DEVELOPMENT OF A FORWARD LINK OPPORTUNITIES MODEL
FOR OPTIMIZATION OF TRAFFIC SIGNAL PROGRESSION
ON ARTERIAL HIGHWAYS
By
CHARLES EDWARD WALLACE
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1979
ACKNOWLEDGEMENTS
I am indebted to many who assisted in this work. The U.S. De
partment of Transportation, the Florida Department of Transportation and
the State University System of Florida all provided support which con
tributed directly or indirectly to this effort. My gratitude to these
sponsors of several research projects, as well as the University of
Florida which provided financial support, cannot be adequately expressed.
I am particularly grateful for the support and assistance given by
the members of my supervisory committee.
Dr. Joseph A. Wattleworth has been my teacher, senior colleague
and, I am proud to say, friend for over a dozen years. His support in
all these capacities, as well as his honoring me by chairing my supervi
sory committee, have given me opportunities that otherwise would have
been impossible. For this I am eternally grateful.
When we first arrived in Gainesville, Professor Kenneth G. Courage
and his family took us into their home, and what had been a close pro
fessional friendship six years ago, became a true friendship that my
family and I cherish. But Professor Courage was far more than a friend
and associate, he was critic, counselor, inspirationalist, temperist and
he sparked that which I humbly hope will be a small but meaningful
contribution to our profession.
Dr. Gary Long has been a source of inspiration and motivation. His
technical and, more significantly, personal support was invaluable to
the success of this research.
Dr. James H. Schaub, Chairman of the Department of Civil Engi
neering has helped me far more than being a member of the committee. He
has provided financial, moral and professional support that leave me
behumbled.
Dr. Dennis D. Wackerly served as the outside member and his assis
tance is sincerely appreciated.
Debbie Reaves, Faye Sullivan, Jean Wollenberg, Kathy McCurley,
Craig Kirkland and Lillian Pieter all assisted with the preparation of
the dissertation and I am extremely grateful to all of them.
Not lastfor many unnamed: parents, associates, family and
friends have helpedbut here the bottom line. My wife Pat and chil
dren, Ryan and Shannon, have given me so much love and understanding.
This is dedicated to them, for they gave the most.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS .............................................. ii
LIST OF TABLES ................................................ vi
LIST OF FIGURES ................................................ viii
ABSTRACT ........................ ............................... x
CHAPTER 1. INTRODUCTION .................................... 1
Need for the Research ...................... 1
Purpose, Objectives and Scope .............. 6
Organization ............................... 8
CHAPTER 2. REVIEW OF ARTERIAL PROGRESSIVE SIGNAL
CONTROL STRATEGIES .............................. 10
Introduction ............................... 10
Theory of Traffic Progression .............. 10
Past Research .............................. 14
Existing Models Pertinent to the Forward
Link Opportunities Model Development ....... 25
CHAPTER 3. DEVELOPMENT OF THE FORWARD LINK
OPPORTUNITIES MODEL ............................. 28
Introduction ............................... 28
Concept of the Forward Link
Opportunities Model ........................ 31
Model Development .......................... 44
Model Implementation ....................... 48
CHAPTER 4. COMPARISON OF THE FORWARD LINK OPPORTUNITIES
AND OPTIMAL BANDWIDTH OPTIMIZATION STRATEGIES ... 54
Introduction ............................. 54
Experimental Design ........................ 54
Analysis of Alternative Arterial
Configurations ............................. 58
Summary .................................... 76
Page
CHAPTER 5. EXTENDED APPLICATIONS OF THE FORWARD
LINK OPPORTUNITIES MODEL ........................ 79
Introduction ............................... 79
Weighting by Physical and Traffic Aspects .. 80
Alternative Objective Functions ............ 92
Extended Analyses .......................... 97
Summary .................................... 103
CHAPTER 6. CONCLUSIONS AND RECOMMENDATIONS ................. 108
Conclusions ................................ 108
Recommendations ............................ 112
APPENDIX A. DESCRIPTION OF THE PASSER II MODEL .............. 117
Overview ................................... 117
Purpose and Applications ................... 117
Background ................................. 118
Functional Description ..................... 119
Input Requirements ......................... 125
Program Outputs ............................ 125
APPENDIX B. DESCRIPTION OF THE TRANSYT6C MODEL .............. 128
Overview ................................... 128
Purpose and Applications .................. 128
Background ................................. 129
Functional Description ..................... 130
Input Requirements ......................... 137
Program Outputs ............................ 140
APPENDIX C. MODIFICATION TO TRANSYT6C TO IMPLEMENT
THE FLOS MODEL .................................. 143
General .................................... 143
Description of Additions/Modifications ..... 143
Comment on Program Structure ............... 161
BIBLIOGRAPHY ................................................ 163
BIOGRAPHICAL SKETCH ............................................ 167
LIST OF TABLES
Table Page
2.1 SUMMARY DESCRIPTIONS OF THE PASSER II AND
TRANSYT6C MODELS ....................................... 26
3.1 APPLICATION OF EQUATION (3.3) TO TWO EXAMPLES .......... 40
4.1 CHARACTERISTICS OF TEST ARTERIAL SYSTEMS ............... 56
4.2 COMPARATIVE RESULTS OF MAXIMAL BANDWIDTH
AND FLOS OPTIMIZATIONS FOR BUFFALO AVENUE
(SIXSIGNAL SYSTEM) .................................... 59
4.3 COMPARATIVE RESULTS OF MAXIMAL BANDWIDTH AND
FLOS OPTIMIZATIONS FOR S.R. 26
(EIGHTSIGNAL SYSTEM) ................................. 62
4.4 COMPARATIVE RESULTS OF MAXIMAL BANDWIDTH
AND FLOS OPTIMIZATIONS FOR S.R. 7
(TWELVESIGNAL SYSTEM) ................................. 67
4.5 COMPARATIVE RESULTS OF MAXIMAL BANDWIDTH
AND FLOS OPTIMIZATIONS FOR BEECH DALY ROAD
(SIXTEENSIGNAL SYSTEM) ................................ 69
4.6 COMPARATIVE RESULTS OF MAXIMAL BANDWIDTH
AND FLOS OPTIMIZATIONS FOR S.R. 7
(TWENTYSIGNAL SYSTEM) ................................. 71
4.7 COMPARATIVE RESULTS OF MAXIMAL BANDWIDTH
AND FLOS OPTIMIZATIONS FOR S.R. 7
(TWELVESIGNAL SYSTEM WITH ONE DISTANCE
REDUCED) .............................................. 77
5.1 COMPARISON OF MAXIMAL BANDWIDTH, UNWEIGHTED FLOS AND
LINK LENGTHWEIGHTED FLOS OPTIMIZATIONS ................ 82
5.2 COMPARISON OF MAXIMAL BANDWIDTH, UNWEIGHTED FLOS AND
PDFWEIGHTED FLOS OPTIMIZATIONS ........................ 82
5.3 COMPARISON OF MAXIMAL BANDWIDTH, UNWEIGHTED FLOS AND IN
FLOW PATTERNWEIGHTED FLOS OPTIMIZATIONS ............... 87
5.4 COMPARISON OF MAXIMAL BANDWIDTH, UNBIASED FLOS AND
LEFTBOUND BIASED FLOS OPTIMIZATIONS ................... 87
5.5 COMPARISON OF MAXIMAL BANDWIDTH, UNWEIGHTED FLOS AND
FLOS/PI OPTIMIZATIONS .................................. 90
5.6 COMPARISON OF PASSER II AND TRANSYT6C/FLOS
OPTIMIZATIONS OF BANDWIDTHS ............................ 95
5.7 SUMMARY OF RESULTS OF EXTENDED INVESTIGATIONS OF THE
FORWARD LINK OPPORTUNITIES OPTIMIZATION POLICY
(STATE ROAD 26) ........................................ 104
A.1 INPUT REQUIREMENTS FOR THE PASSER II MODEL ........... 126
B.1 INPUT REQUIREMENTS FOR THE TRANSYT6C MODEL ........... 138
C.1 TRANSYT6C/FLOS PROGRAM SPECIFICATIONS .................. 144
C.2 ADDITIONAL INPUT REQUIREMENTS FOR THE
TRANSYT6C/FLOS MODELCARD TYPE 7 ...................... 145
LIST OF FIGURES
Figure Page
2.1 EXAMPLE OF THE TIMESPACE RELATIONSHIP OF
UNCOORDINATED TRAFFIC SIGNALS .......................... 12
2.2 EXAMPLE OF THE TIMESPACE RELATIONSHIP OF
PERFECT PROGRESSION .................................... 13
3.1 TIMESPACE DIAGRAM OF A MAXIMAL BANDWIDTH
SOLUTION ILLUSTRATING UNUSED PARTIAL
PROGRESSION OPPORTUNITIES .............................. 29
3.2 TIMESPACE DIAGRAM OF PREVIOUS EXAMPLE
ADJUSTED TO MAXIMIZE FORWARD LINK
OPPORTUNITIES ......................................... 30
3.3 TIMELOCATION DIAGRAM OF MAXIMAL BANDWIDTH
SOLUTION ILLUSTRATING PARTIAL
PROGRESSION OPPORTUNITIES .............................. 34
3.4 TIMELOCATION DIAGRAM OF PREVIOUS EXAMPLE
ADJUSTED TO MAXIMIZE FORWARD LINK
OPPORTUNITIES ........................................ 35
3.5 FLOS DIAGRAM ILLUSTRATING THE OPTIMAL
OFFSETS FOR MAXIMAL BANDWIDTH ONLY ..................... 38
3.6 FLOS DIAGRAM ILLUSTRATING THE OPTIMAL OFFSETS
FOR FORWARD LINK OPPORTUNITIES ......................... 39
3.7 GRAPHICAL ILLUSTRATION OF HILL CLIMBING
TECHNIQUE ............................................ 50
3.8 GENERALIZED FLOW DIAGRAM OF THE
TRANSYT6C/FLOS MODEL ................................... 52
4.1 ARRIVAL AND DEPARTURE PATTERNS ON LINK 72
OF THE S.R. 26 SYSTEM UNDER MAXIMAL
BANDWIDTH AND FLOS OPTIMIZATIONS ....................... 64
4.2 FLOS DIAGRAMS FOR THE LEFTBOUND DIRECTION
ON S.R. 26 FOR THE MAXIMAL BANDWIDTH
AND FLOS OPTIMIZATIONS ................................. 66
viii
Figure Page
4.3 FLOS DIAGRAM FOR THE TWENTYSIGNAL
SYSTEMMAXIMAL BANDWIDTH SOLUTION ..................... 72
4.4 FLOS DIAGRAM FOR THE TWENTYSIGNAL
SYSTEMMAXIMAL FLOS SOLUTION .......................... 73
4.5 TENDENCY BETWEEN IMPROVEMENT IN FLOS
AND COEFFICIENT OF VARIATION OF SIGNAL
SPACING .............................................. 75
5.1 FLOS DIAGRAM FOR UNBIASED FLOS OPTIMIZATION
ON STATE ROAD 26 ....................................... 90
5.2 FLOS DIAGRAM FOR FLOS OPTIMIZATION FAVORING
THE LEFTBOUND DIRECTION ON STATE ROAD 26 .............. 91
5.3 COMPARISON OF UNWEIGHTED FLOS,
FLOS/PI AND PI ................ ........................ 98
5.4 COMPARISON OF INFLOW PATTERNWEIGHTED
FLOS ............................................... . 98
5.5 COMPARISON OF UNWEIGHTED FLOS AND PI OPTIMIZATIONS
WITH MAXIMAL BANDWIDTH OPTIMIZATION, SPLITS
VARYING ............................................. 102
5.6 COMPARISON OF INFLOW PATTERNWEIGHTED FLOS
AND PI OPTIMIZATIONS WITH MAXIMAL BANDWIDTH
OPTIMIZATION, SPLITS VARYING ........................... 102
6.1 CONCEPTUALIZATION OF A COMPLETE OPTIMIZATION
MODEL FOR COORDINATED ARTERIAL TRAFFIC
SIGNAL TIMING ................... ...................... 116
B.1 EXAMPLE OF STOPLINE FLOW PATTERN PRODUCED
BY TRANSYT6C ........................................... 141
C.1 TYPICAL ILLUSTRATION OF LINKNODE CODING
SCHEME FOR TRANSYT6C/FLOS .............................. 150
C.2 EXAMPLE OF THE FLOS MOE OUTPUT TABLE ................... 156
C.3 TYPICAL TRAVEL TIMENORMALIZED PLOT OF
FORWARD LINK OPPORTUNITIES ............................. 157
C.4 TYPICAL TIMESPACE DIAGRAM ............................. 158
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of
the Requirements for the Degree of Doctor of Philosophy
DEVELOPMENT OF A FORWARD LINK OPPORTUNITIES MODEL
FOR OPTIMIZATION OF TRAFFIC SIGNAL PROGRESSION
ON ARTERIAL HIGHWAYS
By
Charles Edward Wallace
December, 1979
Chairman: Joseph A. Wattleworth
Major Department: Civil Engineering
Improved control of motor vehicle traffic on urban streets and
highways has become increasingly important in recent years. Declines in
freeway construction have placed an ever increasing burden on signalized
arterial highways and, thus, the traffic signal control systems thereon.
The design of optimal control strategies to provide for progressive
movement of traffic on these major highways has been a subject of inten
sive research by the traffic engineering profession. The most success
ful methodology currently used is the maximization of progression for
through traffic, or the maximal bandwidth theory.
This dissertation proposes a new concept for the design of traffic
progression which expands upon the maximal bandwidth approach by consid
ering the progression opportunities which present themselves within an
arterial route, but do not necessarily extend throughout the full length
of the route. The concept has been named the forward link opportunities
method. When system optimization is based on the new measure as a
maximization objective function, improvements in both progression oppor
tunities and system traffic operations can be realized.
This dissertation is concerned with the introduction of the forward
link opportunities concept and the development of an optimization model
with which to implement the new strategy.
An existing optimization model is modified to incorporate the
proposed concept, and comparisons are made between the forward link
opportunities optimization strategy and the maximal bandwidth optimi
zation strategy. Investigations indicated that reasonable improvements
can be realized in a variety of typical arterial system configurations.
Enhancements and expanded uses of the concept and the model indi
cate that the theory and model serve as viable design and evaluation
enhancements to the stateoftheart.
CHAPTER 1
INTRODUCTION
Need for the Research
Since the first electric traffic control signal for street traffic
was introduced in 1912 (Sessions, 1971), the complexity of control
hardware and the sophistication of control strategies has increased to
maintain pace with the capabilities of more advanced automotive vehi
cles, higher traffic densities and the improved knowledge of driver
behavior.
Technological advancements in traffic controller hardware have
maintained a reasonable parity with advancements in mechanical and, in
recent years, electronic technology. As signal hardware has grown more
sophisticated, the need for effective design of signal timing to insure
the best operation has increased. The costeffectiveness of digital
computers has led to a number of analytical models whose purpose is the
design of the optimum signal timing.
The need for such design tools varies according to the geographic
and traffic environment. Design for low density areas with widely
spaced signalized intersections generally only needs to be concerned
with signal performance at each individual intersection. For central
business districts, grid systems are often coordinated to provide or
derly flow through the network of signals. In the latter situation,
there is little that can be done to accommodate all travel patterns, and
networks are generally constrained by the physical spacing of signals
and the relatively low travel speeds that are possible.
One of the most challenging aspects in the field of traffic en
gineering is concerned with the control of arterial highways. As the
urban growth pattern has extended from the inner city to the suburbs,
the impact on arterial highways connecting the residential and employ
ment areas has become severe. Excluding hightype highways (freeways
and expressways) arterial highways carry the majority of urban traffic
in terms of vehiclemiles of travel, thus they are the most important
nongradeseparated facilities in any major travel corridor. With in
creasing public resistance to expanding freeway systems, arterials hold
an ever increasing significance in the highway system.
For decades, traffic engineers have sought to provide the best
quality of traffic flow on arterial highways. Despite the fact that
elimination of all control on such important routes is one way to move
through traffic, this is impractical in most instances because cross
street and other conflicting demands must be satisfied as well. Thus,
in most urban and suburban areas, arterial highways must be controlled
by traffic signals to ensure access to the arterial for cross street
traffic and safety to all traffic.
As early as the 1940's, traffic engineers recognized that one of
the most effective means of providing a high quality of travel to through
traffic was to coordinate the timing of traffic signals to provide a
window of green time through a series of signals, within which the
through traffic can travel without interruption by the signals. This
traffic control technique is commonly known as progression. A band of
green time is propagated through the system such that vehicles traveling
within its limits progress through the system without being stopped.
Early signals were unsophisticated and traffic demand was often
heavily oriented toward one direction during periods of congestion, so
it was a fairly trivial task to determine the proper signal offsets (the
time difference between the start of the green signal phase between
adjacent signals) by manually plotting timespace diagrams. As noted
initially, however, the expanding urban demand and sophistication of
hardware has rendered simple design techniques obsolete.
The advent of the digital computer enabled engineers to develop
more sophisticated strategies for designing coordinated traffic control
systems. A variety of models has been offered for both offline (i.e.,
prior to implementation) design and realtime control. Realtime con
trol (i.e., online, usually traffic responsive) is generally adaptive to
traffic conditions and is therefore not as critical from a design stand
point. Realtime control is also extremely expensive, making it imprac
tical as a widespread control tool for arterials that are dispersed
throughout an urban area.
Thus, the more challenging area of interest is the offline design
of systems which operate on a recurring, cyclical basis during particu
lar periods of the day. Virtually all strategies for progression con
trol are based on the objective of maximizing through progression. As
discussed in greater detail in Chapter 2, the most popular approach to
the design of progressive control has centered around the maximal band
width concept. This strategy simply determines the signal timing which
will provide the maximum width of the throughgreen bands (generally bi
directional), subject to providing sufficient time to nonthrough move
ments to avoid oversaturation on their approaches.
The U.S. Department of Transportation, Federal Highway Administra
tion (FHWA), has recognized the need for providing the traffic engineer
ing community with a single design tool that will enable engineers to
effectively analyze arterial traffic flow and design for optimal con
trol. An Ad Hoc Committee on Arterial Traffic Control (MacGowan et al.,
1977) investigated a number of potential computerized models for con
sideration in an integrated arterial control package. The candidate
models which were considered represented, in the opinion of the Ad Hoc
Committee, the most significant computer models for traffic signal
system design and analysis. Models were examined from the perspectives
of system optimization and system evaluation. The models investigated
for their optimization capabilities are as follows:
1) TRANSYT a network optimization model,
2) SIGOP II a network simulation and optimization model,
3) PASSER II an arterial progression optimization model,
4) EXPRESS an arterial progression optimization model,
5) PASSER III a diamond interchange optimization model and
6) SOAP an intersection optimization model.
Three models were investigated solely on their analysis capabili
ties, for use in system evaluation. These were TRANSYT and SIGOP II
from above, plus the TRANS model, which is a traffic simulation model.
Four models were selected for inclusion in the, socalled, Arterial
Analysis Package because, in the opinion of the Ad Hoc Committee, these
represented the best design and analysis capabilities. These models are
SOAP, PASSER II, PASSER III and TRANSYT6C. The FHWA undertook to devel
op a software implementation package, which will enable these models to
be coded according to a unified input standard, and to standardize the
outputs for ease of interpretation. The selection of PASSER II as the
prevalent progression design model for the Arterial Analysis Package
clearly established the importance placed on the validity of the maximal
bandwith strategy, in comparison with models which minimize delay and/or
stops.
However, maximum bandwidth does not address the totality of the
progression optimization problem. With multiple phasing and differing
distributions of green time at various intersections, there are poten
tially numerous opportunities of (at least) partial progression which
are not explicitly recognized by the maximal bandwidth approach. These
progression opportunities (called forward link opportunities) may be
available to through traffic over a subsection of the artery or to cross
street traffic entering (turning onto) the artery within the control
system. Indeed, the progression opportunities available within the
through band constitute a subset of the totality of forward link oppor
tunities available within the system.
It is intuitively evident that all forward link opportunities
should be considered in the design of coordinated systems; yet this as
pect has never been explicitly addressed by researchers in developing
design strategies.
The maximal bandwidth optimization strategy is well implemented in
the PASSER II model. This model is extremely flexible in its considera
tion of design options, permitting virtually any feasible combination of
design parameters for multiphase signal systems. The design capabili
ties notwithstanding, PASSER II is a poor analysis model because it
provides only limited estimates of traffic engineering measures of
effectiveness. Some important measures such as delay, stops and queuing
are not provided. For this reason, the above mentioned Arterial Analy
sis Package includes TRANSYT6C to analyze designs produced by PASSER
II.
Such a marriage of two disjoint computer programs is a rather awk
ward means of achieving an optimal design and obtaining estimates of the
system's performance for evaluation purposes. A more realistic approach
would seem to suggest a single model which optimizes design and provides
the required figures of merit. Furthermore, if such a model improves
upon the maximal bandwidth optimization strategy, a far more powerful
design tool would be available to the traffic engineering profession.
The development of a model incorporating the stated improvements to
the stateoftheart progression design technology and having an anal
ysis capability is the subject of this research.
Purpose, Objectives and Scope
This dissertation presents the methodology of the forward link
opportunities concept of traffic signal control. The basic concept is
developed in terms of a measure of effectiveness of progressive signal
timing design, and a methodology for optimization of signal timing upon
this measure is presented. Investigations are made to determine whether
the proposed concept is a useful tool in the traffic engineering profes
sion.
The specific objectives of the research are as follows:
1. Review the stateoftheart with respect to design strategies
for coordinated traffic signal systems on arterial highways.
2. Develop the measure of effectiveness referred to previously as
"forward link opportunities."
3. Formulate and develop a methodology for modeling the opti
mization of progressive signal timing based on the new measure
of effectiveness and produce an operational model.
4. Compare the effectiveness of the proposed strategy with the
wellaccepted maximal bandwidth technique.
5. Investigate alternative variations of the forward link oppor
tunities model and identify other applications of the model.
6. Formulate guidelines for developing a complete signal design
model.
The primary emphasis of this research deals with a fixed time,
common cycle, coordinated arterial highway traffic signal control sys
tem. To this end the system of interest is a onedimensional (linear)
system of signalized intersections.
In developing the optimization model, a substantial computer pro
gramming effort has been undertaken to modify the TRANSYT6C program.
This approach was used to facilitate demonstration of the new optimiza
tion strategy and is not necessarily suggested as the most effective
modeling approach from the standpoint of data coding or computer pro
cessing time. Accordingly, presentation of computer coding is not with
in the scope of this work; and, as implied in objective number six
above, the model developed is not intended to represent a completely
final computer model. Nonetheless, the model developed is fully opera
tional and can be used immediately within certain minor constraints.
The maximal bandwidth design strategy presented herein is based
upon the PASSER II model which has been well accepted by the traffic
engineering community in this country. In the case of the PASSER II
modelas well as the TRANSYT6C model, which is the foundation of the
forward link opportunities optimization techniqueit is assumed, for
the purposes of this research, that the theory, model structure, analyt
ical methodologies and program logic are soundly based on accepted engi
neering and programming principles and conventions, unless specifically
stated otherwise. This is particularly cogent in the case of the re
sults of comparative investigations. These investigations will be based
on analyses produced by the TRANSYT6C model, which can simulate reason
ably well conditions suggested by other traffic system models.
Organization
The dissertation is structured along the objectives enumerated
earlier. The next chapter covers a review of coordinated traffic con
trol theory and models that are pertinent to this research. Brief
overviews of the PASSER II and TRANSYT6C models are presented, and are
augmented by more detailed descriptions contained in Appendices A and B.
Chapter 3 contains the theoretical basis for the forward link op
portunities concept and the optimization model development. Technical
details on the new model are given in Appendix C.
Chapter 4 covers the experimental comparison of the new strategy
with the commonly accepted maximal bandwidth strategy.
A comprehensive series of investigations into alternative varia
tions of the forward link opportunities model is presented in Chapter 5.
Investigations cover weighting of forward link opportunities by various
physical and traffic characteristics, alternative explicit objective
functions and optimization of several timing functions.
Conclusions and recommendations emanating from this research effort
are given in the final chapter. These include assessments of the forward
link opportunities (FLOS) concept as a measure of effectiveness and as
9
an effectiveness function for optimization. Recommendations include
areas for further developing a complete FLOS optimization model and
areas for further research.
CHAPTER 2
REVIEW OF ARTERIAL PROGRESSIVE SIGNAL CONTROL STRATEGIES
Introduction
A number of techniques have evolved over the past several decades
for the coordinated progression of traffic on arterial highways (as well
as other street systems). The underlying philosophy of all the tech
niques has been to move traffic along the facility with as few interrup
tions as possible.
This chapter contains a review of various theoretical approaches to
signal progression and modeling approaches for the design of progressive
systems.
Theory of Traffic Progression
A typical signalized arterial highway can be construed as a system
of contiguous links connected to nodes, which are the intersections.
Vehicles traveling from node to node along the links follow trajectories
dictated by several factors. These factors include the desired speeds
of the drivers, the relationship of demand to capacity of the roadway,
environmental characteristics (e.g., nature of surrounding land use) and
institutional constraints such as the speed limit. The factor having
the greatest influence on progression is, however, the status of the
traffic signals at each intersection at the time of the vehicles' arrivals
at the intersections. When signals operate randomly with respect to one
another, vehicles will be stopped in proportion to the amount of green
available for their respective movements and, further, according to a
timespace relationship between adjacent intersections. This is illus
trated in Figure 2.1, which is a timespace diagram of a typical, unco
ordinated signal system. The trajectories of four vehicles are shown in
the figure, from which it is evident that only one vehicle was fortunate
enough to traverse the entire system without being stopped.
If the relative start times of the cycles at each signal are ad
justed to match the desired speed of the traffic, perfect progression of
traffic in both directions of travel results, with no other changes to
the signal timing. This is shown in Figure 2.2. Such a progression
scheme enables all vehicles, once traveling within the socalled green
bands, to proceed unimpeded. This is the timespace relationship of
traffic progression. Unfortunately, perfect progression cannot always
be achieved, and the provision of the best progression under (the more
common) less than perfect conditions is the subject of this research.
Before exploring the various strategies developed for the design of
progressive timing, it may be useful to define the elements of progres
sive traffic signal timing. The quality of progression will ultimately
be a function of the following elements (Bleyl, 1967):
1. Cycle length the recurring amount of time available for the
servicing of all required traffic movements. In progressive
systems, this time must be constant for any given period of
time. (Nonfixedtime signal controllers can certainly be
employed in progressive systems, but they must conform to a
fixed time, recurring pattern. Actuated and semiactuated
control are not pertinent to this research and are therefore
excluded.)
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2. Phasing pattern the recurring sequence, or order, in which
the several signal phases are displayed.
3. Splits the distribution of the cycle length among the sev
eral phases, including green, amber and any allred phase. A
split is said to contain the total of these times for a parti
cular movement and the sum of the splits equals the cycle
length.
4. Offset the relative start time of adjacent signal cycles.
Offsets are commonly stated in terms of the number of seconds
within a cycle (or percentage of the cycle) relative to a
singular reference basis.
As stated above, all of these elements will be constant for a given
operating period and for a given design. Bleyl (1967) also includes
speeds in his list of pertinent elements, but in this research, it is
assumed that progression speeds are not variable. This is due to the
fact that drivers do not readily adjust their speeds to imposed progres
sion speeds (Lai et al., 1977).
The subsections below review the theoretical development of arte
rial progression and modeling. The two are inexorably related, partic
ularly in recent decades.
Past Research
The earliest techniques for the design of progressive timing of
coordinated systems necessarily employed rather unsophisticated analy
tical or graphic representations of simple progression which merely pro
jected green band(s) through the system. Such designs were often based
on simultaneous or alternating phases of equal length balanced with
respect to direction, if twoway progression, or totally devoted to a
primary direction. The analytical techniques were based on travel time,
without regard to queuing, turning movements or other exogenous effects.
Graphic techniques involved manually "balancing" bands by means of trial
and error plots of timespace relationships among signals. More "ad
vanced" approaches to this technique employed the use of strips of paper
or other material which could be moved in the time dimension along an
axis corresponding to the individual intersection locations.
In the early 1950's more sophisticated algorithmic techniques were
introduced which allowed variations of speed, cycle length and irregu
larly spaced intersections to be more easily resolved. Such advance
ments were usually perpetuated by signal controller manufacturers to
enhance the attractiveness of their equipment. For example, the Eagle
Signal Company distributed a design technique which employed a nomograph
relating cycle length to link length and speed and an algorithmic deter
mination of resulting offsets (Fieser, 1951).
By the 1960's, traffic engineering researchers were actively in
volved in the development of improvements to the progressive design pro
cess which recognized the advances in control technology, the analytical
capabilities of digital computers and the behavioral patterns of drivers.
Since then, the theory of coordinated traffic control has evolved essen
tially in three directions: (1) maximal bandwidth, (2) minimal delay
(often along with other measures of disutility) and (3) policies de
signed to separate stationary queues from moving platoons. The first
two techniques usually involve offline design, while the last virtually
always involves real time, online control.
Because the several policy methodologies are functionally separa
ble, they are discussed individually.
Maximization of Bandwidth
Of the various policies governing the control of signals on arte
rial highways, the concept of maximizing the through bands has been, and
continues to be, the most popular (Lai et al., 1977 and MacGowan et al.,
1977). The concept of signal timing to achieve the widest possible
throughgreen bands was explicit in the earlier, primarily, graphic
techniques using timespace diagrams. However, increasing use of multi
phase signals and the increasingly irregular relative placement of
signals (due to suburbanization) demanded more sophisticated, analytical
approaches.
A significant advancement in the stateoftheart, with respect to
computational capability, came with the work by Yardeni (1964 and 1965).
His maximal bandwidth model was based on the ratio of green time to
cycle length, the cycle length itself and offsets designed to allow the
maximum vehicle throughput in a system. The model minimizes the devia
tions of the center of green times at each intersection from the center
of the through bands. Extensive inputs were required for this model.
The level of computational sophistication was quite high, due to overly
optimistic expectations in the technique, which had a somewhat faulty
underlying theoretical basis. As a result, this model proved no more
effective than conventional techniques (Wagner and Gerlough, 1969), but
it served to inspire more theoretically complex design strategies.
The minimal deviation of split to bandwidth remained active for
over a decade. Leuthardt's NO STOP1 model (1975) employs the essential
theory, but uses a Techebyscheff approximation to effect the minimization
of the maximum deviations. This model has been analytically "proven"
but experience and basic intuition suggest that balancing through greens
about the center of the through band does not properly consider all the
germane aspects of arterial progression.
The Metropolitan Toronto Traffic Control System (1965) was a source
of several innovations in both this area and the delaybased theory,
discussed later. The SIGART model is a flexible model because it can
consider a variety of control parameters, namely cycle lengths, splits
and progression speed. The major drawback of SIGART is that the model
is highly sensitive to changes in speeds and, thus, encourages designers
to use such variations in the design speed to obtain wider progression
bands. Unfortunately, drivers do not willingly adjust their speeds to
conform to progression speeds, because such parameters are often trans
parent to them, and the urge to travel at their desired speed is more
pertinent (Lai et al., 1977). A more realistic model based on an other
wise similar concept, but without the inherent problems existing in
SIGART, was proposed by Morgan and Little (1964) and Little et al.
(1966). For relatively homogeneous, welldefined systems, this model
(called EXPRESS) produced realistic offsets which enhance progression.
A good deal of preliminary engineering is required to establish all the
other timing parameters, however. The model is based on mixedinteger
linear programming.
Brooks (1965) first proposed the process which has ultimately
become the underlying policy of current progression control from the
perspective of analytical design. His maximal bandwidth model is based
on the timespace relationship between the most critical intersections)
(i.e., that with the least amount of available green time) and the
remaining intersections which, if not properly offset, could interfere
with progression through the critical intersectionss. Computationally,
the approach of minimizing interference with the critical intersection
greatly reduces the number of combinations that have to be tried in an
iterative computer solution. This allows very efficient testing of a
wide range of cycle lengths (and ultimately other timing elements) to
achieve an optimal solution.
A further simplification of computational complexity in timespace
relationships was introduced by Bleyl (1967). The SIGPROG model con
verts speed and link distances to equivalent travel time diagram. This
model tends to favor the direction with the greater traffic demand, but
its optimal solutions, like SIGART, often require speed changes along
the route.
All of the above maximal bandwidth models are somewhat restricted
in the options they can consider in a single run, namely phase patterns,
cycle lengths and splits, to varying degrees. A significant advancement
was made when a maximal bandwidth optimization model was introduced,
which internally optimizes cycle length (over a specified range), pat
terns and offsets, along with a realistic apportionment of green times
among the various phases. Messer and his associates first introduced
the Progression Analysis and Signal System Evaluation Routine (PASSER)
model in 1963 (Messer et al., 1973) and later improved the model which
is now called PASSER II (Messer et al., 1974 and Fambro, 1979).
PASSER II represents the stateoftheart in maximal bandwidth
optimization models (MacGowan et al., 1977) and is used extensively in
the present research. Because of the latter, it is discussed later in
this chapter.
DelayBased Methodologies
A second active area of research in coordination of traffic signals
utilizes various means of minimizing delay in the system, or minimizing
some combination of delay and stops. These strategies are usually di
rected more specifically at two dimensional (grid) networks than at ar
terials (Lai et al., 1977), but most have been applied to arterials in
practice. Minimization of delay and stops is clearly an underlying
objective of maximal bandwidth models; however, the concepts discussed
below use these variables as explicit objective functions.
In 1960 and 1964, Newell first reported the results of his theoret
ical studies of delay at coordinated intersections. His early research
was purely analytical and was restricted to special cases of (1) equally
spaced signals, (2) closely spaced signals and (3) widely spaced sig
nals. Multiple phasing or wide ranges of other parameters were not
included. Later, he and Bavarez (1967) developed a computer program
which employed early findings in a model for minimizing various objec
tive functions based on stops and delays for oneway streets. This
early model assumed uniform arrivals with no platoon dispersion. This
latter assumption was to be an area of intense research in this area.
Meanwhile another of the important early models was being developed
for the Metropolitan Toronto Traffic Control System. The SIGRID network
model (Marrus and Main, 1964) uses a disutility function comprised of a
quadratic function to express delay and stops. The model determines
offsets which minimize this function. Although SIGRID requires very
extensive calibration to give realistic solutions, this model was consi
dered a significant advancement in the stateoftheart in 1964.
Realizing some of the drawbacks of SIGRID, the basic policy was ad
vanced by the Traffic Research Corporation (1956) who developed the
SIGOP model. Despite advancements over SIGRID, however, SIGOP is also
highly sensitive to certain input parameters (e.g., its platoon coher
ence factor and minimum average headways); thus, the model remains
difficult to calibrate. Selection of the appropriate design, in view
of the sensitivity problems, is highly judgemental. Several comparisons
of SIGOP with the TRANSYT model (see below) found the latter to be
superior (Whirting, 1972; Kaplan and Powers, 1973; and MacGowan and Lum,
1975). Until TRANSYT was introduced, SIGOP was widely used in spite of
the difficulties in preparing data and interpreting results.
Further improvements were made to SIGOP by Lieberman and Woo (1975),
resulting in the SIGOP II model. This model retains the objective func
tion policy of SIGOP, but bases the delay and stops estimates on more
realistic (and analytically less complicated) relationships similar to
those used in the TRANSYT model. In tests, SIGOP II has been demonstra
ted as being superior to SIGOP, and it is currently undergoing compara
tive tests against other network models.
A somewhat different approach to the minimal delay policy was first
suggested by Hillier (1965 and 1966). The delay/difference of offset
method is an extension of the stops and delay concepts introduced by
Webster (1958). The model assumes that wellformed platoons move be
tween adjacent signals at the freeflow speed. Given this assumption,
it can be further assumed that the through traffic primarily occupies
the link during the green interval. Thus, by associating the offset
difference between adjacent signals with the expected queue length
(which expresses delay), the latter can be minimized by adjusting the
offsets.
The major difficulty of the delay/difference of offset method is
that it deals only with connected pairs of signals. However, by combin
ing links in series or parallel, the network can ultimately be condensed
to a single link. The offsets which resulted in the minimum expression
of delay in the single (condensed) link is the optimal solution. This
approach was developed further by Hillier and Rothery (1967) and a com
puter model to simplify the computations was written by Wagner and
Gerlough (1969).
Some networks are not completely condensable, however, and the
delay/difference of offset model fails to provide a complete solution.
To overcome this problem, Allsop (1968a and 1968b) formulated the so
called British Combination Method. The network is first condensed as
far as possible using the delay/difference of offset method. Then
Allsop's graph theory is applied to rebuild the network, link by link,
where each link is optimized at each step. The delay/difference of
offset policy remains active today (Lai et al., 1977), although it is
used with less enthusiasm, because of validation findings which favor
other techniques, particularly TRANSYT (Rach et al., 1974 and 1975).
Little et al. (1974) have also used a mixedinteger linear program
ming model (MITROP) to minimize delay in networks. The model utilizes
IBM's Mathematical Programming System Extended (IBM, 1971) package to
minimize an objective function which is a disutility function consisting
of flow and queue length. Flow patterns are periodic, rectangular pla
toons with uniform arrivals. Stochastic effects are represented in
terms of an overflow queue on each link, and these effects are incorpor
ated into the objective function.
The major drawback of the delay (and stops) reduction models de
scribed above has been unrealistic estimates of the effectiveness func
tions. This is often due to unrealistic treatments of platoons, namely,
the assumption of wellformed distributions. The classic work by
Robertson and his associates has largely overcome the deficiencies of
earlier modeling policies, as discussed below.
The most widely accepted analytical simulation of traffic opera
tions at signalized intersections has been based on Webster's work in
the timing of signals and the delays and stops occurring at them
(Webster, 1958 and Webster and Cobbe, 1966). Robertson (1968 and 1969)
developed the TRANSYT model around the theory of Webster's delay equa
tions. Webster's method has, of course, been used in other works, but
Robertson added a more realistic treatment of platoons. Recognizing the
observations of Hillier and Rothery (1967), which demonstrated that
platoons disperse according to a predictable recurring pattern,
Robertson included a platoon dispersion factor (PDF) in his model so
that the delays and stops would be responsive to realistic arrivals and
queuing at the stop line of a link. The validity of the PDF has been
verified by Seddon (1972). This significant advancement led to an
effectiveness function (e.g., a linear combination of stops and delays)
which, when optimized, produces predicted results that correlated well
with field measurements when the design signal settings are implemented.
The optimization methodology employed by Robertson is a socalled
"hill climb" search technique that produces a true optimum, given suffi
cient iterations. (It should be noted, however, that Robertson's algo
rithm is actually a "valley descent" method, as explained in Appendix
B.) The original TRANSYT model was improved over the next ten years
(indeed, further refinements are presently being made). The most com
monly used version today is TRANSYT6 (Robertson and Gower, 1977). The
Institute of Transportation Studies at the University of California at
Berkeley has modified this model to include estimates of fuel consump
tion and vehicle exhaust emissions, as well as demand responses to the
optimal design visavis the base (usually existing) condition. This
version is referred to as TRANSYT6C (Jovanis et al., 1977). Version
seven has been developed (Hunt and Kennedy, 1979) but is not readily
available at this time.
TRANSYT6C is an integral part of the present research and, there
fore, is discussed in more detail later in this chapter and in Appendix
B. It should also be reiterated that the delay (and stops, where ap
propriate) reducing models were all specifically directed at grid net
works, rather than linear arterial highways.
RealTime Control Models
Several realtime, dynamic control models for coordinated signal
systems are pertinent to the present research. The models of interest
have in common the fundamental strategy, or policy, of minimizing inter
ference to through platoons moving through the system.
Among the earliest of these is the "smooth flow theory" introduced
by Chang (1967). The model dynamically controls a network in a manner
designed to minimize congestion by releasing stopped queues ahead of on
coming through platoons. Queues at downstream signals are measured and
offsets are timed to release the downstream queues such that the queues
have just dissipated when the upstream platoons arrive. The strategy
may apply well to small networks (or better still, at isolated critical
intersections), but it quickly degenerates in larger systems and through
progression is eliminated over several signals (Lai et al., 1977). A
similar strategy, called the PLIDENT model (Holroyd and Hillier, 1971)
was also found to increase delay and reduce progression.
Another similar policy operates primarily in oversaturated net
works. The "queueactuated signal control" technique (Lee et al., 1975)
may operate either in coordinated or uncoordinated networks with actu
ated controllers. The approach employs detectors near the input of each
link that detect the extension of the queue over the entire length of
the link. When this "spillover" is detected, the signal on the criti
cal link is switched to green to clear the offensive queue. This strat
egy works well only in oversaturated, uncoordinated networks where the
variation of offsets is not as critical, because there is no progression
per se.
A third policy lies between the foregoing extremes. Rosdolsky's
model (1973) does not require interconnect (coordination) within a net
work. Detectors measure the momentary degree of concentration to sense
the relative positions of stationary queues and moving platoons. As in
the case of the smooth flow theory, the offset is advanced to clear the
queue to avoid interference with the moving platoon. Thus, the neces
sary control information is "carried" by the traffic stream, negating
the need for coordination within the system. The major difference in
Rosdolsky's strategy is that signals on the perimeter of the network are
coordinated to stagger offsets to traffic entering the system in order
to avoid "collisions" of conflicting moving platoons. However, this
technique also fails in larger systems (Lai et al., 1977), for the same
reasons as before.
From the perspective of arterial control, these dynamic models
would appear to have a certain degree of appeal, particularly on oneway
streets. That is, the dissipation of queues ahead of through platoons
would enhance the propagation of the platoons and minimize stops and
delay. The common drawback is that the variation of cycle lengths and
splits, and, therefore, offsets, tends to have a degenerative throttling
effect on through movements. This aspect is investigated as part of the
present research.
Existing Models Pertinent to the Forward Link
Opportunities Model Development
As noted in the previous section, the maximal bandwidth optimiza
tion policy is presently the most widely accepted basis for arterial
progressive control. A stateoftheart model embodying this policy is
the Progression Analysis and Signal System Evaluation Routine, Version
Two (PASSER II). The maximal bandwidth strategy is directed at through
traffic, as is the forward link opportunities strategy (developed in the
next chapter), thus PASSER II is an appropriate model to use for compa
rative purposes.
For reasons discussed more thoroughly in the next chapter, the
TRANSYT6C model has been selected to form the framework of the proposed
forward link opportunities model.
Both of these models are described in some detail in Appendices A
and B, respectively, for the benefit of readers who are unfamiliar with
the models. Table 2.1 contains a summary of the models in terms of
their computer requirements, inputs, outputs and optimization tech
niques.
TABLE 2.1 SUMMARY DESCRIPTIONS OF THE PASSER II
AND TRANSYT6C MODELS
PASSFR TT
TRANSYT6C
1. Computer Requirements
Language
Computer System
Program Length
(incl. comments)
Execution Time
2. Inputs
Geometric Descrip
tors
Traffic Volumes
Capacities
Minimum Greens
Cycle Length
Phasing
Splits
FORTRAN IV
IBM 370
1425 (approx.)
1 second (max.)
None explicit, except
link length
Total per link
(vehicles/hour)
Vehicles/hour of green
Seconds
Specify range
Optional, multiphasing
Optional (if specified,
dubbed in as minimum
greens)
Offsets
Speed or Travel
Time
By link
FORTRAN IV
IBM 370/CDC 6400
5870 (approx.)
Varies with the square
of the number of nodes
Implicit in linknode
formulation, and link
length
Total per link and up
stream inputs (vehi
cles/hour)
Vehicles/hour of green
Seconds or fractions
of cycle
Fixed
Fixed, multiphasing
Optional, direct entry
or computed internally
Optional, may be input
for analysis
By link
PASSER II TRANSYT6C
TABLE 2.1 (CONTINUED)
PASSER II
3.
OutputsMOE
Saturation Ratio (v/c)
Delay
Stops
Fuel Consumption
Emissions
Progression Speed
Average Speed
Vehiclemiles of Travel
Vehiclehours of Travel
Bandwidth
Attainability
By link
Uniform plus random
Yes (may be weighted
by length of delay)
Yes
HC, CO, and NO
No
Yes
Yes
Yes
No
No
4. Optimization
Cycle Length
Phasing
Splits
Offsets
Progression Speed
Objective Function
Yes
Yes, two to six
Yes, distribution
of v/c ratios
Yes
Yes( +2mph of
average)
Max. bandwidth
efficiency
No
No
Yes, distribution of
v/c ratios
Yes
No
Min. ftn(delay, stops,
fuel and emissions)
IT
EM
By movement
No
No
No
No
Yes
No
No
No
Yes
Yes
TRANSYT6C
CHAPTER 3
DEVELOPMENT OF THE FORWARD LINK OPPORTUNITIES MODEL
Introduction
In the previous chapter, the maximal bandwidth strategy was identi
fied as the most common methodology used in the design of optimal pro
gression on arterial highways. A widely accepted model embodying this
concept is the Progressive Analysis and Signal System Evaluation Rou
tine, Version Two (PASSER II). The focus of this research is to deter
mine whether improvements in the design of traffic signal progression
can be made over the maximal bandwidth method.
The maximal bandwidth strategy considers only the relative align
ment of signal offsets which produces the maximum widths of the bidirec
tional bands. No explicit consideration is given to intersections that
are not critical to the bounding of the through bands. The uncertainty
of how to treat signal timing at the noncritical intersections intro
duces a form of entropy into the system design. That is, some available
portion of the cycle may not be transformed into useful progression.
This is illustrated in the timespace diagram shown in Figure 3.1. Note
the outlined areas, which represent partial progression opportunities
outside the through bands.
If the offsets are adjusted somewhat, these additional (partial)
progressive opportunities can be increased as illustrated in Figure 3.2.
This is the purpose of the forward link opportunities concept, which is
described in the following section.
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Concept of the Forward Link Opportunities Model
Forward link opportunities (FLOS) derive from the timespace rela
tionship of traffic signal timing along the arterial highway. As the
name implies, there is no explicit consideration given to the presence
of traffic demand to accept the progressive opportunities (although in
Chapter 5 a strategy which recognizes the periodic demand is presented).
Forward link opportunities are defined as the number of successive
links along an arterial roadway, from an intersection displaying a green
signal indication (including amber and any allred phase), that will
have green signal indications at their downstream ends, in progression;
that is, when encountered at times dictated by given link travel speeds.
In other words, FLOS are the number of successive links downstream of
any green signal over which progression could occur, during a finite
increment of time, without interruption by a red signal indication.
The variables that determine the availability of FLOS are time,
progression speed and the timing of the traffic signals. In addition,
the parameter, link length, must be included. Time is a continuous
variable, but events which conveniently describe traffic signal oper
ations are discrete (e.g., the start and end times of the signal phases).
Therefore time can be divided into discrete intervals of uniform dura
tion for the purposes of conceptual development. These intervals can be
sufficiently short to yield a realistic representation of time and
effect the identification of events that describe the status of the
traffic signals.
For a given system, progression speeds are assumed to be constant
(by link) for the reasons stated in Chapter 2. Signal status is cycli
cal for any given set of timings being considered.
Forward link opportunities are derived from these variables as
described below.
A fundamental element in the determination of FLOS is dependent on
the event that signals are in the green interval at the appropriate
times (where, "green" is defined to include the actual green, plus amber
and any allred time). This is described in terms of a binary event
function as follows:
D.
True, if mod[(t + j + T), T] < P.T, or
Et LFalse, otherwise;
(3.1)
where E = the event that signal j is green at time increment t, and
jt
t ranges from one (1) to T;
D. = the distance in feet from the first intersection to
intersection j;
V. = the average progression speed (mph) between intersections
one (1) and j which is determined as follows:
V = D /[' =1 (di/vi)], where di and vi are linkspecific
lengths and speeds, respectively;
F = a factor to convert from mph to feet per time increment;
4. = the offset (i.e., the positive difference in start times
of the throughgreen phases) of intersection j, relative
to the first intersection, in time increments;
T = the cycle length in time increments; and
P. = the fraction of the cycle that the through phase at
intersection j is green.
intersection j is green.
A single forward link opportunity exists at intersection j when
ever Ejt and E(j+1)t are true at any time increment, t. This relation
ship can be illustrated graphically. When the signal timings are ad
justed for travel time according to the modulo function in Eq. (3.1),
the timespace diagram (such as Figures 3.1 and 3.2) can be adjusted
such that the progression speed has zero slope with respect to the time
axis. When this is done, the distance between intersections is no
longer relevant (at the progression speed) and the distance scale can be
"collapsed" into a unitless scale where only the relative order of
intersections is pertinent. When Figures 3.1 and 3.2 are transformed in
this manner, the timelocation diagrams shown in Figures 3.3 and 3.4,
respectively, result. The through bands and partial progression op
portunities outside the through bands are also identified in these
diagrams. The partial progression opportunities conform to the defi
nition of single forward link opportunities given above, if it is
envisioned that the blank "cells" of the diagram represent that Ejt is
true and the stars represent the fact that Ejt is false. All conditions
where adjacent events (with respect to locations at any time t) are true
are outlined in the figure. These outlined areas represent the presence
of FLOS.
Figure 3.4 clearly shows more potential FLOS than Figure 3.3. For
example, a vehicle entering the arterial (leftbound) from a side street
at intersection eight at time t1 can expect to travel the remainder of
the system without encountering a red signal, if it travels at the
progression speed.
The above has described the FLOS concept in qualitative terms. The
quantification of FLOS is now presented.
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The logical function, E t, is readily converted to a numerical
function as follows:
11 E or
S5t : (3.2)
0I Et;
where Sj = a binary status function, which is equivalent to the event
function Ejt, and describes the status of signal j at time
interval t.
To complete the quantification of actual FLOS for each intersection
and at each time interval, the products of the binary status variable are
summed over all intersections for each increment of time as follows:
N j
FLOSit = Sit + Skt ; (3.3)
j=i k=i
where FLOSit = the forward link opportunities from intersection i at
time interval t, and
the other variables are as defined before.
The product of terms in the above equation is necessary to count
only those successive intersections for which all of the binary status
variables are unity (for time increment t). Furthermore, it is nec
essary to decrement the sum by one (if Sit is equal to unity) to indi
cate that the value of FLOS represents the number of downstream forward
link opportunities from intersection i. [An example of the function of
Eq. (3.3) is given later, following the introduction of the diagrams
which result from its use.]
When Eq. (3.3) is exercised separately for both directions of
travel on the arterial, a "FLOS diagram" can be produced to illustrate
the FLOS for each signal at each time increment. The FLOS diagrams for
Figures 3.3 and 3.4 are shown in Figures 3.5 and 3.6, respectively.
Note that the individual FLOS at each signal, and for each time inter
val, fill out the areas previously outlined in Figures 3.3 and 3.4 (and,
for that matter, Figures 3.1 and 3.2).
Having illustrated the results of Eq. (3.3) in Figures 3.5 and 3.6,
an example of the application of Eq. (3.3) may clarify the meaning of
the equation. In Figures 3.5 and 3.6 one cell or, the respective
values of FLOSit in each figure, are circlednamely, the values for
intersection five (5) at time increment twenty (20). Table 3.1 gives
the detailed solution of Eq. (3.3) applied to these FLOS5,20 calcu
lations.
The improved number of FLOS in the 5,20 cell is demonstrated in the
example as they increased from two (2), in the maximal bandwidth solu
tion, to three (3) in the solution which maximizes FLOS.
The FLOS concept thus recognizes and quantifies the availability of
both continuous and discontinuous bands which can serve trips that
originate within the system and can also serve trips which exit the
arterial within the system. Another important aspect of this concept is
that it can consider the quality of progression from the driver's per
spectivenamely, how many additional signals will permit continuous
passage of vehicles which have already traveled some distance within the
system.
FLOS is a useful measure of effectiveness, but more significantly,
when used as an optimization variable, it can apportion available green
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time and assign offsets in such a manner that these forward link oppor
tunities are maximized, thus providing the best progression from the
driver's perspective. This is demonstrated in a comparison of Figure
3.5, which is the FLOS diagram of the maximal bandwidth solution, and
Figure 3.6, which is the FLOS diagram of the maximal FLOS solution.
The aggregate FLOS is found by summing Eq. (3.3) over all time
intervals and all signals. For one direction (say rightbound), this is
accomplished as follows:
N T
FLOS = C/T z E FLOSit; (3.4)
i=l t=l
where FLOSr = the aggregate number of forward link opportunities for the
rightbound direction;
C = the cycle length in seconds;
T = the cycle time in time increments; and
FLOSit is defined by Eq. (3.3).
The ratio C/T converts the aggregate FLOS from time increments to
seconds.
The aggregate number of FLOS for the leftbound direction is compu
ted similarly, but it must be recognized that the scan for individual
FLOS in Eq. (3.3) is done from intersection N to the first intersection,
or in the left direction. The total FLOS is simply the sum of the FLOS
in each direction.
The generalized model for total aggregate FLOS is given below,
2 N T i k=1 (rightbound), or
FLOS = C/T E z FLOSjt, j = (3.5)
k=l i=l t=l haj b v d
kN+1ilk=2 (leftbound);
where all variables have been previously defined.
FLOS are represented on a percycle basis, aggregated over all time
increments (corrected to seconds if required).
In the example which has been used in Figures 3.1 through 3.6, the
total FLOS increased from 1,978 in the maximal bandwidth solution, to
2,131 when offsets were changed to maximize FLOS. This is about an 8%
improvement.
Two additional measures related to forward link opportunities are
useful. In the absence of any signalized traffic control, the full
cycle would be available as forward link opportunities. This hypotheti
cal measure (referred to as cycle forward link opportunities, or CFLS)
can be computed directly for unweighted CFLS as follows, for both direc
tions:
C NI
CFLS = 2 E E k = 2 CN(N1) ; (3.6)
t=1 k=1
where C = the cycle length in seconds and
N = the number of intersections.
The ratio FLOS/CFLS, called a progression quality ratio (PQR ),
is analogous to the bandwidth efficiency (see Chapter 2 and Appendix A),
which is the PASSER II objective function, but in this case for FLOS.
By the nature of its definition, PQRc will always be less than unity.
In the example considered earlier, the bandwidths did not change in the
two solutions; however, the PQRc increased from 0.35 in the maximal
bandwidth solution, to 0.38 in the maximal FLOS solution.
The throughonly FLOS are those forward link opportunities within
the through bands. For any given solution the through forward link
opportunities (TFLS) is simply a fraction of the CFLS, or,
TFLS = CFLS x BW/C; (3.7)
where BW = the sum of the bandwidths in both directions of travel, in
seconds (see Appendix A for a detailed description of the
bandwidths); and
the rest is as before.
A progression quality ratio based on FLOS/TFLS (or PQRt) is a
measure of the partial progression opportunities outside, or in addi
tion to, the through bands.
Since more partial progression opportunities resulted from maximiz
ing FLOS (e.g., opportunities outside the through bands), the PQRt
increased similarly (from 1.35 to 1.45). Examining only the FLOS out
side the through bands, a net increase of 153 FLOS resulted, or about
30%.
In summary, when consideration is given to partial progression op
portunities, offsets can be shifted to improve shortterm progression
over the solution derived by the maximal bandwidth optimization policy.
This example has demonstrated the FLOS concept, which serves as the
basis of a new traffic progression optimization model. Outwardly, the
FLOS concept is neither analytically complex, nor overly sophisticated
from a theoretical perspective. It is, however, an innovative approach
to signal progression design which has a good deal of intuitive appeal.
Since, at the surface level, the concept embodies the principles of the
maximal bandwidth methodology,it is consistent with the current direc
tion the traffic engineering profession is taking with regard to arte
rial control. That this concept can potentially improve the quality of
traffic progression beyond that provided by existing maximal bandwidth
models, suggests that model development and comparative investigation in
this area are warranted.
The remainder of this chapter covers the development of a candidate
FLOS optimization model.
Model Development
The basis of the FLOS formulation was given in Eq. (3.3). Forward
link opportunities are a function of location, time, velocity and signal
status. Location is simply a discrete function equivalent to signal po
sition. Time refers to displacement within the cycle. Velocity can
generally be considered to be a scalar quantity, because satisfactory
progression will generally be responsive to the desired speed of the
drivers (as noted in Chapter 2). Signal status is a function of the
signal settings and, of course, time in the cycle.
Fundamental Elements of Progression
Signal progression is a composite of four signal timing parameters.
These were defined in Chapter 2 and are (1) cycle length, (2) phasing
pattern, (3) green splits and (4) offsets. All four of these parameters
affect the quality of progression to varying degrees.
Offsets explicitly define the speeds of the progression bands. The
progression band speed is determined by the desired speed of travel,
which defines travel times between adjacent signals. But irregular
phasing and splits require an offset that will position the green phases
to best accommodate the progression bands. Splits determine the propor
tion of the cycle available for progression. The PASSER II maximal
bandwidth model distributes green time in proportion to the demand/capa
city ratio as discussed in Appendix A. Splits favoring the arterial
traffic would provide more time to move traffic on the main street, to
the detriment of the cross street.
Since the progression band, or partial bands, may only be a frac
tion of the cycle, the cycle length dictates the widths of the bands.
While over a long period, say an hour, the total time available for
arterial green will be proportionately the same, regardless of the cycle
length, shorter cycles tend to restrict through bandwidths, resulting in
increased stops and delays. Longer cycles tend to accommodate more
through traffic, but queuing can become excessive during the longer red
signal periods.
Patterns dictate the flexibility of the timing plan which enables
phase arrangements to accommodate the through bands. Patterns which
maximize green time for through movements will permit wider through
bands, but it is not always ideal to have the opposing through traffic
moving simultaneously. Often, better twoway progression can be ob
tained by allowing one through movement and the parallel leftturn to go
first, followed by an overlap of both through movements, then the
through and leftturn in the second direction.
Formulation of the Model
Optimization upon forward link opportunities is based on the fol
lowing objective function:
2 N T ilk=1 (rightbound), or
max FLOS = C/T E E E FLOS.t, j = (3.8)
k=1 i= t=1 +lik=2 (leftbound).
This objective function is subject to certain constraints on the
decision variables, which are discussed below.
Minimum Green and Demand Satisfaction
The minimum green on any approach is governed by safety considera
tions. Common practice dictates that the green display should be no
less than five seconds, followed by a four second amber and, often, a
one second all red interval. Thus, ten seconds is generally an appro
priate minimum green interval, provided pedestrians are not a considera
tion (e.g., for exclusive leftturn movements). When pedestrians do
move with traffic on a particular phase, the crossing time is generally
three seconds times the number of lanes to cross, plus four seconds
clearance. The minimum green constraint is, for the purposes of this
research, defined as
SLN + 4, if pedestrians are a consideration, or
MGi = (3.9)
LM otherwise;
where MGi = minimum allowable green time for phase i and
LN = number of lanes the pedestrian must cross.
In an optimal solution, traffic demand on all approaches should be
satisfied without recurring congestion. Webster's method (Webster and
Cobbe, 1966) is generally used to calculate required green time to sat
isfy their demand according to the following (Fambro, 1979).
Gi = [(yi/Y) x (CL)] + li ; (3.10)
where Gi = length of green, including amber and all red, for
phase i;
yi = ratio of actual demand (veh/hr) to saturation flow
(veh/hr of green) for phase i;
Y = sum of the yi for all phases;
C = cycle length;
1. = lost time due to starting up the gueue at the begin
ning of phase i; and
L = total lost time.
If green periods are allowed to vary, the design value of minimum
green shall be subject to the following constraint:
Min Greeni = max (Gi, MGi) ; (3.11)
where Gi = a value of green for which yi in Eq. (3.10) is no
greater than a given upper limit, and
MG. = as defined above.
The subject of absolute capacity (i.e., maximum vehicles per hour
of green time per lane) is not addressed in this research. A reason
able, and accepted, value is assumed to be valid, namely 1,800 vehicles
per lane, per hour of green time (Courage and Landmann, 1978).
Finally, the sum of all design minimum green times for p phases
must be less than or equal to the cycle length, or,
p
z Min Green. < C. (3.12)
i=1
The sum of the actual greens (where amber and allred intervals are
included in the "green" times) must equal the cycle length.
Design Speed
Signal timing designs which provide progression at speeds substan
tially different than the desired speed of the traffic do not adequately
serve the motoring public. Slight variations in speed are certainly
permissible since desired speed is a stochastic function. However, for
the purposes of this research, it is assumed progression speeds will be
constant for a given link, but may vary among links as required.
Model Implementation
The aggregate forward link opportunities expressed in Eq. (3.8) is
a multidimensional function of the disjoint elements defined earlier,
which are by their nature either nonlinear (e.g., location with respect
to time), or linearly dependent (e.g., proportionately equivalent pe
riods of time). Thus, computation of the objective function is nec
essarily algorithmic in nature and its solution requires an iterative,
search and find approach.
An exhaustive search approach is computationally prohibitive. For
example, in a system of n intersections, with a fixed set of signal pat
terns and fixed green splits, the number of trials would equal Cn
where C is the cycle length, just to examine all possible offsets. A
six signal system would require over 46 billion iterations for a 60
second cycle. Clearly, such large numbers of computations would tax
even the largest of modern digital computers.
On the other hand, Eq. (3.8) is a bounded concave function since,
for a given set of signal timings, the objective function will vary
between some minimum value representing the worst possible progression
and a maximum representing the optimal solution.
A variety of technologies exist for solving nonlinear concave
functions (Wagner, 1969); however, the present objective function is
discontinuous with respect to the multiplicity of possible signal set
tings. Stated otherwise, a change in one setting to influence the FLOS
in, say, one direction of travel, may have a correspondingly opposite
effect on the opposite direction. The nature of this effect (e.g.,
beneficial or adverse) is analytically unpredictable without using
extremely complex mathematical formulations.
Faced with this same dilemma, the developer of the TRANSYT model
(Robertson, 1969), formulated a search technique he referred to as hill
climbing. As stated in Chapter 2, "hill climbing" is a misnomer, for in
the TRANSYT model the objective is minimization of a combination of
stops and delay, thus "valley descent" is more appropriate. In the
proposed model, however, the objective is to maximize forward link
opportunities so the algorithm is truly a hill climb in this applica
tion.
The principle is illustrated idealistically in Figure 3.7. The
ordinate axis is aggregate forward link opportunities, or the objective
function, Eq. (3.8). The abscissa is a representation of the universal
set of timing elements which produce the FLOS. Although the plot sug
gests a continuous function, it must be stressed that this is not the
case. The optimization technique proceeds as follows:
1. The initial settings are evaluated for FLOS, represented in
Figure 3.7 by the set of conditions, S1.
2. The offset of each link in the system is varied in turn by a
small amount and the objective function is recalculated for
each new set. When disimprovement is noted, the direction of
the search reverses until a local optimum is found, depicted
by set S*
3. Offsets are varied by medium or large increments specified in
the input data (generally 15% or 40% of the cycle length, re
spectively) to force the investigation in other regions of the
solution space. This is depicted by the medium jump to S2 and
the large jump to S3 which resulted in improved local optima
at S2 and S**, respectively.
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tive function was lower and the solution converged on S**.
The above process completely describes the optimization technique
employed in the forward link opportunities model. To implement this
model, the TRANSYT6C model was modified to substitute the forward link
opportunities objective function for the TRANSYT6C objective function
(e.g., the Performance Index, see Chapter 2 and Appendix B) in the hill
climbing routine.
Four subroutines were added to the existing program to accomplish
the calculation of the FLOS objective function. Another subroutine was
added to calculate a new measure of effectivenessnamely, percentage of
arrivals on the red signal (derived from Courage and Parapar, 1975). A
sixth subroutine was added to plot a timespace diagram, which was not
in the existing program. A generalized flow chart of the program is
shown in Figure 3.8.
A detailed description of all program modifications to achieve the
FLOS model is given in Appendix C. As noted therein, certain other mod
ifications were implemented to improve the output formats of the exist
ing program.
The TRANSYT6C/FLOS model is fully operational. The new model fully
retains all of the existing capabilities of the TRANSYT6C model and
includes either analysis of, or optimization upon, the forward link op
portunities. There are no restrictions on the model's use as a network
model (TRANSYT6C) without consideration of forward link opportunities.
FIGURE 3.8 GENERALIZED FLOW DIAGRAM OF THE TRANSYT6C/FLOS MODEL
When FLOS are included in the analysis, the following restrictions
apply to the present model:
1. Only a onedimensional (e.g., linear) network configuration
may be modeled.
2. Linknode numbering must conform to a specified format, but
such a practice is generally preferred in any case.
3. No grouping of nodes or sharing of links by various distinct
classes of traffic is allowed (both are available in the
normal network model).
4. Variation of green splits is permitted, but minimum design
greens must be generated externally to conform to the con
straint expressed in Eq. (3.11).
5. No priority lane or demand response functions of the original
model may be exercised.
The following limitations, inherent to the original TRANSYT6C
model, remain similarly in the new model:
1. Cycle length may not be internally varied. Multiple runs,
with corresponding changes to input data, are required to
examine different cycle lengths.
2. Phasing of signal displays may not vary within a given run;
however, any reasonable phasing can be modeled.
A variety of FLOS weighting factors or functions and alternative
objective functions are available. These are identified in Appendix C.
Since Chapter 5 is devoted to investigations of such alternatives, fur
ther discussion in the text is deferred to that chapter. Further poten
tial improvements are noted in Chapter 6.
CHAPTER 4
COMPARISON OF THE FORWARD LINK OPPORTUNITIES AND
OPTIMAL BANDWIDTH OPTIMIZATION STRATEGIES
Introduction
This chapter contains a detailed comparison of the forward link
opportunities (FLOS) optimization strategy with the maximal bandwidth
strategy. Its purpose is to demonstrate that, for similar conditions,
the FLOS optimization strategy produces offsets which provide progressive
opportunities equal to or superior to a maximal bandwidth optimization.
Equivalent physical and traffic conditions for several typical
arterial configurations are tested. The tools for optimizing under the
two methods utilize the newly developed TRANSYT6C/FLOS model, described
in Chapter 3 and in Appendix C, and the PASSER II model described in
Chapter 2 and Appendix A.
All comparative measures of effectiveness (MOE) are produced by the
TRANSYT6C/FLOS model, which calculates MOE for both the maximal band
width optimal solution and the FLOS optimal solution.
Experimental Design
Both the maximal bandwidth and FLOS optimization strategies are
timespace solutions to a progressive design. In Chapter 3, four ele
ments were identified as components of a progression design. These are
signal offsets, cycle length, splits and phasing pattern. In a direct
comparison of the two methodologies, the only parameter which should be
allowed to vary is offsets. To allow the remaining parameters to vary
would result in a biased comparison. This is not to say that a FLOS
optimization of these other parameters would not result in their differ
ing from a maximal bandwidth optimization. Indeed, variation of green
splits is treated in Chapter 5.
The principal question addressed in this chapter is whether the
signal offsets can be realigned to provide improved forward link oppor
tunities compared to a maximal bandwidth solution, all other considera
tions being equal. Accordingly, all tests in this chapter are limited
to fixed conditions with respect to cycle length, splits and patterns,
as well as the fixed physical and traffic characteristics.
Test Systems
The number of signals in a given system is likely to have a percep
tible effect on the relative solutions from these two strategies. Five
typical arterials are tested. All five are actual arterial highways for
which data were readily available to the author. The characteristics of
the five systems are summarized in Table 4.1.
All test facilities are geometrically and functionally similar, but
they differ in the average spacing of intersections, configuration of
intersections and traffic demands. They represent a typically diverse
set of test conditions.
Evaluation Methodology
For a given system configuration, the following sequence is exercised:
1. The base geometric and traffic conditions are exercised for the
maximal bandwidth solution using the PASSER II model.
2. The cycle length, splits and phasing patterns, along with com
parable base data, are input to the TRANSYT6C/FLOS model.
Since this model analyzes the initial input conditions, the
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offsets from the maximal bandwidth solution are input as well;
thus, the PASSER II solution is analyzed completely.
3. TRANSYT6C/FLOS then optimizes for unweighted forward link op
portunities by varying the offsets, and produces the resulting
MOE.
Results of the tests are presented in tabular form. The MOE of
interest are identified below.
Unweighted forward link opportunities (FLOS) are reported in their
actual values, as calculated by Eq. (3.5). Improved progression should
increase the FLOS.
The progression quality ratios (PQR) corresponding to the ratios of
actual FLOS to the several control values are calculated according to
Eqs. (3.6 and 3.7). The PQR defined by FLOS/CFLS, where CFLS are the
cycle forward links, is similar to the bandwidth efficiency (see Chapter
3). But here, it is the ratio of actual FLOS to the total forward link
opportunities potentially available in a cycle, in the absence of any
signal control. The PQR defined by FLOS/TFLS, where the latter are the
throughonly forward links, is a measure of the partial progression
opportunities existing outside the through bands. As their names imply,
the ratios should increase with improved progression.
Delays in vehiclehours per hour (vehhr/hr) are reported for the
throughonly movements on the artery and for the total system, including
cross street traffic. With improved progression, delay should decrease
on the artery, and thus, in the entire system.
Stops are presented in terms of percentages of vehicles stopped,
first for only the through movements on the artery and secondly, for the
entire system. As in the case of delay, the number (or percentage) of
stops should decrease as progression is improved.
The percentage of arrivals on the red signal phase is related to
the number of stops, although the relationship is not linearly propor
tionate. Still, improved progression should result in a reduced propor
tion of the through traffic arriving during the red phase. (The percent
red arrivals is reported in lieu of the more positive percent green
arrivals to reduce confusion in the comparisons. Thus, all three traf
fic measures, delay, percent stops and percent red arrivals should
decrease with improved progression.)
Because it is theoretically possible that optimization upon FLOS
may cause a change in through bandwidth, these are reported as well,
along with the bandwidth efficiency and attainability, as defined for
the PASSER II model. Other measures are discussed as needed to qualify
specific results, and several peculiarities inherent to either the FLOS
concept, or the limitations of the TRANSYT6C (and thus, TRANSYT6C/FLOS)
model, are noted as appropriate.
In the following sections, references are occasionally made to
direction of travel. For simplicity, all systems should be viewed as
being laid out from left to right, regardless of actual orientation,
where "rightbound" is taken as the "A" direction as defined in the
discussion of the PASSER II model in Appendix A (also see Figure C.1 in
Appendix C).
Analysis of Alternative Arterial Configurations
SixSignal System
A typical sixsignal system is Buffalo Avenue in Tampa, Florida
(see data set BUF in Table 4.1). Results of the maximal bandwidth and
FLOS optimizations for this system are presented in Table 4.2.
TABLE 4.2 COMPARATIVE RESULTS OF MAXIMAL BANDWIDTH
AND FLOS OPTIMIZATIONS FOR BUFFALO AVENUE
(SIXSIGNAL SYSTEM)
MAXIMAL NORMAL PERCENT
MOE BANDWIDTH FLOS DIFFERENCE
Cycle Length (sec) 60 60
Bandwidths (sec) 42 43 2.4
Efficiency (%) 35 36 _a
Attainability 0.98 1.00 2.0
FLOS 675 684 1.3
PQR = FLOS/CFLS 0.37 0.38
PQR = FLOS/TFLS 1.07 1.06 0.9
Delay (vehhr/hr):
Through on Artery 21.0 21.1 0.5
Total System 46.0 46.0 0
Percent Stops:
Through on Artery 52.6 51.5 2.1
Total System 62.2 61.3 1.4
Percent Red Arrivals 41.0 40.4 1.5
a. Percent differences for these values are not given in this
and subsequent tables because roundoff errors would pro
duce different numerical values. The percent changes are
actually as indicated for bandwidths and FLOS, respectively.
This example resulted in very limited improvement in the number of
FLOS when the maximal bandwidth solution was further optimized for the
new measure. Indeed, there was effectively no improvement in the pro
gression quality ratio of FLOS to cycle forward links (CFLS).
The FLOS optimization resulted in one interesting occurrencethe
bandwidth was increased by one second, specifically in the rightbound
band (from 20 to 21 seconds). This increase, of a supposedly already
maximum bandwidth, does not suggest that PASSER II fails to achieve a
maximal bandwidth solution, but is due to the fact that the PASSER II
model deals internally in real values of time, while TRANSYT6C/FLOS
deals in integer values of time. This occasionally allows the TRANSYT6C/
FLOS model to produce bands that vary slightly from PASSER II because of
roundoff. In this and all subsequent tests, if in the first simulation
of a PASSER II solution, the bandwidth calculated by TRANSYT6C/FLOS is
less than that calculated by PASSER II, offsets are changed to correct
for roundoff error so that the timing ultimately used produces at least
the same bandwidths as PASSER II.
The increased bandwidth explains the slight decline in the ratio of
total FLOS to through forward link opportunities (FLOS/TFLS). The
through FLOS were proportionately higher than the increase in total FLOS
for the system in the rightbound band.
Regarding the more traditional traffic engineering measures, a very
slight increase in delay to through traffic on the artery is observed,
despite slight decreases in both the percentages of stops and arrivals
on red. Total delay in the system remained unchanged, however, suggest
ing that increased delay to some through traffic was offset by decreased
delay to some turning traffic. Delay to crossstreet traffic is unaffected
by any changes in offsets because their proportion of green time remains
the same. But why has delay increased for some through traffic despite
marginally improved forward link opportunities and a slightly wider
through band in one direction?
The phenomenon suggested by the above question is significant to
this research. Considering that FLOS optimization considers only the
relative alignment of green phases (as does PASSER II in determining
offsets), it is entirely possible that a shift in an offset can result
in greater delay because a more highly concentrated portion of a platoon
may arrive on the red signal, particularly at the leading edge of the
through band. Indeed, this occurred in the present example on one link
which experienced an 8% increase in delay due to a 2% increase in the
proportion of red arrivals. This increase in red arrivals was not
offset by other declines in delay to through traffic; although, as
stated earlier, it was offset by decreases to turning traffic which
resulted in no change in the total delay in the system.
In summary, the comparison of maximal bandwidth and forward link
opportunity optimizations on a small system of six signals produced
virtually no differences in the two solutions despite the occurrence of
an adverse phenomenon in the FLOS optimization. The quality of progres
sion was negligibly improved by the FLOS optimization.
EightSignal System
A slightly larger system of eight signals on State Road (S.R.) 26
in Gainesville, Florida, was tested similarly. The results of this
study are summarized in Table 4.3.
In this example, a more discernible improvement in the quality of
progression was realized after the maximal bandwidth solution was further
TABLE 4.3 COMPARATIVE RESULTS OF MAXIMAL BANDWIDTH
AND FLOS OPTIMIZATIONS FOR S.R. 26
(EIGHTSIGNAL SYSTEM)
MAXIMAL NORMAL PERCENT
MOE BANDWIDTH FLOS DIFFERENCE
Cycle Length (sec) 98 98
Bandwidths (sec) 52 52 0
Efficiency (%) 27 27
Attainability 1.00 1.00 0
FLOS 1,978 2,131 7.7
PQR = FLOS/CFLS 0.36 0.39
PQR = FLOS/TFLS 1.35 1.46 8.1
Delay (vehhr/hr):
Through on Artery 33.5 33.8 0.9
Total Systems 67.8 66.8 1.5
Percent Stops:
Through on Artery 35.8 34.5 3.6
Total Systems 46.0 45.0 2.2
Percent Red Arrivals 26.7 23.7 11.2
optimized by maximizing FLOS. The raw number of FLOS increased about
8%, thus FLOS/CFLS increased from 0.36 to 0.39. The improvement of
partial progression opportunities outside the through bands increased
similarly in their aggregate. If only nonthrough FLOS are considered
(i.e., counting only those FLOS outside the through bands, or FLOS 
TFLS), these additional opportunities increased from 515 to 668, or 30%.
The proportions of through vehicles stopped or arriving during the
red signal declined, as did all stops in the system. But, once again,
the delay to through traffic increased slightly (1%) despite a nearly 2%
decrease in delay systemwide. The phenomenon described earlier was
repeated in this case, but even more dramatically. On a single link
delay increased from 0.85 vehhr/hr to 5.78 vehhr/hr, or 680%. This
was due to an increase in red arrivals from 16% to 40%. Because this
case is far more dramatic than the previous one on Buffalo Avenue, a
graphical illustration of the effect of such a shift in red arrivals is
warranted. Figure 4.1 shows the arrival and departure patterns at the
end of the subject link (link 72) for the maximal bandwidth solution
(top) and the maximal FLOS solution (bottom), from the patterns produced
by TRANSYT6C. Shifting the start of green on this approach relative to
the upstream signal resulted in substantially more queuing as the depar
ture pattern indicates, despite very little change in the arrival pat
tern. According to TRANSYT6C's analysis, the average maximum queue
increased from five to thirteen vehicles per cycle. This is despite a
significant increase (from 146 to 196, or 34%) in the FLOS at this link.
Unfortunately, the additional FLOS occur during a period when virtually
no traffic is arriving (i.e., the later portion of the green period
which has negligible arrivals).
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Maximal Bandwidth Solution
LINK 72 MAX FLOW 3600 VEH/H
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Phase
Maximal FLOS Solution
FIGURE 4.1 ARRIVAL AND DEPARTURE PATTERNS ON LINK 72 OF THE S.R. 26
SYSTEM UNDER MAXIMAL BANDWIDTH AND FLOS OPTIMIZATIONS
1
Thus, generally reduced delays on the remaining through links did
not offset this extremely high increased delay on link 72. Additional
time savings on other nonthrough links (e.g., those assigned to left
turning traffic) did compensate for the single major problem.
Most of the increased FLOS actually occurred in the leftbound
direction and the FLOS diagrams for this direction are given in Figure
4.2 for both the maximal bandwidth and FLOS solutions, for comparison.
As noted above, link 72 (intersection seven) experienced a large part of
the increased FLOS.
Summarizing this example, a more appreciable improvement in the
quality of progression was observed on this system when FLOS were opti
mized beyond the maximal bandwidth solution. A severe problem resulted
at one intersection, however. This problem notwithstanding, all sys
temwide MOE improved under FLOS optimization.
TwelveSignal System
The test facility discussed below is a section of State Road (S.R.)
7, also U.S. 441, in Ft. Lauderdale, Florida. This test raises an issue
that constitutes a minor limitation of the TRANSYT6C model, namely the
computational resolution of the model. Since the new model is based
upon the frame of the original TRANSYT6C model, this limitation carries
forward. The issue is discussed with the description of the test re
sults, which are summarized in Table 4.4.
First reviewing the results of this test outright, there was a
negligible increase in FLOS and the corresponding progression quality
ratio under FLOS optimization, but a slight increase in all the traffic
related MOE, including total delay in the system. These results suggest
that the PASSER II solution, as analyzed by the TRANSYT6C/FLOS model, is
very close to an ideal optimum.
66
1 2 3 4 5 6 7 8 1 2 3 4 6 7 8
1 : 1 : : 1:
S2 3 : : : ::
1 2 3 4 : : : :
1 : : : : : :
1 2 3 4 : : : : 1
1 1 1 : : : : : : :
1 2 3 4 : : : : : :
1 2 3 4 : : : : :
1 2 3 4 : : :
1 2 3 4 5 : : :
1 2 3 4 5 6
1 2 3 4 5 6 7 2 3 4 5 6 7
1 2 3 4 5 6 7 2 3 4 5 6 7
1 2 3 4 5 6 7 2 3 4 5 6 7
1 2 3 4 5 6 72 3 4 5 6 7
1 2 3 4 5 6 71 2 3 4 5 6
1 2 3 4 5 6 7 1 2 3 4 5 6 7
1 2 3 4 5 6 7 1 2 3 4 5 6 7
1 2 3 4 5 6 7 1 2 3 4 5 6 7
1 2 3 4 5 6 7 1 2 3 4 5 6 7
1 2 3 4 5 6 7 1 2 3 4 5 6 7
1 2 3 4 5 6 7 1 2 3 4 5 6 7
1 2 3 4 5 6 7 1 2 3 4 5 6 7
1 2 3 4 5 6 7 1 2 3 4 5 6 7
1 2 3 4 5 6 7 1 2 3 4 5 6 7
1 2 3 4 5 6 7 1 2 3 4 5 6 7
1 2 3 4 5 6 7 1 2 3 4 5 6 7
1 2 3 4 5 6 7 1 2 3 4 5 6 7
1 2 3 4 5 67 1 2 3 4 5 6
1 23 4 5 6 1 2 3 4 5 6
1 2 3 4 5 6 7 1 2 3 4 5 6 7
1 3 4 6 1 2 3 4 5 6
1 2 3 1 2 3 4 5 6
1 2 1 2 3 4 5 6
S2 3 1 2 3 4 5 6
1 2 1 2 1 2 3 4 5 6
1 1 1 2 3 4 5 6
1 1 1 2 3 4 5 6
1 1 12 3 4 5 6
S1 1 2 3 4 5 6
1 1 1 2 3 4 5 6
S1 1 2 3 4 5 6
1 2 3 4 5 6
S1 2 3 4 5 6
1 1 2 3 4 5 6
1 1 2 3 4 5 6
1 1 : 1 2
11 31i 1 2
: 1 : : 1 1 2
: 1 1 *: 1 : 1 2
: 1 : 1 : 1 2
1 1 2
: : : : : : : : 1 2
: : : : : : : : 1 2
Maximal Bandwidth Maximal FLOS
FIGURE 4.2 FLOS DIAGRAMS FOR THE LEFTBOUND DIRECTION ON S. R. 26
FOR THE MAXIMAL BANDWIDTH AND FLOS OPTIMIZATIONS
TABLE 4.4 COMPARATIVE RESULTS OF MAXIMAL BANDWIDTH
AND FLOS OPTIMIZATIONS FOR S.R. 7
(TWELVESIGNAL SYSTEM)
MAXIMAL
MOE BANDWIDTH
Cycle Length (sec)
Bandwidths (sec)
Efficiency (%)
Attainability
FLOS
PQR = FLOS/CFLS
PQR = FLOS/TFLS
Delay (vehhr/hr):
Through on Artery
Total System
Percent Stops:
Through on Artery
Total System
Percent Red Arrivals
102
34a
17a
0.74a
1,720
0.20
1.15
90.9
210.6
63.8
72.3
48.5
a. These values are based directly on
remainder of this column are based
NORMAL
FLOS
102
36
10
0.75
2,762
0.21
1.17
92.2
212.0
65.9
73.4
48.8
PASSER
on the
PERCENT
DIFFERENCE
5.9
1.4
1.5
1.7
1.4
0.7
3.3
1.5
0.6
II output, the
TRANSYT6C/FLOS
analysis of this condition with the bandwidth, efficiency,
and attainability as indicated for the FLOS optimization.
L
As noted in Table 4.4, the bandwidths calculated by PASSER II to
taled 34 seconds (16 seconds rightbound and 18 seconds leftbound).
Since TRANSYT6C operates in steps rather than seconds (see Appendix B),
with a maximum of 60 steps per cycle, the resolution of a 102 second
cycle is reduced. Indeed, neither 16 nor 18 seconds can be achieved
exactly, given a conversion factor of 1.7 seconds per step. The signal
timing input to the TRANSYT6C/FLOS model thus produces bandwidths one
second greater than the PASSER II solution, in each direction.
It would not be meaningful to simulate a condition wherein the ini
tial bandwidths resulting from a TRANSYT6C/FLOS analysis were less than
those produced by PASSER II (namely, 15 and 17 seconds, respectively).
This could be easily accomplished by shifting one or two offsets, but
such a move is arbitrary, and further, the FLOS optimization would con
verge on the same solution reported earlier.
The salient conclusion resulting from this test is that the maximal
forward link opportunities solution was effectively no better than the
maximal bandwidth solution in terms of the progression opportunities;
and indeed, the shifts in offsets had a minimally detrimental effect on
stops and delay.
SixteenSignal System
A somewhat larger arterial system, in terms of the number of inter
sections, was tested for data from Beech Daly Road in Detroit, Michigan,
although the signal spacing is 21% less than the previous example.
The comparative results are summarized in Table 4.5. These results
are not particularly dissimilar from the previous test in terms of the
absolute magnitudes of the proportional changes. But, here all indica
tors tended in the direction of total improvement, albeit the improvements
TABLE 4.5 COMPARATIVE RESULTS OF MAXIMAL BANDWIDTH
AND FLOS OPTIMIZATIONS FOR BEECH DALY ROAD
(SIXTEENSIGNAL SYSTEM)
MAXIMAL NORMAL PERCENT
MOE BANDWIDTH FLOS DIFFERENCE
Cycle Length (sec) 87 87
Bandwidths (sec) 35a 36 2.9
Efficiency (%) 21a 21
Attainability 0.60 0.60 0
FLOS 5,574 5,574 1.6
PQR = FLOS/CFLS 0.27 0.27
PQR = FLOS/TFLS 0.52 0.53 1.9
Delay (vehhr/hr):
Through on Artery 94.8 93.6 1.3
Total System 209.0 208.3 0.3
Percent Stops:
Through on Artery 51.1 49.9 2.3
Total System 61.6 60.8 1.3
Percent Red Arrivals 35.8 34.9 2.5
a. These values are based directly on PASSER II output, the
remainder of this column are based on the TRANSYT6C/FLOS
analysis of this condition with the bandwidth, efficiency,
and attainability as indicated for the FLOS optimization.
were minor. Once again the roundoff error due to the lower resolution
of the TRANSYT6C/FLOS model has occurred. Indeed, the original band
widths in the PASSER II solution were 17 and 18 seconds for the right
and leftbound directions, respectively. These were converted to 17 and
19 seconds, respectively, in the PASSER II solution, as simulated by
TRANSYT6C/ FLOS. The final bandwidths were 16 and 20 seconds after FLOS
optimization. In this case, the FLOS optimization caused a slight (1
second) decrease in bandwidth, which, as was stated earlier, is not an
unexpected occurrence.
TwentySignal System
The largest system that can be analyzed by PASSER II is a twenty
signal system. Because of this, the TRANSYT6C/FLOS model was written
with a similar maximum (although it is expandable, up to 50 signals, at
considerable increase in run time, as explained in Appendix C).
The site used for this test is also a section of State Road (S.R.)
7, or U.S. 441, in Ft. Lauderdale, but this section is mutually ex
clusive of the twelvesignal system. The results of the test are given
in Table 4.6.
In this case all measures showed discernible improvement when FLOS
optimization was applied to the PASSER II solution. The most signifi
cant improvements were realized in the rightbound direction where FLOS
increased by 8% (5,254 to 5,658); delay and the fraction of red arrivals
decreased for through traffic on the artery by 9% and 11%, respectively;
and the number of through stops decreased by 6%.
The FLOS diagrams for the two solutions are shown in Figures 4.3
and 4.4. The staggered alignment of the rightbound greens, particularly
TABLE 4.6 COMPARATIVE RESULTS OF MAXIMAL BANDWIDTH
AND FLOS OPTIMIZATIONS FOR S.R. 7
(TWENTYSIGNAL SYSTEM)
MAXIMAL NORMAL PERCENT
MOE BANDWIDTH FLOS DIFFERENCE
Cycle Length (sec) 106 106
Bandwidths (sec) 51 51 0
Efficiency (%) 24 24
Attainability 0.88 0.88 0
FLOS 11,150 11,719 5.1
PQR = FLOS/CFLS 0.28 0.29
PQR = FLOS/TFLS 0.56 0.59 5.4
Delay (vehhr/hr):
Through on Artery 158.1 146.8 7.1
Total System 384.7 374.9 2.5
Percent Stops:
Through on Artery 47.8 46.1 3.6
Total System 58.6 57.2 2.4
Percent Red Arrivals 34.9 32.4 7.2
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from intersections twelve through twenty of the maximal bandwidth solu
tion (Figure 4.3), has been smoothed considerably in Figure 4.4 to
account for most of the improvement. The net increase in nonthrough
FLOS in this direction was 72%. The change in nonthrough FLOS left
bound was 19%, for a net total increase of 40%.
Trend Analysis
Comparing the relative results of the foregoing tests revealed no
discernible trends in the relationships between FLOS, delay, stops or
the percentage of red arrivals. The five case studies were sufficiently
diverse in nature that the results would appear to depend largely upon
the particular geometric and traffic compositions of the various test
sites.
An apparent trend emerged which relates the percentage change in
forward link opportunities (between the maximal bandwidth and FLOS
optimizations) with the dispersion of signals in the system. Table 4.1
contains the average intersection spacings of the five sites, and obser
vation of these indicates a good deal of variance among them. Average
spacing itself had no discernible correlation with the comparative
optimal solutions; however, if the variation of signal spacing within
the systems is considered, a trend is observed. Since the average
spacings differ in magnitude, the coefficient of variation (ratio of the
standard deviation to the mean) is used as a normalized measure of
signal dispersion. Plotting the coefficients of variation against the
percent change in FLOS results in the trend shown in Figure 4.5.
The plot suggests that in systems having more uniformly spaced
signals, the improvement that may be achieved under FLOS optimization,
compared to maximal bandwidth optimization, will tend to increase. This
8
0 8 (Number of signals)
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FIGURE 4.5 TENDENCY BETWEEN IMPROVEMENTS IN
FLOS AND COEFFICIENT OF VARIATION
OF SIGNAL SPACING
is logically explained in the nature of timespace relationships.
Namely, highly irregular spacings would tend to constrain the flexibili
ties of the offsets, thus, reducing the potential for improving signal
offsets for forward link opportunities over the maximal bandwidth solu
tion.
A specific test supports this contention. The example for twelve
traffic signals, State Road 7 in Ft. Lauderdale, Florida, experienced
the poorest degree of improvement among the five sites, all measures
considered. This arterial system had the largest average signal spacing
as well as the largest variation of signal spacing (1410 feet, compared
to a range of 410 feet to 1000 feet for the remaining sites). One link
in particular is extremely long (6170 feet), which is 2.5 times the
length of the remaining links. For the purpose of demonstration, this
excessively long link was reduced by onehalf and the system was tested
as before, with the results shown in Table 4.7. Comparing these results
with Table 4.4, which is the analysis based on actual geometric conditions,
the adjusting of one excessively long link length to reduce the variation
of link lengths has reversed all the earlier negative trends in traffic
operations and the improvement in FLOS was doubled, relatively speaking.
No mathematical relationship is hypothesized about the above findings.
Extensive testing would be required with additional data to establish a
firm relationship. On the other hand, it appears both reasonable and
rational to conclude that forward link opportunities are likely to be
more productive in more uniformly spaced signal systems.
Summary
The foregoing tests demonstrated that, for various sets of condi
tions, optimizations of signal offsets upon forward link opportunities
TABLE 4.7 COMPARATIVE RESULTS OF MAXIMAL BANDWIDTH
AND FLOS OPTIMIZATIONS FOR S.R. 7
(TWELVESIGNAL SYSTEM WITH ONE DISTANCE REDUCED)
MAXIMAL NORMAL PERCENT
MOE BANDWIDTH FLOS DIFFERENCE
Cycle Length (sec) 102 102
Bandwidths (sec) 34a 35 2.9
Efficiency (%) 17a 18 
Attainability 0.74a 0.75 1.4
FLOS 2,686 2,780 3.5
PQR = FLOS/CFLS 0.20 0.21
PQR = FLOS/TFLS 1.20 1.18 1.8
Delay (vehhr/hr):
Through on Artery 84.6 84.2 0.5
Total Systems 208.5 205.9 1.2
Percent Stops:
Through on Artery 62.2 61.4 1.3
Total System 70.7 69.9 1.1
Percent Red Arrivals 47.1 46.8 0.6
a. These values are based directly on PASSER II output, the
remainder of this column are based on the TRANSYT6C/FLOS
analysis of this condition with the bandwidth, efficiency,
and attainability as indicated for the FLOS optimization.
were virtually equal to or superior to the maximal bandwidth optimiza
tions in terms of the timespace relationships among signals. When
compared with the maximal bandwidth optimizations, FLOS optimizations
increased the number of forward link opportunities from over 1% to
nearly 8%. Monodirectional improvements were naturally higher.
In the majority of cases, the overall quality of traffic flow,
measured in terms of stops and delays, was similar or improved. In
isolated cases (and at one location in particular, the twelvesignal
system), the traffic measures declined slightly; but a rationale was
presented to explain this phenomenon. It was demonstrated that shifts
in offsets may have a deleterious effect on certain approaches which may
result in unsatisfactorily large numbers of stops and, thus, increased
delay. This may occur even if the percentage of arrivals during the red
phase is unaffected or reduced. This phenomenon is covered further in
the next chapter.
Several computational limitations of the TRANSYT6C/FLOS model in
its present form have been identified. These may affect the resolution
of the signal timing, particularly under conditions of long cycle
lengths.
On balance, however, substantial improvements were generally gained
in the regions outside the through bands, where the only areas for real
improvements exist. Considering the five cases collectively, as a
measure of overall potential of the new model, both the quality of
progression as well as the quality of traffic operations improved under
the FLOS optimization concept. The FLOS optimization concept would,
therefore, appear to have merit as a design strategy for progressive
signalization.
CHAPTER 5
EXTENDED APPLICATIONS OF THE FORWARD LINK
OPPORTUNITIES MODEL
Introduction
The previous chapters have demonstrated that the concept of maxi
mizing forward link opportunities to effect progressive signal designs
is conceptually and functionally feasible. Chapter 4, which included
tests of five typical system configurations, demonstrated that the
alteration of signal offsets alone can provide improved progression
opportunities. In addition, these evaluations have demonstrated that
the FLOS optimization strategy can improve system operations in most
circumstances tested. Finally, the TRANSYT6C/FLOS model clearly has
potential as a useful design and analysis tool in the traffic engi
neering profession.
In this chapter, the capabilities of the FLOS concept are expanded
through the use of capabilities existing in the TRANSYT6C/FLOS model.
The major concepts covered are the weighting of forward link oppor
tunities (FLOS) by various physical and traffic aspects, variations in
the explicit FLOS objective function and extended applications of the
model. Certain of these extended applications of the TRANSYT6C/FLOS
model identify solutions to the problem observed in Chapter 4 of shift
ing offsets that serve to increase delay and stops.
To insure continuity and consistency with earlier investigations,
the precepts of the PASSER II and TRANSYT6C models and their theoretical
bases continue to be promulgated in these investigations.
Weighting by Physical and Traffic Aspects
Forward link opportunities or their relationship to other system
opportunities are important measures of the overall quality of pro
gression on an arterial highway. Other considerations, however, might
suggest the favoring of various other elements of the system. The
weighting policies are discussed below (also see Appendix C).
Weighting by Link Length
The distance of unimpeded travel on a given link is important to
the motorist who values total time spent traversing a section of high
way. Specifically, a timeconscious driver may tend to be more satis
fied having successfully passed through an intersection which enabled
him to travel a more significant portion of his trip, than on a short
section where the potential of being stopped shortly downstream is more
evident. Consequently, the weighting of individual intersection FLOS by
the lengths of the downstream links would seem to be of importance. The
formulation of this weighting for one direction only (and assuming time
is in seconds) is,
N C
FLOS(linklength)r = E E FLOSit x li+1 ; (5.1)
i=l t=1
where li+1 = length of link i+1, and
the rest as before.
This weighting strategy was applied on three of the data sets that
were examined in Chapter 4 (the six, eight and twelvesignal systems).
These three conditions were used because the results of normal FLOS
optimizations in Chapter 4 were not totally consistent. The two larger
systems of sixteen and twenty signals yielded more consistent results in
the earlier application and thus, would not be expected to benefit as
much from weighting. Summarized results from weighting the three small
er systems, comparing the percentage change in aggregate MOE from the
maximal bandwidth solution to the FLOS solution, with and without
weighting, are listed in Table 5.1. The optimal solution for FLOS
weighted by link length was virtually the same as the solution for
unweighted FLOS in the two smaller systems. In the twelvesignal sys
tem, the adverse results reported earlier diminished slightly. But the
small increase in FLOS under unweighted FLOS optimization was decreased
even more under distanceweighted FLOS.
It would thus appear that this weighting strategy has little prac
tical value in the systems examined.
Weighting by Platoon Dispersion Factor
The complement of the above weighting strategy is one that weights
FLOS by the inverse of link length. This weighting accounts for the
motorist's desire to successfully traverse several closely spaced sig
nals, as opposed to successfully passing through only one intersection,
even though the passage enabled a perceived time savings. This policy
would tend to have a more rational basis in theory as well. As spacing
between intersections increases, dispersion of traffic increases (Hillier
and Rothery, 1967). There is also a greater tendency for traffic to
operate in a manner more common to isolated intersections than coordina
ted intersections where there are lengthy separations between intersec
tions (Papapanou, 1976).
A significant advancement in deterministic simulation of traffic
operations was made by Robertson (1969) in his use of the Platoon Dis
persion Factor (PDF), as discussed in Chapter 2 and Appendix C. The re
lationship is repeated below,
TABLE 5.1 COMPARISON OF MAXIMAL BANDWIDTH, UNWEIGHTED
FLOS AND LINK LENGTHWEIGHTED FLOS OPTIMIZATIONS
PERCENT CHANGE FROM MAX. BANDWIDTHa
6SIGNALS 8SIGNALS 12SIGNALS
MOE UWF WF UWF WF UWF WF
FLOS 1.3 1.3 7.7 7.7 1.5 1.0
Total Delay 0.5 0.2 1.5 1.5 0.7 0.3
Total Stops 1.4 0.5 2.2 2.2 1.5 0.4
a. The abbreviated column headings in this and subsequent tables
of this type are: UWF = unweighted FLOS optimization,
WF = optimization under FLOS, weighted as indicated.
TABLE 5.2 COMPARISON OF MAXIMAL BANDWIDTH, UNWEIGHTED
FLOS AND PDFWEIGHTED FLOS OPTIMIZATIONS
PERCENT CHANGE FROM MAX. BANDWIDTH
6SIGNALS 8SIGNALS 12SIGNALS
MOE UWF WF UWF WF UWF WF
FLOS 1.3 1.3 7.7 7.7 1.5 1.5
Total Delay 0.5 0.5 1.5 1.5 0.7 0.5
Total Stops 1.4 1.4 2.2 2.2 1.5 0.7
PDF = (1 + kt)1 ; (5.2)
where t = link travel time and
k = a coefficient, generally 0.5.
Equation (5.2) is essentially an inverse function of link length, if the
assumption of constant progression speed on the link is held. Because
this factor is recognized as an important component of the TRANSYT6C
simulation of traffic flow, and expresses the desired FLOS weighting
strategy, it is logical to use the PDF as the explicit weighting factor.
The formulation is similar to Eq. (5.1), except that the PDF for
link i+1 is substituted for li+1 in Eq. (5.1).
Tests using this weighting factor were also performed on the three
data sets used above (see Table 5.2). In the cases of the six and
eightsignal systems, the optimal solutions were virtually identical to
the solutions under unweighted FLOS.
It may be recalled that all tests of the twelvesignal system (a
section of S. R. 7 in Ft. Lauderdale, Florida) have exhibited less than
expected results under FLOS optimizations, except when the single exces
sively long link was shortened for a demonstration. When PDFweighting
is applied to the FLOS optimization of this system in its actual config
uration, the negative trend is reversed, as shown in the summary results
given in Table 5.2.
While the actual improvements in FLOS are identical, the net differ
ence in total delay between normal FLOS optimization and PDFweighted
FLOS optimization is slightly over 1%. Although all the tests on this
site have demonstrated very minor changes in the pertinent MOE, this ex
ercise illustrates the fact that FLOS optimization results can be im
proved somewhat by applying the PDF weighting factor.
Weighting by Total Demand
Demand weighting is common in traffic engineering analysis. Since
the FLOS concept in its simplest form is purely a timespace function,
no consideration is given to the relative demand when optimizing off
sets. It should be noted, however, that signal splits are proportioned
on the basis of demand, specifically the demand to capacity ratio; thus
demand is indirectly considered.
Weighting the FLOS according to demand would seem to have some
intuitive appeal. The function is formulated similarly to Eq. (5.1),
except the flows on the individual links are substituted for the down
stream link lengths in the equation.
When this weighting was tested on the three cases presently being
considered, the results indicate that the strategy does not improve the
solution. No FlOE were superior to those produced by normal FLOS optimi
zation, and in several cases, both FLOS and traffic operations were
adversely affected. Thus, direct weighting by link demand does not
appear, in these tests, to be a significant optimization strategy. This
strategy would probably be of greater significance under conditions
where traffic demands are highly imbalanced with respect to direction of
travel.
Weighting by Stopline Arrival Pattern
In Chapter 2, and above, the concept of platoon dispersion has been
discussed. Robertson's (1969) modeling of platoon dispersion is speci
fically described in Appendices B and C. The effect of platoon disper
sion is to predict an arrival pattern at the downstream end of a link,
considering the release patterns from the upstream inputs. The release
patterns are determined by the phasing (i.e., the order of release from
several approaches to the link) and by the travel time to the downstream
stopline, smoothed by the PDF discussed above, also see Eq. (C.3) in
Appendix C. As a result of this modeling approach, realistic momentary
flow rates at the stopline may be predicted.
Another important aspect of Robertson's flow model is that, as
stated previously, demand from cross streets which turn onto the artery
and become part of the through traffic stream are also included in the
arrival patterns. In a progressive system, linktolink through traffic
tends to form platoons which propagate downstream within the through
band. Other inputs (e.g., from side streets) tend to fall between these
through platoons in platoons of their own. Weighting of FLOS by the
actual arrival pattern thus considers the microscopic aspects of traf
fic, simulating, in effect, a "realtime" condition, as influenced by
activities occurring upstream of each signal.
Finally, consideration of the moving platoons in an optimization
policy is analogous to the realtime, online control strategies de
scribed in Chapter 2. If offsets can be set to insure that the largest
concentrations of traffic arrive on the green, preferably after the
queue has dissipated, good operations should result. As stated in
Chapter 2, this strategy often fails in realtime control systems where
the cycle lengths and offsets are changed dynamically. It could be
theorized, however, that the strategy would work better under fixedtime
control, provided the desired progression of platoons was achievable,
and given the constraints imposed by providing twoway bands.
To accomplish this, it is necessary to propagate traffic on each
link, from upstream to the signal of interest, in small slices of time
and to project these, more individualized, forward link opportunities
downstream from their time of arrival.
The arrival, or inflow, pattern (IFP) model is formulated for one
direction as,
N C
FLOS(IFP)r = z FLOSit x q ; (5.3)
i=1 t=l
where qit = the microscopic flow rate at signal i
at time interval t, a function of the upstream
demand, travel time and the platoon dispersion
factor (see Appendix C).
To illustrate the use of this objective function, data from State
Road 26 in Gainesville, Florida, were used. The objective function of
Eq. (5.3) was applied to the initial timings from the PASSER II optimal
bandwidth solution. Summary results are shown in Table 5.3. Because
bandwidths are quite likely to change under this optimization strategy,
the changes in bandwidths are also reported, as are the percentage of
through red arrivals on the artery, which are significant to this study.
The values presented are the relative effectiveness of unweighted
FLOS optimization and inflow patternweighted FLOS, both compared
against the maximal bandwidth solution. As observed in the table, the
quality of progression (measured by the number of FLOS), and the propor
tion of stops and through red arrivals decreased using the weighting
function. This demonstrates that the proposed strategy will not fully
produce the desired effects discussed above, even though total delay was
reduced further by this optimization.
The constraints upon the offsets were simply too severe to permit
improvements in the system design using this weighting factor alone.
TABLE 5.3 COMPARISON OF MAXIMAL BANDWIDTH, UNWEIGHTED
FLOS AND INFLOW PATTERNWEIGHTED FLOS OPTIMIZATIONS
PERCENT CHANGE FROM MAX. BANDWIDTH
MOE UNWEIGHTED WEIGHTED
Bandwidths 0 0
FLOS 7.7 2.5
Total Delay 1.4 2.5
Total Stops 2.2 3.5
Through Red
Arrivals on
Artery 11.2 1.9
TABLE 5.4 COMPARISON OF MAXIMAL BANDWIDTH, UNBIASED FLOS
AND LEFTBOUND BIASED FLOS OPTIMIZATIONS
PERCENT CHANGE FROM MAX. BANDWIDTH
UNBIASED LEFTBOUND BIASED
MOE RIGHT LEFT TOTAL RIGHT LEFT TOTAL
Bandwidths 0 0 0 21.7 0 9.6
FLOS 1.1 12.7 7.7 5.1 13.6 5.7
Total Delay 1.4 1.2
Total Stops 2.2 2.7
Through Red Arrivals 17.4 4.9 11.2 16.6 11.6 2.6
The same tendency was found in similar tests on other systems. In Chap
ter 4, it was noted that one link in particular (see Figure 4.1) had
experienced an increase in red arrivals from 16% to 40% under unweighted
FLOS optimization. In the present test, the weighted FLOS optimization
produced 23% red arrivals on this same link. However, this improvement
was offset by disimprovements elsewhere.
These trends observed would tend to support the conclusions re
ported in Chapter 2 with regard to the several dynamic, realtime con
trol systems. On the other hand, the inflow patternweighted FLOS
optimization strategy does have merit when considered in concert with
other objective functions. This is discussed later in this chapter.
Weighting by Direction of Travel
Finally, from the designers' point of view, it may be desirable to
favor one direction of travel over another, such as during peak periods,
irrespective of other considerations. This policy would, for example,
recognize the more critical needs of the commuter predominantly traveling
in the peak direction visavis the more casual traveler driving in the
direction of nonpeak period prevailing flow. The formulation is simply
a modification to Eq. (5.1), or any of its other variations, as follows:
FLOS = W1 x FLOS + W2 x FLOS2 ; (5.4)
where W = a weighting factor for the two directions of travel and
the subscripted FLOS are as before, without the directional
summation.
To illustrate this weighting strategy, the State Road 26 example
was optimized on FLOS with the coefficients of the rightbound and left
bound FLOS set to unity and ten, respectively. This approach would be
used to favor the leftbound direction which is the primary direction
of travel in the afternoon peak period. The FLOS diagrams for the
system with unbiased FLOS optimization and directionally biased optimi
zation are shown in Figures 5.1 and 5.2, respectively. Summary results
are given in Table 5.4. An unexpected outcome resulted from this test.
While FLOS were increased significantly leftbound (as seen in Figures
5.1 and 5.2) and the systemwide totals of stops and delay changed only
imperceptibly, the changes in percentages of red arrivals were reversed
from what would be expected. Examination of the arrival patterns for
the leftbound links reveals that all platoons arrived at the intersec
tions coincidentally with the start of green, when stationary platoons
were just being released. Thus, the platoons were stopped and delayed.
The nearly horizontal edge of the beginning of the through band left
bound (seen in Figure 5.2) accounts for this, and this phenomenon sug
gests that the queue clearance strategies of the realtime models dis
cussed in Chapter 2 would have an application in this approach, at least
for the one direction.
In further support of the queueclearance theory, it is observed
that the rightbound progression pattern is funnel shaped with a large
diverging band which does advance the start times of the green phases,
thus clearing the queues ahead of the platoons.
This effect is not necessarily universally applicable, since the
resulting FLOS for this case are constrained by the splits provided.
However, it is clear that, in this example, the leftbound greens could
be shifted by advancing offsets increasingly from right to left, which
would rotate the leftbound funnel to a more horizontal orientation.
The desired effect would thus be achieved.
