute the stop probabilities and

average delays for phase 2 of the above pretimed signal. The stop

probabilities and average delays of phase 2 when phase 3's volumes are

50 vph and 300 vph are shown in Figure 3.7. In this figure, the results

from the two methods show some differences. The deviations are highly

significant at very low volumes (such as less than 150 vph), but they

are small when volumes of both phases 2 and 3 approach 300 vph.

One observation stands out clearly in this figure, which is that

the Pretimed Model Method is unable to differentiate isolated and

coordinated controls. In the Analytical Method, isolated and

coordinated controls yield different stop probabilities and average

delays. But the two different controls yield the same results in the

Pretimed Model Method.

The different results can be explained by investigating the

intermediate results from the Analytical Method. The intermediate

results from this method are shown in Table 3.3 for phase 2 under

coordinated control, with phase 2 having a volume of 150 vph, and phase

3 of 300 vph. In Table 3.3 (a), there are four patterns, which fall

into two types of cycles. The first two patterns have a red time of 40







Phase 3's volume = 300 vph


e garevA de l ay ( sec/veh )


27.

25-

23-

21.

19.

17.

15-

13.

11.

91


41


50 100 150 200 250 300
Phase 2's volume

Phase 3' volume = 50 vph


Stop probability


delay (sec/veh)



0


50
in vph


100 150 200 250 300


Phase 3's volume = 300 vph


Stop probability


0.9

o 0.8.
o 0
00 0.7

0.6

0.5-


50 1CO 150 200 250 300
Phase 2's volume


results
results
results


from
from
from


Figure 3.7.


50
in vph


100 150 200 250 300


Sthe Analytical Method, under coordinated control
Sthe Analytical Method, under isolated control
the Pretimed Model Method

Stop Probability and Average Delay Comparisons,
the Analytical Method vs. the Pretimed Model Method.


0 0 0 u



1 1 1 1 1


1.0-

0.9-

0.8

0.7.

0.

0.5-


SI I I I I


j j A


I I I I I I I I I


I _


Phase 3' volume = 50 vph


Average

27-

25-

23

21

19-

17

15-

13

11-

9


I _I I_ I I 1





64


Table 3.3. A Further Comparison Between the Analytical Method
and the Pretimed Model Method.




(a) Results from the Analytical Method, under coordinated control
(Phase 2's volume=150 vph, phase 3's volume=300 vph)


Type of pattern
(sec.)

r=g3+Gmin=40

g=g2=20

r=Gsla3+Gmin=40

9=92=20

r=g +Gsla2+Gmin

g=g2=20

r=Gsla+Gmin=60

9g=g92=20


Pattern
prob.


0.653



0.020



0.271



0.056


Pattern
stop prob.


0.779



0.779



0.833



0.833


Pattern
avg. delay
(sec/veh)


16.57



16.57



24.11



24.11


Phase stop probability 0.80

Phase average delay (sec/veh) 19.54





(b) Results from the Pretimed Model Method
(phase 2's volume=150 vph)


Phase
Green time
(sec)


20


Phase
stop probability



0.727


Phase
average delay
(sec/veh)


14.55


Phase









2


Pat.



1



2



3



4


Phase
Red time
(sec)


40









seconds and a green time of 20 seconds, while the last two patterns have

a red time of 60 seconds and a green time of 20 seconds. As explained

previously (with the aid of Figure 3.5), the second cycle type results

from phase skipping. The pattern probabilities in Table 3.3 (a) show

that 67.3% of cycles are of the first type, while 32.7% of cycles are of

the second type. In comparison, the Pretimed Model Method treats all

the cycles as the first cycle type, since it has no way to figure out

what percentage will be of the second cycle type. In other words, the

Pretimed Model Method is unable to handle phase skipping.

Furthermore, the stop probability and average delay in the first

cycle type are different in the Analytical Method from those computed in

the Pretimed Model Method. This is because the Analytical Method uses

equations which were developed by considering phase skipping and

dwelling. The equations used in the Pretimed Model Method, however, do

not have the capability to handle phase skipping and dwelling.

From the above discussions, it can be seen that the deviation of

the two methods' results is due to the inability of the Pretimed Model

Method to deal with phase skipping and dwelling. Thus, models for

pretimed phases are not applicable for actuated phases under low volume

conditions.

Summary and Discussions

This chapter presents the Analytical Method to estimate stop

probabilities and average delays for semi-actuated signals at low

volumes, under both isolated and coordinated controls. The method

investigates various cycle patterns, and computes their probabilities

using Markov chain recurrence and the Poisson distribution. The method

is validated against simulation, confirming that the mathematics in the









method is correct. Stop probability and average delay from this method

are compared to those from a model for pretimed signals. The

comparisons show that when traffic volumes approach 300 vph, they are

close. When traffic volumes are much lower than 300 vph, the difference

is significant.

The method is easy to apply when there are only two phases. When

the number of phases increases, the complexity increases significantly.

For instance, a four-phase signal (one non-actuated phase and three

actuated phases) under isolated control has 23 1 = 7 states in one

cycle. Hence, there are total of 2 72 = 98 sub-transitions to

consider.

The method is capable of handling single ring operations without

overlap, but it would be too involved to deal with either dual ring or

overlap situations. The difference between single and dual rings is

illustrated in Figure 3.8. Figure 3.8 (a) is a single ring operation

without overlap, which is characterized as having each movement appear

in only one phase. Figure 3.8 (b) is a dual ring operation with

overlap, in which phase 1 may change to any of phases 2, 3 or 4,

depending on the traffic demand of those three phases.

Because of these disadvantages, a second method is considered using

a macroscopic approach. The second method, which is presented in the

next chapter, does not analyze the various situations of individual

cycles. Instead, it uses average red and green times.

Equations Development

This section develops the stop probability and average delay

equations used in this chapter. The non-actuated phase is treated first.
















1 2 3








(a) Single ring operation without overlap


(b) Dual ring operation with overlap


Figure 3.8.


Actuated Signal Operation Examples of Single Ring
without Overlap and Dual Ring with Overlap.








The Non-Actuated Phase

Since the non-actuated phase appears in every cycle, regardless of

whether it has traffic demand or not, it generally acts like a pretimed

phase, except that its red and green lengths may vary from cycle to

cycle. For given red and green times, the non-actuated phase can be

regarded as a pretimed phase. Hence, the stop probability and average

delay equations for pretimed phases can be applied to the non-actuated

phase.

Although a variety of stop probability and average delay equations

have been proposed for pretimed signals, the ones used in the HCM [19],

SOAP [52], or TRANSYT [45] are most popular. Since Hagen and Courage

[17] have demonstrated that their models agree closely at volume levels

below the saturation point, the stop probability equation used in SOAP

and the average delay equation used in TRANSYT are adopted in this

study. For convenience, the two equations are shown in the following:

1 g/C
Ps = (3.29)
1 xg/C

d = dl + d2 (3.30)

C(1 g/C)2 Bn x2 Bn
= + [(__)2 + ]1/2
2(1 xg/C) Bd Bd Bd

where C = r + g

x2
Bn = 2(1 x) +
1800v
4x x
Bd = ( )2
1800v 1800v

Equation (3.29) can be derived to another form for easy

application.









1 g/C
Ps =
1 xg/C

C(1 g/C)

C(1 xg/C)

C-g

C(1 v/s)

rs

C(s v)

r v
(1 + )
C s -v

1 rv
(r + )
r + g s- v

r + gs (3.31)
r + g
r+g

rv
where gs = (3.32)
s v

Equation (3.31) means that the stop probability is the sum of the red

and saturation green times divided by the cycle length. This form is

consistent with the intuition, since a vehicle will be stopped if it

arrives either in the red time or in the saturation green time joining

the end of a discharging queue. Hence, the stop probability is the

proportion of the sum of the red and saturation green times to the cycle

length.

Equation (3.30) consists of two terms, dl and d2. The first term

represents the average delay per vehicle under the assumption of uniform

arrivals with an average arrival rate of v throughout the cycle. The

second term is often called random delay or overflow delay since it









attempts to account for the fact that the vehicles arrive randomly [24].

The second term can be verified to be very small relative to the first

term when traffic volume is less than 300 vph. Hence, the second term

can be neglected under low volume conditions.

The first term of Equation (3.30) can also be derived to another

form for easy application.

C(1 -g/C)2
dl =
2(1 xg/C)

C(1 g/C) C(1 g/C)

1 xg/C 2

r + gs C g

r+g 2

r + gs r
(3.33)
r+g 2

Ps (3.34)
2

Moreover, since Equation (3.34) is approach delay, it is divided by 1.3

to reflect only the stopped delay portion of the total approach delay

[19].

r
dl = Ps (3.35)
2.6

Equation (3.35) means that the average delay is the red time multiplied

by the stop probability, and then divided by 2.6.

Equations (3.31), (3.32), and (3.35) are used in the Analytical

Method as Equations (3.3), (3.9), and (3.4), respectively.

Recall that uniform arrivals with an average rate of v throughout

the cycle are assumed in Equations (3.31), (3.32), and (3.35). Thus,









the number of vehicle arrivals is proportional to the length of red

time. But this property is not always true for actuated phases.

Actuated Phases

Unlike the non-actuated phase appearing in every cycle, an actuated

phase receives green time only after it is activated by a vehicle

arrival. After the actuated phase is activated, it may receive green

time immediately, depending on whether dwell is occurring or not. The

actuated phase will receive green time immediately if the green light is

dwelling on the non-actuated phase. Otherwise, the actuated phase will

have to wait to receive green time. In the former case, there is only

one arrival, which triggers the end of the red time, regardless of how

long the red time has been. In the latter case, the actuation(s) may

occur at any point of the red time. In this case, similarly to pretimed

or non-actuated phases, a longer red time can be assumed to include more

arrivals. Because of the difference between the two cases, red time is

not applicable in calculating arrivals. Effective red time (which

subtracts the red time without arrivals from the actual red time) is

used to substitute for the actual red time for actuated phases. It is

clear that effective red time is zero when the actuated phase terminates

the dwell; otherwise, the effective red time is the same as the actual

red time. The equation development is also split into the two

situations.

Positive Effective Red Times Consider a model which has a cyclic time

period with two components r and g. R represent a red time, but g can

be red or green depending on whether the actuated phase is activated in

r or not. If a vehicle arrives in r, g will be green; otherwise, if no

vehicle arriving in r, g remains red.








First of all, let's consider stop probability. A vehicle can

arrive either in r or g. For a given vehicle, the probability of

arriving in r is r/(r + g), and the probability of arriving in g is

g/(r + g). Hence, the stop probability of the vehicle is
r g
Ps =- (Ps in r) + -(Ps in g) (3.36)
r+g r+g

Where (Ps in r) and (Ps in g) denote the stop probability of the vehicle

arriving in r and in g, respectively. When the vehicle arrives in r, it

will be stopped, i.e.,

(Ps in r) = 1 (3.37)

When the vehicle arrives in g, there are two possibilities. If there is

no other vehicle arriving in r, g will be red, and the vehicle will be

stopped. The probability of this possibility is e-vr. In the other

possibility, if any vehicles have arrived in r, g will be green. The

probability of this possibility is (1 e-vr). When this occurs, there

are two sub-situations: either (1) the vehicle arrives in the saturation

green time (gs), or (2) the vehicle arrives in the remaining green time

(g gs). The probability of sub-situation (1) is gs/g and the vehicle

will be stopped (since the vehicle will join the end of the dissipating

queue), i.e., Ps = 1. The probability of sub-situation (2) is

(g gs)/g,
but the vehicle will not be stopped (since the queue has already

dissipated), i.e., Ps = 0. In conclusion, the stop probability of the

vehicle arriving in g is









gs g gs
(Ps in g) = e-vr 1 + (1 e-vr) [ 1 + 0 ]
g g

g

Substituting (3.37) and (3.38) into (3.36), the stop probability is

r g gs
Ps = 1 + [ e-vr + (1 e-vr) ]
r+g r + g g

r + g g g -vr
+ e
r + g r + g

r + gs r + gs
= + (1 ) e-vr
r + g r+g


= e-vr + (1 e-vr) r + gs (3.39)
r + g
r+g

Equation (3.39) can be further examined by investigating the

extremes. When v approaches zero, Ps approaches unity because e-vr

approaches unity. This result is consistent with intuition. Remember

that any vehicles arriving in the unsaturated green time will not be

stopped. When traffic demand is very low, vehicle arrivals are

generally sparsely distributed, so that almost every vehicle will arrive

in a red time, and very few will follow immediately to arrive in the

green time. Hence, the stop probability is close to unity. On the

other hand, when v is high, Ps approaches (r + gs)/(r + g) because e-vr

approaches zero. In this case, the actuated phase will be activated in

almost every cycle, like the pattern of a pretimed phase. Hence, the

stop probability will approach that of a pretimed phase.

Average delay can be estimated by the similar fashion. Since the

average delay per vehicle can be regarded as the total delay experienced








by any vehicle, the equation will be developed by considering one

vehicle.

A vehicle can arrive either in r or g. For a given vehicle, the

probability of arriving in r is r/(r + g), and the probability of

arriving in g is g/(r + g). Hence, the total delay of the vehicle is

r g
d = (d in r) + (d in g) (3.40)
r+g r+g

Where (d in r) and (d in g) denote the expected waiting time of the

vehicle arriving in r and in g, respectively. When the vehicle arrives

in r, its expected waiting time is half the red time, i.e.,

(d in r) = r/2 (3.41)

When the vehicle arrives in g, there are two possibilities. If there is

no other vehicle arriving in r, g will be red, and the vehicle will have

to wait till the next cycle. Its expected waiting time is half of g

plus r, i.e., r + g/2. The probability of this possibility is e-vr. In

the other possibility if any vehicles have arrived in r, g will be

green. When this occurs, there are two sub-situations: either (1) the

vehicle arrives in the saturation green time (gs), or (2) the vehicle

arrives in the remaining green time (g gs). The probability of sub-

situation (1) is gs/g and the expected waiting time of this sub-

situation is gs/2. The probability of sub-situation (2) is

(g gs)/g,
but the vehicle will not be delayed (since the queue has already

dissipated), i.e., d = 0. Combining these two sub-situations, the

expected waiting time of the vehicle arriving in g is

g gs gs
(d in g) = e-vr (- + r) + (1 e-vr) (3.42)
2 g 2








Substituting (3.41) and (3.42) into (3.40), the average delay is

r r g g gs
d = + [ e-vr (- + r) + (1 e-vr) ]
r+g 2 r + g 2 2g

r2 g(g + 2r) gs2
S+ e-vr g + + (1 e-vr) (3.43)
2(r + g) 2(r + g) 2(r + g)

Please note that approximations are used in the equation

development. A good example of approximations is the treatment of the

saturation green time (gs). The saturation green time is the first

portion of the green time, during which the vehicular queue is

discharging. Conversely, the remaining green time is called the

unsaturation green time (gu), during which the queue has been fully

discharged, and the arriving vehicle passes the intersection without

being stopped or delayed.

The approximation can be better explained by the aid of a typical

queue length diagram as illustrated in Figure 3.9. In this figure, gs

is further split into two parts: gs1 and gs2. The gs1 is the time to

discharge the queue accumulated in r, while gs2 is the time to discharge

the additional vehicles arriving in the green time and joining the end

of the moving queue. In the extreme case if the vehicle arrives at the

beginning of r, it will be released at the beginning of g. Conversely,

if the vehicle arrives at the end of r, it will be released at the end

of gs1. Therefore, the expected waiting time for a vehicle arriving in

r is more than r/2, but for simplicity, only r/2 is used in Equation

(3.40).

When the vehicle arrives in gs, it will be released in gs2. In the

extreme case if the vehicle arrives at the beginning of gs, it will be

released at the beginning of gs2. If the vehicle arrives at the end of



















Vehicles








\--- I Time
I- 9s1 -- gs2 -I

I~I- g- -I
I r I


Figure 3.9. Queue Length Example at a Signalized Intersection.










gs, it will have no delay. Hence, the expected waiting time for a
vehicle arriving in gs is less than gs/2, but for simplicity, gs/2 is

used in Equation (3.40).

The error due to the above simplified treatment for gs is expected

to be small for two reasons. The first is that the two errors will be

counterbalanced. The second is that the saturation green time is

usually very short under low volume conditions.

Equations (3.39) and (3.43) are used in the method as Equations

(3.5) and (3.6), respectively.

Zero Effective Red Time Zero effective red time refers to the situation

where the actuated phase terminates the dwell. When this occurs, the

vehicle which activates the actuated phase will come to a stop and

experience a delay of yellow time of the non-actuated phase plus some

lost time. Since lost and yellow times are not considered in this

method, the vehicle is treated as stopped without any delay. Moreover,

since any further vehicles arriving in the green time also have no

delay, the average delay is

d = 0 (3.44)

Based on the same reasoning, the vehicle which terminates the dwell

is the only one stopped in this phase. Since the stop rate depends on

the total number of vehicles in the phase, and only the first vehicle is

stopped, the probabilities of all possible numbers of vehicles arriving

in the green time must be considered in order to arrive at the stop

probability.

Assuming Poisson arrivals, the probability of i vehicles arriving

in the green time (g) is









(vg)ie-vg
(3.45)
i!

where i=O, 1, 2, ...


For a given i, the total vehicle arrivals (N) is the i vehicles plus the

one which triggers the phase, i.e.,


N = i + 1 (3.46)

The expected N across i is

o (vg)ie-vg
N = Z (i + 1)
i=O i!

= vg + 1 (3.47)

Since there is only one vehicle stopped, regardless of how many

vehicles arrive in the green time, the expected number of vehicles

stopped (Nstop) across i is

o (vg)ie-vg
Nstop = *1
i=0 i!

= 1 (3.48)

Thus, the stop probability is


Ps = Nstop
N

1
(3.49)
vg + 1

Equations (3.44) and (3.49) are used in the method as Equations

(3.8) and (3.7), respectively.














CHAPTER FOUR
APPROXIMATION METHOD FOR PERFORMANCE EVALUATION
OF SEMI-ACTUATED SIGNALS AT LOW VOLUMES


Introduction

The Analytical Method presented in the previous chapter provides a

microscopic approach to estimate stop probabilities and average delays

for semi-actuated signals when volumes are low. The method investigates

various cycle patterns in terms of red and green times, and computes

their probabilities. Also, stop probability and average delay of each

cycle pattern are estimated. As discussed previously, however, applying

the method would be complicated for signals with more than three phases

or with special control capabilities, such as permissive periods.

Because of this disadvantage, a second method is considered using a

macroscopic approach.

Unlike the Analytical Method, which uses the individual red and

green times, the method in this chapter uses average red and green times

in the stop probability and in the average delay equations developed in

the Analytical Method. Since this method does not analyze the various

situations of individual cycles, it should be easier to apply where

conditions are unfavorable to the Analytical Method. This second method

will be referred to as the Approximation Method, in contrast to the

first Analytical Method.

Because the Approximation Method uses average times, there is no

need to consider individual cycles and their probabilities, which is

generally complicated when conducted under the Analytical Method.
79










However, since the average times must be supplied to the method, they

have to be either collected in the field or generated by a simulation

model. In other words, the ease with which the Approximation Method is

applied is at the expense of collecting additional data.

This chapter first addresses the concept of the Approximation

Method, and develops the method. Then the method is validated against

both the Analytical Method and the microscopic simulation model

developed previously. Finally, the summary and discussions are

presented.

Concept of the Method

As reviewed in Chapter Two, HCM, SOAP, and TRANSYT use average

cycle lengths in models for pretimed signals to estimate average delays

for actuated signals. This method produces reasonable approximations

where traffic volumes are moderate to high. When traffic volumes are

low, however, caution must be taken in using this method. As

demonstrated earlier, this method could produce good approximations for

semi-actuated signals where traffic volumes approach 300 vph. When

traffic volumes are lower than 300 vph, however, the estimates are no

longer reliable. The reason has been identified as phase skipping and

dwelling. When traffic volumes approach 300 vph, phase skipping and

dwelling rarely occur. The semi-actuated signals will functionally act

like pretimed signals except that the phase lengths may vary from cycle

to cycle. Hence, using the average times in models for pretimed phases

should produce reasonable approximations. However, when traffic volumes

are much lower than 300 vph, phase skipping and dwelling occur

frequently. Models for pretimed phases are no longer valid for actuated

phases, since they don't take phase skipping and dwelling into account.










The above findings suggest two points: (1) stop probability and

average delay equations for pretimed phases cannot be directly applied

to actuated phases when traffic volumes are low, and (2) using average

times to substitute individual times could work under low volumes if

equations consider phase skipping and dwelling. Since the equations

developed in the Analytical Method have considered phase skipping and

dwelling, it would be a reasonable approach to use average red and green

times in these equations. The development of such a method is the

subject of this chapter.

Method Development

Since the approach of this method is to apply the average values in

the equations developed in the Analytical Method, the intuitive way to

do this is to determine the average values, then to substitute them into

the equations. However, actuated phases cause problems, both in

applying the equations and in determining the averages, since there are

two sets of equations, one for the case of a zero effective red time,

and the other for an effective red time greater than zero. In the

following, the problems will be explained and the treatments will be

discussed. The non-actuated phase will be treated first.

The Non-Actuated Phase

Since the non-actuated phase will not be skipped regardless of

whether it has traffic demand or not, it generally acts like a pretimed

phase, except that its red and green lengths may vary from cycle to

cycle. Because there is no phase skipping, the average red and green

times can be easily determined by simply taking the averages of all the

red and green times of individual cycles. Then they are input into

Equations (3.3) and (3.4) to calculate stop probability and average










delay. For convenience, Equations (3.3) and (3.4) are rewritten as

functions of R and G, where R and G are average red and green times,

respectively.

R + Gs
Ps = (4.1)
R + G

R
d =- Ps (4.2)
2.6

vR
where Gs = (4.3)
s v

Actuated Phases

Actuated phases may be skipped due to lack of traffic demand. When

a phase is skipped, there is no vehicle arrival during the corresponding

red time. Because of this, the Analytical Method uses effective red

time (which subtracts the red time without arrivals from the actual red

time) in Equations (3.5) and (3.6) to calculate stop probabilities and

average delays for the actuated phases. To comply with this in the

Approximation Method, the red time corresponding to a skipped phase is

also excluded from the total red time in determining the average red

time.

When dwell occurs, the actuated phase following the non-actuated

phase terminates the dwell. Similarly to phase skipping, the actuated

phase has no arrival until it is activated by a vehicle (which also

terminates the dwell). Hence, the actuated phase has a zero effective

red time in the cycle. In conclusion, when an actuated phase appears,

its effective red time is normally positive, but is zero when the phase

terminates the dwell.









Recall that in computing stop probabilities and average delays,

Equations (3.5) and (3.6) are for a positive effective red time and

Equations (3.7) and (3.8) are for a zero effective red time. Again, in

accordance with this difference, in the Approximation Method, the green

occurrences of an actuated phase are also split into two parts according

to their effective red times.

The required logic is summarized in the flow chart shown in Figure

4.1 to determine the average parameters. The chart determines the total

effective red time (TER), the number of green occurrences (Ng), and the

green occurrences with a positive effective red time (Nr). In the

figure, an accumulator called red increment is used to accumulate the

red time. When the phase is skipped, the red increment is reset to zero

and the procedure goes to the next cycle. When the phase appears, Ng is

increased by one. When dwell occurs, and is terminated by the phase,

the red increment is reset to zero again. Otherwise, the red increment

is added to TER, and Nr is increased by one.

After TER, Ng, and Nr are obtained, the other parameters can be

computed in the following way:

R = TER/Nr (4.4)

Nd = Ng Nr (4.5)

Ns = Nc Ng (4.6)

where Nd = green occurrences with a zero effective red time,

Ns = times of the phase being skipped, and

Nc = the total number of cycles.

It should be noted that there is no need to determine the average

green time (G) since its value will be the same as the initial green




















































where NG = number of green occurrences
TER = total effective red time


Figure 4.1.


Determining the Total Effective Red Time
and the Number of Green Occurrences.








time (g). This is because that the actuated phases are assumed to

terminate the green times after the initial green time completes.

After the parameters are obtained, the next task is to input them

into the equations. The task is easier where dwell does not happen. In

this situation, Equations (3.5) and (3.6) are first changed to the

following forms:

Ps = e-vR + (1 e vR s (4.7)
R + G

R2 G(G + 2R) G 2
d = R- + e-vR G + + (1 e-vR) (4.8)
2(R + G) 2(R + G) 2(R + G)


Since e-vR is the probability of no vehicle arriving in the time

interval of R, it can be regarded as the probability of the phase being

skipped. Since the phase is skipped Ns times out of Nc cycles, the

percentage of phase skipping is Ns/Nc. Hence, e-vR can be replaced by

Ns/Nc. Similarly, (1 e-vR) can be regarded as the probability of the

phase not being skipped, and can thus be replaced by Ng/Nc. As a

result, Equations (4.7) and (4.8) can be changed to

Ns Ng R + Gs
Ps = + (4.9)
Nc Nc R + G

R2 Ns G(G + 2R) Ng Gs2
d = + ---- + ---- (4.10)
2(R + G) Nc 2(R + G) Nc 2(R + G)

The estimation procedure ends at this point if dwell does not

happen, i.e., Nd = 0. If on the other hand Nd is greater than zero,

more computations are required. Since green occurrences (Ng) have been

split into Nr (positive effective red time) and Nd (zero effective red

time), the computations are also split accordingly.









For positive effective red time, the total vehicle arrivals (N) in

a cycle can be estimated by multiplying the average arrival rate and the

sum of the red and green times, i.e., v(R + G). The number of vehicles

stopped (Nstop) in one cycle can also be estimated by multiplying the

stop probability and the total arrivals, i.e.,

Nstop = v(R + G)Ps
Ns Ng R + Gs
= v(R + G)[-- + -- ] (4.11)
Nc Nc R + G

Because positive effective red time occurs Nr times out of Ng green

occurrences, the total Nstop and the total N in the Nr cycles are

Ns Ng R + Gs
Stop = Nr v(R + G)[-- + -- (4.12)
Nc Nc R + G

N = Nr v(R + G) (4.13)

On the other hand, for zero effective red time Nstop = 1, since the

only vehicle stopped in one cycle is the one which activates the

actuated phase. In addition, the total arrivals in this case can be

estimated by multiplying the average arrival rate and the average green

time, i.e.,

N = vG (4.14)

Because zero effective red time occurs Nd times out of Ng green

occurrences, the total Nstop and the total N in the Nd cycles are

Nstop = Nd (4.15)
N = Nd vG (4.16)

Combining (4.12) and (4.15), the total Nstop in the Ng green

occurrences is

Ns Ng R + Gs
Nstop = Nr v(R + G)[-- + -- ] + Nd (4.17)
Nc Nc R + G








Combining (4.13) and (4.16), the total N in the Ng green occurrences is

N = Nr v(R + G) + Nd vG (4.18)

The stop probability therefore is


Ps = stop
N

Ns Ng R + Gs
Nr v(R + G)[-- + ] + Nd
Nc Nc R + G
= (4.19)
Nr v(R + G) + Nd vG

The computations of average delay follow the same logic. For

positive effective red time, the total delay in one cycle can be

estimated by multiplying the average delay per vehicle and the total

arrivals, i.e.,

D = v(R + G)d

R2 Ns G(G + 2R) Ng Gs2
= v(R + G)[- + -- + --] (4.20)
2(R + G) Nc 2(R + G) Nc 2(R + G)


Because positive effective red time occurs Nr times, the total delay in

the Nr cycles is

R2 Ns G(G + 2R) Ng Gs2
D = Nr v(R + G)[ + + ] (4.21)
2(R + G) Nc 2(R + G) Nc 2(R + G)


For zero effective red time, since the average delay is zero, there

is no extra delay created. Therefore, Equation (4.21) is the total

delay in the Ng green occurrences. To divide the total delay by the

total arrivals, the average delay is









R2 Ns G(G + 2R) Ng Gs2
Nr v(R + G)[ + + -- ]
2(R + G) Nc 2(R + G) Nc 2(R + G)
d = (4.22)
Nr v(R + G) + Nd vG


For clarity, the above equations and the conditions for their use

are summarized in Figure 4.2. Equations (4.1) and (4.2) are for the

non-actuated phase. Equations (4.9) and (4.10) are for the actuated

phase where there is no green occurrence with zero effective red time.

Otherwise, Equations (4.19) and (4.22) should be used.

Tests of the Approximation Method

The Approximation Method was tested against both the Analytical

Method and the simulation model developed previously. The method was

developed into a computer program called DELVACS, which is an acronym

for Delay Estimation for Low Volume Arterial Control Systems. The

DELVACS program will be described in more detail in the next chapter.

Briefly speaking, the program allows the user the choice of entering the

volumes alone, or of entering both the volumes and the computational

parameters (such as the average red and green times). When

computational parameters are not entered, they will be generated by the

program, using simulation. Then the computational parameters are input

to the equations in Figure 4.2 to arrive at stop probabilities and

average delays.

A three-phase signal under both isolated and coordinated controls

was tested using the DELVACS program. The only inputs supplied were the

volumes, and the computational parameters were generated by the program.

As in the test of the Analytical Method, 216 sets of volumes were tested

for both controls-the three phases were given volumes from 50 vph to














Non-Actuated


R+G
S= 0 \ Ps R + Gs (4.1)
Nd -------,R + G

R
d = --- Ps (4.2)
2.6


Ns Ng R + Gs
Ps = -- + ---- (4.9)
Nc Nc R + G

R2 Ns G(G + 2R) Ng Gs2
d= -+ + (4.10)
2(R + G) Nc 2(R + G) Nc 2(R + G)


Ns
Nr v(R + G)[-
Nc


Ng

Nc
Nc


R + Gs
S+ Nd
R + G


Ps =


Nr v(R + G) + Nd vG


R2
Nr v(R + G)[- +
2(R + G)


Ns G(G + 2R)

Nc 2(R + G)


Ng
Nc
Nc


Gs2
2(R + G)
2(R + G)


(4.19)


(4.22)


Nr v(R + G) + Nd vG


Cigure 4.2. Stop Probability and Average Delay Equations
and the Conditions for Their Use under the
Approximation Method.


> 0
------


Actuated









300 vph with 50 vph increments. Each set of volumes was tested for

6,000 seconds.

Comparison with Simulation

Part of the results are shown in Figure 4.3, which presents four

comparisons: stop probabilities and average delays of phase 2 under both

isolated and coordinated controls when phase 3's volume is 50 vph.

Since other sets of results show the same pattern, they are omitted.

The stop probabilities and average delays from the simulation model

shown in Figure 3.6 are duplicated in this figure. An inspection

suggests that the results from the Approximation Method and simulation

follow the same pattern. For average delay, about half of the data

points from the Approximation Method fall within one standard deviation

of the mean from the simulation model. For stop probability, the data

points from the two methods adhere very closely when the signal is under

coordinated control. When the signal is under isolated control, the

data points still follow the same pattern, but with much more

dispersion.

Comparing Figures 3.6 and 4.3, the Analytical Method agrees more

closely to the simulation model than the Approximation Method does.

This suggests that the Analytical Method is a better predictor than the

Approximation Method. However, considering the relative ease with which

the stop probabilities and average delays may be determined in the

Approximation Method, it is reasonable to conclude that it could be used

as a cost-effective substitute for the Analytical Method for the

purposes of stop probability and average delay estimation.




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