thesis wu en sch 2008

8/11/2019 Thesis Wu en Sch 2008

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Coordination of TrafficSignals in Networks

and

Related Graph Theoretical Problems on SpanningTrees

vorgelegt vonDipl.-Math. Gregor Wunsch

Von der Fakultat II Mathematik und Naturwissenschaftender Technischen Universitat Berlin

zur Erlangung des akademischen Grades

Doktor der Naturwissenschaften Dr. rer. nat.

eingereichte Dissertation

Vorsitzender: Prof. Dr. Rainer WustBerichter: Prof. Dr. Rolf H. Mohring

Prof. Dr. Ekkehard G. Kohler

Berlin 2008D 83


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ACKNOWLEDGEMENTS

This work would not have been possible without the help of a number of people. Iwish to thank everybody who has been involved, directly or indirectly.

First of all, I thank my supervisor Rolf Mohring. I am grateful for his support andencouragement and I am especially indebted to him raising my interest for discreteapplied mathematics during my years as an undergraduate.

In addition, I wish to thank Ekkehard Kohler for taking the second assessmentof this thesis and for numerous discussions and stimulating ideas on traffic modelsinvolving traffic signals.

Next, I wish to thank Klaus Nokel from PTV AG in Karlsruhe. During the lastapproximately four years he and the PTV supported the work on an optimizationtool for coordinating signals in various ways. Not only that we were equipped withsoftware, but also I enjoyed our discussions and profited from valuable suggestions.

Also, I am very grateful for the financial support that I received within theDFG research training group MAGSI (GK-621). I thank all my colleagues fromMAGSI for a fruitful interdisciplinary research environment. Moreover, I thankArmin Zimmermann for organizing almost everything concerning the activities withinMAGSI.

Furthermore, I am indebted to my coauthors Ekki Kohler, Rolf Mohring, KlausNokel, Alexander Reich and Romeo Rizzi and, especially, Christian Liebchen, whoraised my interest for strictly fundamental cycle bases. Yet, from all of them, Ilearned a lot during intensive discussions.

I also want to thank all members of the research groups of Rolf Mohring, GunterZiegler and Stefan Felsner for the excellent working atmosphere. Here, special thanksgo to my office mates Felix Konig, Heiko Schilling and Bjorn Stenzel. I very muchenjoyed our numerous chats on various things like sports, elections, statistics anddiscrete mathematics.

Moreover, I would like to thank Christian Liebchen, Nina Brenner, Tobias Harks,Richard Lutjens, Nicole Megow, Guido Schafer, Izaskun Seara Tejados, Bjorn Sten-zel, and Sebastian Stiller for their careful proofreading of the manuscript. The ex-position of this thesis improved from their valuable suggestions and comments.

Last, but surely not least, I am grateful to my friends and my family for theirsupport and their interest in my work.

Berlin, February 2008 Gregor Wunsch

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CONTENTS

Introduction 1

1 The Network Signal Coordination Problem 7

1.1 Introduction to Coordination of Signals in Networks . . . . . . . . . . 7

1.1.1 The Language of Traffic Signals. . . . . . . . . . . . . . . . . . 7

1.1.2 Optimizing Traffic Lights . . . . . . . . . . . . . . . . . . . . . 9

1.1.3 Software to Optimize Coordination . . . . . . . . . . . . . . . . 11

1.2 The Network Signal Coordination (NSC) Problem . . . . . . . . . . . 13

1.2.1 Definition of the NSC . . . . . . . . . . . . . . . . . . . . . . . 13

1.2.2 The Relation Between the NSC and the PESP . . . . . . . . . 15

1.2.3 NP-completeness of the c-NSC problem . . . . . . . . . . . . . 17

1.3 A Revised MIP Formulation for the NSC problem . . . . . . . . . . . 19

1.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.3.2 The Offsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.3.3 The Cycle Equations . . . . . . . . . . . . . . . . . . . . . . . . 24

1.3.4 A Variable Phase Sequencing . . . . . . . . . . . . . . . . . . . 27

1.3.5 The Objective Function . . . . . . . . . . . . . . . . . . . . . . 28

1.3.6 Non-uniform Cycle-lengths . . . . . . . . . . . . . . . . . . . . 31

1.3.7 The Mixed-Integer Linear Program . . . . . . . . . . . . . . . . 36

1.4 Application of the NSC Model in Practice . . . . . . . . . . . . . . . . 37

1.5 Conclusion and Open Questions. . . . . . . . . . . . . . . . . . . . . . 39

2 Strictly Fundamental Cycle Bases on Grids 41

2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.2 Prelimenaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.3 Lower Bounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.3.1 A New Asymptotical Lower Bound . . . . . . . . . . . . . . . . 45

2.3.2 The Challenge of Small Grids . . . . . . . . . . . . . . . . . . . 57

2.3.3 The Amaldi MIP for the General MSFCB Problem . . . . . . . 60

2.3.4 A new MIP formulation . . . . . . . . . . . . . . . . . . . . . . 66

2.3.5 A Tight Bound forG8,8 . . . . . . . . . . . . . . . . . . . . . . 70

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vi CONTENTS

2.4 Upper Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

2.4.1 A New Asymptotical Upper Bound. . . . . . . . . . . . . . . . 792.5 Experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 842.6 Conclusions and Open Questions . . . . . . . . . . . . . . . . . . . . . 87

3 Classification of Tree Spanner Problems 913.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913.2 A Unified Notation for Tree Spanners (UNTS) . . . . . . . . . . . . . 933.3 Maximum StretchProblems . . . . . . . . . . . . . . . . . . . . . . 96

3.3.1 Coincidences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963.3.2 Anticoincidences . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3.4 Average Stretch Problems. . . . . . . . . . . . . . . . . . . . . . . 1013.4.1 Coincidences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013.4.2 Anticoincidences . . . . . . . . . . . . . . . . . . . . . . . . . . 102

3.5 Max-Stretch AndAverage-Stretch Problems Never Coincide . . 1083.6 First Benefit of the UNTS . . . . . . . . . . . . . . . . . . . . . . . . . 110

3.6.1 An Open Complexity Status . . . . . . . . . . . . . . . . . . . 1103.6.2 Inapproximability of the MMST Problem . . . . . . . . . . . . 114

3.7 Conclusions and Open Questions . . . . . . . . . . . . . . . . . . . . . 114

4 Experiments 1174.1 A MIP Solver Comparison on Selected NSC Instances . . . . . . . . . 1174.2 The Influence of Cycle Bases on the MIP Performance . . . . . . . . . 1 2 34.3 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

4.3.1 Evaluating an NSC model . . . . . . . . . . . . . . . . . . . . . 1284.3.2 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . 1304.3.3 Portland. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1324.3.4 Denver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

4.4 Conclusion and Open Questions. . . . . . . . . . . . . . . . . . . . . . 139

Bibliography 143


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INTRODUCTION

Are you the type of person that likes waiting at a red light? Do you enjoy beingstopped by uncoordinated signals? If not, you are invited to read on to find out howthe coordination of traffic signals can help to reduce delays and, thus, avoid havingto wait at red lights.

In urban areas there is a strong demand for transportation. Probably, the mostsustainable means of transportation are the public ones, like buses, trams or theunderground. Nevertheless, not all demands for transportation can be covered bythe public sector. A major part of the overall transportation in cities is composed ofindividual drivers.

There are different ways to improve road traffic conditions in city areas. However,infrastructural arrangements like broadening streets or even building new ones tocope with increasing demands are often not an appropriate option, due to high costsor space limitations. Instead, intelligent means of traffic control are required to solvetodays road traffic problems, like high delays, narrow capacities or traffic jams.

When speaking of ways to control traffic, traffic lights or traffic signals are ofprimary importance. A clever adjustment of the signal settings surely helps to re-duce delays and increase capacities, thereby avoiding traffic jams. Today, intelligentcomputer-aided traffic control signals are even capable of reacting to different traf-fic situations. Namely, they adapt their settings to the respective demands at the

junctions.

However, there are traffic scenarios where the traffic responsive signals reachtheir limit. For example, when there is constant and high traffic volume, the re-sponsive signals repeatedly apply similar control strategies. Therefore, they behavecomparably to fixed time traffic signals. These fixed time traffic signals repeatedlyrespond to a prescribed signal timing program and not to the actual traffic condi-tions. Hence, research into fixed time traffic signals and their control strategies is anongoing endeavor.

Operating fixed time signals offers different means of controlling traffic. On theone hand, some signal parameters adjustments influence the traffic flow locally at asingle junction. In many situations, such a local calibration of the signals turns outto be sufficient to cope with the aforementioned problems. On the other hand, in

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2 Introduction

situations with constant high traffic, another non-local control strategy for the fixed

time signals becomes more significant: the coordination of the signals.Coordinating traffic signals means the following: coupling of signals via a param-eter called offset. This quantity specifies how green phases of different signals areshifted (or offset) to each other. Most prominent coordination objectives are so-calledgreen waves, where vehicles travel without being impeded by a signal showing red.Nevertheless, when considering networks of signals instead of arterials of signals, itis often not possible to adjust green waves for the whole network. Instead, the goalgreen wave has to be replaced by a more practical term like minimum possibledelay. Hereby, the item delay refers to waiting times of vehicles facing red at thesignals.

Many approaches and models have been proposed in order to find good coordi-

nations of signals in networks. Still, the majority of them reveal shortcomings eitherway, be it unrealistic modeling of real-world circumstances or the fact that they donot give guarantees for their solution quality.

To summarize, there is a need for a mathematical optimization approach forcoordinating fixed time traffic signals in networks.

This discussion on new required control strategies for fixed time signals is nota theoretical one. Rather, the industry, i.e., traffic companies that plan, manage,and control traffic, demands applicable approaches for coordinating traffic signals innetworks.

As an indication thereof, we briefly report on an industry project that emerged be-tween the TU Berlin and the PTV AG, which is a traffic planning software company

from Karlsruhe, Germany. In this project, the aim was to develop mathematicaloptimization software to coordinate fixed time traffic signals in networks. Duringthe project, we developed a mixed-integer linear programming approach, which min-imizes the delay of vehicles in a network by adjusting optimal offsets. However,several other functionalities were incorporated in the model. The outcome of theproject with the PTV, though, is a concrete implementation of the optimizationapproach, which is about to be included in PTV software soon.

In our mixed-integer linear program (MIP) for the coordination of traffic signals,a particular physical constraint has been modeled. This constraint, which we willtherefore call Cycle Constraint, has to be formulated for all cycles C of thegraphGthat represents the network of signalized junctions. It suffices, however, tostate the cycle constraints for the elements of a cycle basis. This then implies theseconstraints for allcycles ofG. Depending on the respective application, though, ithas to be a cycle basis with a certain property. In our case of a MIP for coordinatingfixed time signals in networks one has to define the cycle constraints for the elementsof an integralcycle basis.

This means that any integral cycle basis can be used to define the cycle constraintsfor our MIP. Although any two integral cycle bases lead to MIPs with equal optimalobjective value, the computational behavior of their MIPs may be different. Observethat this may be of importance, since we are considering networks of large size whereone may not come up with optimal solutions.


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Introduction 3

A quantity one can use to compare cycle bases is the so-called widthof a basis.

Loosely speaking, the width of a basis is defined as the productover all cyclesof a basisof the number of possible values that the integer variable for the cycleconstraint for that cycle can take. Thus, the width of a basis gives an impression ofthe size of the MIPs feasibility region. The hope is that the smaller the width of abasis, the better the corresponding MIP performs.

Among the class of integral cycle bases strictly fundamental cycle bases are aprominent subclass. For a graph G = (V, E), a strictly fundamental cycle basis B isdefined by a spanning treeT ofG. In particular, the cycles ofB are exactly the onesinduced by non-tree edges ofTwith respect to the graph G. Then, in the MinimumStrictly Fundamental Cycle Bases (MSFCB) problem one seeks a spanning tree Tthat induces a basis of minimum length.

In 1982, Deo et al. [DKP82] proved the MSFCB problem to be NP-completefor general graphs. Since then, many heuristics for the MSFCB problem have beenproposed. Nevertheless, for comparing the results of these heuristics, i.e., wheneverconcrete experiments were conducted, sample graph classes were considered. Besidesrandom graphs, grid graphs are the most important such graph class.

Grid graphs are also of interest for the two following reasons. First, consideringthe coordination of traffic signals, many real-world networks have a grid-like struc-ture. One only has to think of the layout of central areas in north american cities.Second, for the MSFCB problem, grid graphs turn out to be computationally tricky.This fact is probably due to an extreme amount of symmetric spanning trees ongrids.

In 1995, Alon et al. [AKPW95] proved that for square grids with n vertices, thesize of an optimal solution to the MSFCB problem is in (n log n). Still, we decidedto investigate bounds on the optimal value of an MSFCB on a square grid havingthe form

c1 n log2 n o(n log n) OPTn c2 n log2 n + o(n log n).We could prove that the above statement is true for c1 = 1/12 and c2 = 0.979,respectively.

An optimization problem that is closely related to the MSFCB problem is theone of finding a t-tree spanner with minimalt, [CC95]. In this problem, one seeks a

spanning treeT for a given general graph G, such that the maximum over all pairsof vertices (u, v) V V\ {(v, v)|vV}of the ratiodT(u, v)/dG(u, v) is minimal.Here, dG(u, v) refers to the length of a shortest path between u and v in G. ThequantitydT(u, v) denotes the length of the path between u and v in T.

The relation between finding a minimal t-tree spanner of a graph and an MS-FCB can be noticed when considering the following unified notation for tree span-ner (UNTS) problems. In the UNTS, a problem is defined through a triple

(goal,domain,term) .

Here,goalis either the maximum stretch or the average stretch. Second, asdomain,either all non-tree edges or all edges or all pairs of vertices are considered. Finally,


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4 Introduction

termmay be one of the following four: dT(u, v) or dT(u, v)/dG(u, v) ordT(u, v)+w(e)

ordT(u, v)/w(e), withw(e) denoting a weight of an edge. Although not all combina-tions ofgoal, domainand termare possible, there remain 20 tree spanner problems,classified by the UNTS. Interestingly, these 20 notationally different problems col-lapse to 12 with a general weight function w and to only five, when considering0/1-weights on the edges.

Among these five problems that do not coincide even in the unweighted case,arebesides the MSFCB problemprominent optimization problems like the Min-imum Average Stretch Spanning Tree Problem[PT01], the Shortest Total PathLength Spanning Tree Problem [DKP82, WCT00] and the Minimum DiameterSpanning Tree Problem [HL02].

Generally speaking, the UNTS provides a classification of related problems, which

had not been realized as such before. Hence, interconnections can be revealed andproperties like complexity status or inapproximability factors can be carried forwardbetween the problems.

The chronology of topics within this introductory part is reflected in the organi-zation of the thesis.

Outline of the Thesis

In Chapter1,we investigate the Network Signal Coordination (NSC) problem. Aftera short introduction of the most important traffic engineering terms related to trafficsignals, we first examine a study of related work. In the case of the NSC problemthis turns out to be of importance in order to clearly restrain the problem fromother optimization tasks regarding traffic signals. Thereafter, we formally define theNSC problem and report on similarities to the related Periodic Event SchedulingProblem (PESP). Moreover, we take advantage of the PESP in order to prove theNSC problem to be NP-complete. Then, in the main part of the chapter, we presenta model for the NSC problem. In particular, we develop in detail a mixed-integerlinear programming (MIP) approach to solve the NSC problem. We conclude thechapter with a discussion of possible applications in practice of an NSC model in

general and the MIP approach in particular.

In the mixed-integer linear programming formulation for the NSC problem, thesub-problem of finding appropriate integral cycle bases arises. In Chapter 2, weconsider the problem of finding Minimum Strictly Fundamental Cycle Bases (MS-FCB) on grid graphs. In particular, we investigate lower and upper bounds for thisproblem. As for the lower bounds, we consider both combinatorial approaches andmixed-integer linear programming formulations of the problem that we enrich withseveral additional cuts. Thereafter, we consider upper bounds for the MSFCB prob-lem on grids. In particular, we construct trees by making intensive use of recursivelydefined sub-structures. We conclude the chapter with an experimental section in


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Introduction 5

which we provide benchmark results for the MSFCB problem on grids which help

evaluating further research.The MSFCB problem can be interpreted as a problem of finding a spanning

tree that minimizes the sum of path lengths between particular pairs of verticesin a given graph. Interestingly, finding minimum average stretch tree spanners ormin-max stretch tree spanners of graphs can be interpreted in a very similar way.In Chapter 3, we provide a classification of several problems that aim at findingspanning trees in a graph, which minimize the average or the maximum value ofcertain distances between particular pairs of vertices in a graph. We propose aunified notation for these problems, which include several prominent problems incombinatorial optimization. With this notation at hand, we identify all coincidencesand anti-coincidences of these problems. Moreover, we provide a missing complexity

status for one of the problems and observe that an inapproximability result of one ofthe problems can in fact be applied to another problem too, where it had previouslybeen unknown.

In Chapter4, the experimental work that is related to the Network Signal Co-ordination (NSC) problem is presented, thereby coming full circle back to the firstchapter. The experiments conducted are threefold: First, we perform a solver com-parison at some example instances for our MIP model. Here, we compare the MIPsolvers CPLEX, MOPS, and SCIP with respect to their ability to find good solutionsin short time. In particular, we run two series of experiments using once the defaultMIP solver settings and once settings that emphasize the finding of good primal

solutions. Second, we report on the influence of cycle bases to the computationalbehavior of the MIP for the NSC problem. This experiment is of general interest,because a positive influence of short bases on computation times of mixed-integerprogramming formulations of practical applications is expected although very fewstudies actually proved it. So, in particular, we investigate the correlation betweenthe width of a cycle basis and the lower bound obtained by a MIP computationof 10 seconds. Finally, the third series of experiments is probably the most impor-tant one: we evaluate our model by carrying out case studies. Namely, we considerthe real-world inner city networks of Portland and Denver and compare the resultsobtained by our optimization approach with results found by other means. For thesecomparisons, we use the microsimulation tool VISSIM.

How to read this thesis

The thesis is chronological in structure. However, the chapters can be followedindependently, too. Chapter 2 and Chapter 3 come with their own introductionand consider related, but individually presented, problems. Furthermore, these twochapters do not explicitly require the reading of the Chapters 1 and 4. On the otherhand, the Chapters1 and 4 are strongly related and we recommend that Chapter 1is read prior to Chapter4.


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6 Introduction

Moreover, we give a chapter outline at the beginning of each chapter. Also,

conclusions are drawn and open questions are raised at the end of each chapter.

A further remark

We assume the reader of this thesis to be familiar with the basic concepts in lin-ear and integer programming, graph theory, and complexity theory. For additionalinformation on linear and integer programming we refer to [Sch86, NW88]. Goodtextbooks on graph theory are for example [Wes96] and[Die00]. Moreover, conceptsin complexity theory that are necessary to follow this thesis are covered by [GJ79]

and [HO02].As for the parts that deal with traffic engineering concepts or with traffic signalterms in particular, we refer to Section 1.1.1 for short textual explanations of themost important terms. Herewith, following most parts of Chapter1and Chapter4should be unproblematic. Of course, while developing our mixed-linear integer pro-gram in Section1.3, we give formal definitions of all relevant terms, too. However,additional information can be found in Richtlinien fur Lichtsignalanlagen RiLSA,Lichtzeichenanlagen fur den Straenverkehr[ril92] and in the Highway CapacityManual [hcm00].


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1THE NETWORK SIGNAL

COORDINATION PROBLEM

In this chapter, we consider the Network Signal Coordination (NSC) problem. TheNSC problem was introduced in 1975 by Gartner et al. [GLG75a], though manysimilar optimization tasks were known and have been defined already a lot earlier.After a brief summary of the most important terms concerning traffic signals in Sec-

tion 1.1.1, we give a short overview of the most significant optimization problemson traffic signals in Sec. 1.1.2, also to be able to clearly define the NSC problemand to restrict it from other problems. Then in Sec. 1.2.1 we formally define theNSC problem and illuminate similarities to the Periodic Event Scheduling Prob-lem (PESP) in Section 1.2.2. We report on the complexity of the NSC problem inSection 1.2.3. Thereafter, in Sec. 1.3 we develop in detail a revised mixed-integerlinear programming (MIP) formulation for the NSC problem. Finally, we explain apossible application of our MIP in practice, see Section1.4. Parts of this chapterwere published in [MNW06].

1.1 Introduction to Coordination of Signals in Networks

Before we define the Network Signal Coordination Problem, we introduce the mostimportant terms related to traffic signals and give an overview of what kinds ofoptimization tasks concerning traffic signals have been considered so far. Othersurveys on the topic are provided for example by[SS95,tft] or contained in[Lam07].

1.1.1 The Language of Traffic Signals

There is no unique language in the field of traffic engineering in general, and neitherin topics related to traffic signals. Rather, the terms and notation depend on the

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8 The Network Signal Coordination Problem

respective country and language. However, since following this thesis requires only

basic knowledge of traffic terms, in this section we give only a short textual descrip-tion of the most important terms. Notice that we do not give formal definitions here.For those, we refer to section 1.3. Nevertheless, whenever it is possible, i.e., whenwe do not work with a term, but only want to give an intuition, we omit a formaldefinition at all. For complete information see[hcm00]and [ril92].

In traffic engineering one considers traffic, i.e., vehicles, moving through a singleand isolatedjunction, anarterial, which is a, possibly bi-directionally traversable,series of junctions, or through a whole network, i.e., an arbitrary set of junctions.Then, one distinguishes different types ofsignals at the junctions. Here, the termsignal refers to all signaling devices at a junction. Roughly speaking, the followingtwo types are the most important ones. On the one hand, there are traffic respon-

sive signals. At these signals, the signal settings react on the present traffic. Onthe other hand,fixed-time (controlled) signals do not react on the actual traffic.Here, after a prescribed time span, called cycle length, the pattern of red phaseand green phase repeats. The particular division of a cycle length into a red and agreen is referred to as (red green) split. At a fixed-time signal, there are usuallydifferent signal groups that control the traffic for particular directions. The greenphases of different signal groups are shifted against each other, since they usuallycontrol competing traffic streams. In addition, the order of the signal groups ata signal is called phase sequencing. See Fig. 1.3 on page 20 for an example ofa signal timing plan in which the relevant data for one fixed-time traffic signal ismerged.

A very important term is the one of an offset. The(inter node) offset deter-mines how different signals are operated or shifted relatively to each other. Thatmeans the following: at each signal there is a marked out reference point, whichsometimes is the begin of the green phase of the first signal group. Then, the offsetdenotes the time span between reference points of two signals at two consecutive

junctions. In this case, there is an offset for each pair of consecutive signals. How-ever, the offset can also be defined for one single signal. Then, it determines thetime span between this signals reference point and a given network-wide zero refer-ence point. See Figure 1.4 for an illustration of both types of offsets. A sketch ofthe intra-node offset, which determines the shifting of different signal groups atone signal, is depicted in Figure 1.5 for example.

Of course, when considering signalized junctions, arterials or networks, severaloptimization tasks come to mind. Generally, one is interested in optimizing signalsettings in order to achieve a certain goal. Such signal settings are the red greensplit, the phase sequencing, the cycle length, and the offset. As for the goals toachieve, for example, minimizing the delay or maximizing the bandwidth have tobe mentioned. Here, the term delay refers to the delay that is due to the signalization,i.e., delay that occurs when vehicles have to wait because of a red. On the otherhand, maximizing bandwidth means that the signals along an arterial or within anetwork are adjusted, such that a preferably wide possible corridor through the greenphases of consecutive signals exists, within which the vehicles do not have to stop atthe signals at all. Such a corridor is sometimes called a greenband. See Figure1.1


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1.1 Introduction to Coordination of Signals in Networks 9

for a visualization of greenbands.

Whenever the offsets are included in the signal settings to be optimized, we saythat we optimize the coordination. In the literature the term synchronizationis sometimes used synonymously. However, we prefer the term coordination andleave the item synchronization to cases where the offsets and the cycle length areoptimized.

When considering traffic flow one distinguishes between amicroscopicview anda macroscopic view. The model is said to be microscopic if each individual vehicle isconsidered. On the contrary, we speak of a macroscopic model or approach, if the ve-hicles are aggregated in some sense. For example, it is popular to considerplatoonsof vehicles, that is, groups of consecutive vehicles close together that are treated asone quantity. However, it has to be mentioned that there are traffic models for which

a classification into one of the two views is not obvious.

1.1.2 Optimizing Traffic Lights

Since the introduction of automatic traffic signals in the 1920s, much work andresearch has been done on modeling, analyzing, and later also on simulating andoptimizing traffic signals. In this section we mention the most important modelingand optimization approaches. Notice, however, that we do not claim to provide acomplete overview.

When talking about optimization in the context of traffic signals, one faces manydifferent optimization tasks. Table 1.1 gives a glimpse of possible differentiations

between them.

One of the first important scientific publications on traffic signals was by Web-ster [Web58]in 1958. In this pioneering work, he prepared the ground for analyz-ing single traffic signals, e.g., by providing delay-estimating formulae that are, in aslightly changed form, still in use today. Using this formulae, Webster also researchedon minimizing the delay by adjusting optimal green proportions at a signal.

Then, during the 1960s research was no more restricted to one single junction, butrather arterials and networks of signals were considered. In 1963, Newell [New64]investigated the coordination of signalswith certain assumptions on the densityof trafficalong an arterial of one-directional traffic. In his macroscopic model he

suggests that best coordination is achieved simply by coordinating two consecutivesignals at a time.

Morgan and Little [ML64] and Little [Lit66] then developed an optimizationmodel that maximizes bandwidth, for the first time using mixed-integer linear pro-gramming. In their approach, the authors adjust optimal values to offsets, a commonsignal cycle length, and progression speeds, considering a bi-directional arterial ofsignals.

In the mid-1960s a first big field-study in coordinating a real traffic networkwas carried out in the city of Glasgow, Scotland. In this experiment, conductedby Hillier [Hil65, Hil66], different types of signal controllings were tested in innercity sub-network in Glasgow to evaluate their benefit. Later, in 1967, Hillier and


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Table 1.1: The table provides criteria to distinguish between mathematical ap-proaches for problems dealing with traffic signals. Observe, however, that not allcombinations are reasonable.

Criteria Possibilities

type of approach optimization, heuristic (genetic algorithms, localsearch etc.)

variables offset, red-green split, cycle length, phase se-quencing, travel speed, routes, almost any com-bination thereof

objective minimizing delay, number of stops, fuel con-sumption; maximizing greenband; combinationsthereof

type of signal fixed-time signals, traffic responsive signals,both

type of approach theoretical, practicalapplication on single junctions, arterials, networkspreconditions on traffic public only, individual only, nonepreconditions on demand high demand only, low demand only, nonepreconditions on signalization common cycle length, nonemodeling perspective macroscopic, microscopic

Rothery[HR67]analyzed the relation between platooning behavior of vehicles andcoordination. In particular, they calculate total delay as a function of the offsets.For this study the authors investigate four signalized junctions in London, England.

At the same time, a graph theoretical model for optimizing the coordinationin order to minimize the delay was developed by Allsop [All68]. In his article heproposed an iterative approach that successively expands the considered sub networkfor which a solution had already been found.

In 1969 the theoretical work for one of the until now most widely used softwaretools for the optimization of coordination was published by Robertson [Rob69]. Werefer to the next section for a more detailed discussion of the properties and attributesof TRANSYT.

Then, in a series of publications Gartner [Gar72] and Gartner et al. [GLG75a,GLG75b,GLG76]developed an mixed-integer linear programming approach for net-work coordination. In fact, the optimization model that we introduce in Section1.3is based on their approach.

At the same time, another approach evaluated the possibilities of linear pro-gramming for optimizing traffic signal settings. Antoniadis [Ant75], though, did notinclude the offset into his model.

A next, also from the theoretic point of view, important approach was the oneby Improta and Sforza[IS82] in 1982. There, the authors developed a mixed-integerlinear program, similar to the one by Gartner et al. [GLG75a], but included a branch


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1.1 Introduction to Coordination of Signals in Networks 11

and backtrack method that relaxed certain assumptions on the delay functions, which

had been made by Gartner et al.Dauscha et al. [DMN85], then introduce in 1985 a very general problem frame-

work for finding cyclic schedules of task systems. Nevertheless, their problem for-mulation is strongly motivated by coordinating traffic signals. In addition, they arethe first who report on the complexity of scheduling periodic events or coordinatingsignals, respectively.

In 1989, Serafini and Ukovich provide two important contributions. In [SU89b],they develop an involved mathematical model specially tailored for the fixed-timetraffic control problem, i.e., for the coordination of fixed-time signals. Second,in [SU89a] they present a widely noticed general approach that schedules periodicevents, which is, though, very similar to [DMN85].

Then in 1991, a second computer-aided approach, named SCOOT1, was publishedby Robertson and Bretherton [RB91]. However, in contrast to TRANSYT, SCOOTapplies to adaptive traffic signals.

Later, in 1996 Hassin [Has96] presents a flow algorithm approach to the synchro-nization of networks with an explicit application to the synchronization of fixed-timetraffic signals. The approach consists of a local search heuristic and, moreover, acharacterization of local optima is given. In addition, comparisons to other heuristicapproaches like TRANSYT were carried out on a network in the city of Tel Aviv,Israel.

A heuristic optimization approach that bases on genetic algorithms was proposedby Almasri and Friedrich[AF05] in 2005. There, the authors use a cell transmis-sion model which was originally introduced by Daganzo [Dag95]. However, onlyadaptively controlled signals are considered.

In 2005, Braun and Weichenmeier[BW05] introduce a second heuristic approach.In their model, the authors consider several signal settings, including offsets, andpropose a genetic algorithm method. Nevertheless, they report on test runs of theapproach on networks of small sizes only.

In order to show that modeling and optimizing traffic signals indeed attractedresearchers from different fields of expertise we give the following two citations, too:

in 1994 Ianigro [Ian94] developed a traffic model by using petri nets. Then, heused this model to implement a simulation that finds optimal signal settings for theconsidered traffic network. A second interesting approach is the one by Gershen-son [Ger05]. In 2005 he considered traffic flow in a signalized network in the contextof self-organizing systems, i.e., he investigates self-organizing traffic lights that adaptto changing conditions.

Other important contributions that we did not mention in detail before are [Sto68,CI88]. For an broader overview we refer the reader to [KN03].

1Split Cycle Offset Optimisation Technique


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1.1.3 Software to Optimize Coordination

Whereas the last section was intended to give an overview of optimization approachesto traffic signal related tasks, this section explicitly reflects some software packagesthat optimize the coordination of fixed-time signals, both on arterials and withinnetworks. However, again, we do not claim completeness, but rather mention the toour knowledge three most important ones: TRANSYT[Rob69], MITROP [GLG76]and SYNCHRO[syn00].

Remark 1.1. Though nowadays, coordinating traffic signals is done mostly com-puter aided, still, some techniques are in use, where computers only provide graph-ical support. For example, especially for arterials, coordinating via a graphic-basedby-hand approach that visualizes greenbands in dependence of offsets is common, see

Figure1.1.

Figure 1.1: A still today quite usual method to determine a good coordination onarterials: a graphical approach3 by hand. Notice the greenbands in green and bluefor opposite traffic.

Unquestionably, the software tool TRANSYT4 is state-of-the-art with respect torelevance for practitioners who want to model, analyze, and optimize traffic signalsettings for a network. The wide acceptance of TRANSYT as well as its importanceis stressed by the following quotation.

The TRANSYT method serves as an unofficial international standardagainst which to measure the efficiency of other methods of coordinatingnetworks of traffic signals. [RB91]

The software tool TRANSYT is based on an approach by Robertson[Rob69], pub-lished in 1969. In this approach, fixed-time signals are considered and signal settingsare improved via a gradient (hill climb) search technique or a genetic algorithm ap-proach. Namely, a so-called performance index (PI) is minimized. An advantage ofTRANSYT is the very detailed objective function, i.e., many aspects can be taken

3By courtesy of the PTV AG.4The acronym TRANSYT stands for TRAffic Network StudY Tool


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1.2 The Network Signal Coordination (NSC) Problem 13

into account for the PI. Since in Section 4.3 we work with TRANSYT, we now

give some detailed information on its mode of operation and the adjusted parametersetting in the following remark.

Remark 1.2. We use version Transyt-7F 10.1 and its genetic algorithm approachto optimize offsets in a network. The genetic algorithm parameters are: crossoverprobability = 40%, mutation probability = 2%, convergence threshold = 0.01%,number of generations = 100, and population size = 25 which are the defaults.Further, we calibrated the performance index PI such that it measures delay only.As the mode we used both the single-cycle mode.

To summarize this paragraph: when developing a model that optimizes the co-ordination of traffic signals in a network, evaluating it by comparing the results tothe ones of TRANSYT is a must.

In 1976 Gartner, Little and Gabbay introduced the software tool MITROP5

that optimizes traffic signal settings [GLG76, GLG75a, GLG75b]. By then, theMITROP was one of the first approaches that used integer programming techniques.In addition, Gartner et al.s model was the first one to simultaneously optimize (onenetwork wide) cycle length, red-green splits at the signal and offsets. However, sincethe relation between these signal settings is quadratic, they use piece-wise linearapproximations. Our approach, which we present in Section 1.3 is actually based onMITROP. Hence, we adopt most of their notation. Nevertheless, shortcomings of theMITROP model are the objective function, see Section1.3.5for more information,

and the assumption of a global cycle length. However, to the best of our knowledge,MITROP has not had commercial success, thus, it may be considered a theoreticalapproach.

A third software tool, which has to be mentioned when speaking of coordinationof signals in a network, is SYNCHRO [syn00]. Very similar to TRANSYT, the toolSYNCHRO models all traffic signal settings and searches heuristically for a solutionwith small average delay.

However, the three briefly introduced software tools, TRANSYT, MITROP, andSYNCHRO are by far not the only computer programs that deal with the coordina-tion of signals in networks. Still, we consider them to be the most relevant ones to

compare our approach with.

1.2 The Network Signal Coordination (NSC) Problem

In this section we consider the Network Signal Coordination (NSC) problem. First,in Section 1.2.1 we give a definition of the problem. Thereafter, in Section 1.2.2,we classify the NSC problem to related problems like the PESP that also deal withperiodically repeated events. Finally, in Section1.2.3we prove the NP-completenessfor the NSC problem.

5MITROP abbreviates Mixed Integer Traffic Optimization


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1.2.1 Definition of the NSC

In this section we will give a mathematical problem formulation for the Network Sig-nal Coordination (NSC) problem. However, the problem formulation will be as whatwe want to be understood by network signal coordination. Obviously, there is nounique possibility to define the problem mathematically. Nevertheless, none of thealready known problem definitions suffices our needs. Either, they are formulated toogeneral, or, they are formulated too narrow, meaning that many specialized assump-tions regarding the signal or the traffic flow, e.g., platoon lengths, are incorporatedinto the problem formulation.

So, we try to find a problem formulation that is tailored to the coordination oftraffic signalsin networks, while not being over-restrictive with special assumptions,

e.g., on lengths of platoons. Still, we want our problem formulation to map thefollowing phenomena that are inherent to the practical problem.

In what we refer to as the practical problem we are considering the followingsetting. We are given a network of junctions which are signalized with fixed-timetraffic signals. The signals in the network have uniform cycle length. Then, theobjective of the practical problem is to adjust offsets at the signals such that delaythat is due to poor coordination, is minimized. Moreover, we want our problemformulation to map a macroscopic traffic scenario. Obviously, this is, in general,a quite restrictive assumption. Nonetheless, under certain conditions, on which wereport in Section 1.3, the macroscopic view is accepted. Then, as a consequence,

we assume the objective function to be separable. This means that for one link, thedelay of the vehicles only depends on the offsets of the two incident signals. Hence,we may consider each link individually, i.e., the objective is a sum of functions thatevaluate the link performance: so-called link performance functions (LPFs). Finally,we think it is both, sufficiently exact and sufficiently general, to assume the LPFs tobe continuous and piece-wise linear.

Mathematically, we model the network as a directed graph D = (V, A), wherewe refer to an element vVsynonymously as junction, signal, node, or vertex. Anelementa Awe call an arc. We use link or edge synonymously. The network widecycle length of the signals is denoted by c with c and the vector V is theoffset vector for the signals.

First, the following observation can be made. In the above requirements, we didnot mention any constraint on an offset v, vV. Hence, any V constitutesa feasible solution, or can be turned into a feasible solution, respectively, by replac-ing v with v mod c. Thus, formulating the network signal coordination problemas a decision problem, which answers the question whether there is a feasible set ofsignal offsets does not make any sense.

Rather, we consider the network signal coordination as an optimization problem.However, we formulate the optimization problem by its corresponding decision prob-lem using a parameter K. We define the Network Signal Coordination (c-NSC)problem as follows.


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Network Signal Coordination (c-NSC) Problem

Instance: A directed multigraphD = (V, A). Continuous, piece-wise linearfunctionsha: + for alla A that fulfillha(x) =ha(x + c)for all x . A number K +.

Task: Find a vector V, such thata=(u,v)A

ha(v u) K

or decide that no such vector exists.

We refer to the problem version where the cycle length c is part of the problem inputas NSC.

The following has to be noticed. We define the NSC problem using a networkwide uniform cycle length c. Actually, for this case one can restrict the functionshaa little more without losing practical relevance. Namely, one can claim piece-wiseconvexity for the ha. In particular, we assume that it holds that for all aA thereexists anx such thathais convex on the interval [x, x+c]. Nevertheless, since weare going to consider the NSC problem with non-uniform cycle lengths, too, we omitthe convexity assumption on the functions ha, because in this case, the piece-wiseconvexity property is hard to reconcile with observations from practice.

Notice that our above definition of the c-NSC is very similar to other problemdefinitions. For example, Hassin [Has96] introduced a problem, which he callednetwork synchronization problem. There, he also optimizes a separable function ofdifferences of node potentials. However, for the problem formulation, Hassin doesnot make any assumptions on the type of functions, although, when developing hisapproach, he also reports on periodic, continuous piece-wise linear function.

Still, for our definition of the c-NSC, we decided to include assumptions on thetype of the objective as well as the periodicity. We think that the periodicity andthe continuous piece-wise linear functions characterize well a general network signalcoordination problem.

1.2.2 The Relation Between the NSC and the PESP

This section is dedicated to illuminate coincidences between the NSC problem andrelated problems such as the Feasible Differential Problem (FDP) and the PeriodicEvent Scheduling Problem (PESP).

We are aware that the problem formulation of the NSC problem overlaps withmany known problems. One major difficulty to properly classify the NSC problemis that differentstreams of research dealt with very similar problems. For example,traffic engineers, as summarized in Section 1.1.2, considermostly motivated frompracticeproblems concerning signal coordination, which are formulated very closeto the NSC problem. Nevertheless, there was no uniform mathematical problemformulation established. Other decisive influences came from the field of operationsresearch and from computer scientists. In 1985, Dauscha et al. [DMN85] proposed


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an approach for cyclic schedules for task systems where they put the problem of

coordinating signals in a much more general framework. In addition, it was thisarticle that reported for the first time on the complexity of a problem closely relatedto coordinating signals.

However, it was not until 1989 that Serafini and Ukovich proposed a problemformulation and a notation that became broadly accepted. In particular, in [SU89b]they introduced an approach specialized for coordinating signals in networks. More-over, Serafini and Ukovich published another article in the same year. In [SU89a]they introduce a mathematical model for periodic scheduling problems. Althoughtheir problem definition is very similar to the one in [DMN85], their formulation of aPeriodic Event Scheduling Problem (PESP) attracted much more attention. ThePESP can be seen as the periodic extension of the so-called Feasible Differential

Problem (FDP).

Feasible Differential Problem (FDP)

Instance: A directed graphD = (V, A) and vectors and u A.Task: Find a vector V such that for every arc a = (v, w) it holds

a w v uaor decide that no such vector exists.

In both problems, the FDP and the PESP, one considers a directed graphD =(V, A) and seeks a vector

V. This setting is equal to the one for the NSC

problem. However, for the FDP and the PESP, one additionally is given vectorsand u A. With these vectors, constraints on are formulated, non-periodicones for the FDP and periodic ones for the PESP. Notice that the absence of suchconstraints is the major difference between the NSC problem and these two problems.However, it is a trivial observation that the c-NSC problem withK= is equivalentto the PESP witha= 0 and ua = T for all arcs aA.

Periodic Event Scheduling Problem (T-PESP)

Instance: A directed graphD = (V, A) and vectors and u A.Task: Find a vector V such that for every arc a = (v, w) it holds

a (w v) mod T uaor decide that no such vector exists.

When comparing the NSC to the PESP, the following facts have to be mentioned.Because of the observation that the feasibility version of the NSC is contained in thePESP, one could think that the NSC problem is simply a special case of the PESP.However, this is not true, as can be seen when considering the optimization problemversions of the two problems. In particular, we consider the optimization problemversion of the PESP as it is used for its main application: periodic timetabling. Then,on the one hand, one assumes separable objectives for both problems. On the other


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hand, different types of functions are considered to measure the link performance for

the two problems. For the NSC problem, we assume the performance function fora link to be continuous and piece-wise linear. When optimizing periodic timetableswith the PESP, though, often non-continuous functions that are periodically linearon an interval of length T, are considered.

Moreover, due to the different origins, different names for the model quantitieshave become standard. The period lengthTof the PESP, for example, is equivalentto the cycle length c for the NSC problem. Moreover, in the PESP context, the

V, are called potentials, whereas when coordinating signals, they are called nodeoffsets.

To summarize, the NSC problem and the PESP can be regarded as quite similar.

The differences lie in the bounds on the vector for the PESP and in the differentobjective functions. Therefore, modeling or solution approaches to the two problems,as for example MIP formulations, differ as well, little, but noticeable. However, wedo not go into further detail for that question.

Another interesting aspect when comparing the NSC problem to similar problemsdoes not become apparent until considering a special model for NSC. It is possibleto model the NSC problem not using variables v for a v V, but exclusivelywith variables a := vu for each a = (u, v) A. When doing so, one hasto add certain constraints to the model in order to be able to recalculate the nodeoffset v out of the link offset a of a solution.

6 An important observation is thatsuch constraints do also appear in the according model for the PESP and, generally,

in every approach that models periodically repeated events using variables on arcsinstead of variables on nodes. Often, these constraints are called cycle constraints,but, actually, they have had various names like congruence equations [SU89b] orloop constraints [GLG75a]. In fact, these cycle constraints were already formulatedby Kirchhoff[Kir47] in an aperiodic manner, i.e., for describing properties of voltagesin electrical networks.

Finally, we give a short resume on this section. In the 1950s and 1960s trafficengineers developed the first models for coordinating traffic signals in networks. Al-though by then the problems were approached more from the practical side, manysophisticated mathematical solution methods were proposed. Then, during the years,

the problem of coordinating signals inspires researches from operations research andcomputer science. First, they come up with own approaches to signal coordinationand provide first complexity studies. Thereafter, they develop much more generalproblem frameworks that cover many different real-world applications like railwaytimetabling[Lie06]. Nevertheless, a backfertilization to the actual origin, i.e., thecoordination of traffic signals, could be observed. For example, new mathematicalexpertise in network flow theory[Has96]entered the signal coordination approaches.

6See Section1.3.3for a detailed discussion of these constraints at the example of modeling thecoordination of traffic signals.


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1.2.3 NP-completeness of the c-NSC problem

In 1975, Gartner et al. [GLG75a] introduced an optimization problem that mini-mizes the delay of vehicles in a network with fixed-time signals by adjusting optimaloffsets. However, although providing a mixed-integer approach for their problem,which they called Network Coordination Problem, Gartner et al. did not reporton the complexity of the problem.

In 1985, Dauscha et al. [DMN85] proved the NP-completeness of their CyclicSchedule Problem, which can be considered to be very close to the problem of coordi-nating traffic signals. They reduced the Graph k-Colourability Problem, see[GJ79],to their problem. Later on, other hardness proofs for slightly more general, problemsfollowed. In 1989, Serafini and Ukovich [SU89a] provided an NP-completeness prooffor the PESP. They reduced the Hamiltonian Circuit Problem, see [GJ79], to theirproblem.

In 1996 Hassin[Has96] proves the NP-completeness of his Network synchroniza-tion approach by a reduction from the Minimum Cluster Problem, see [GJ79].

The c-NSC problem of Section 1.2.1 obviously belongs to the class NP. Has-sin [Has96] reported on the NP-completeness for a variant of his problem whichimplies the hardness of the c-NSC problem. Still, in Theorem 1.3 we provide analternative reduction from T-PESP proving the c-NSC problem to be NP-completefor c 3.Theorem 1.3. The c-NSC problem is NP-complete for c 3.

Proof. It is obvious that the c-NSC problem belongs to NP. We show the NP-hardnessby a reduction from T-PESP, see Section1.2.2for a definition, which is known to beNP-complete forT 3[Odi94,Odi96].

LetI= (G= (V, A), , u A) be an instance ofT-PESP for some T 3. Definean instanceI = (G = (V, A), ha, K) for the c-NSC problem in the following way.First, letG:= G, i.e., we take the graph G with its vertex setVand its edge set A.Further, we set K = 0 and c = T. The functions ha are defined as follows. Forana A we set

ha(t) =

2c(uaa) t + 2uac(uaa) , ift [a c(uaa)

2 , a],

0, ift [a, ua],2

c(uaa) t 2

ua

c(uaa) , ift [ua, ua+ c

(ua

a)

2 ]

(1.1)

which defines ha on the interval

I :=

a c (ua a)

2 , ua+

c (ua a)2

which has length c. For t / I, we set ha(t) = ha(t) with t = (tmodc) +ac(uaa)

2 , where we observe thatt I. Notice that with this definition, the func-

tionsha are continuous, piece-wise linear, and fulfill ha(x) =ha(x + c) for all x .The definition of the ha is motivated as follows. We can assume without loss of

generality that for the PESP instanceI, it holds for all a A that ua a < T.


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1.3 A Revised MIP Formulation for the NSC problem 19

Moreover, the functions ha are defined such that ha(t) = 0 if and only if 0 tmod c ua a. See Figure1.2 for an illustration of one functionha.

0

1

a ua

c

t

ha(t)

. . . . . .

Figure 1.2: We deduce a function ha using the PESP bounds a and ua, suchthat ha(t) = 0 if and only if 0 t mod c ua a.

However, if the instanceI is a Yes-instance, then there exists a vector Vsuch that

a=(u,v)Aha(w v) = 0.

Thus, ha(w v) = 0 for all aA. Hence, 0(w v) modcua a, whichmeans thatIwas a Yes-instance for the PESP.

It is obvious from the construction that the other direction works out the same,i.e., that a Yes-instanceI for the PESP is always recognized by a correspondingYes-instanceI of the NSC problem.

Notice, that an APX-hardness proof from the optimization version of the T-PESPsee[Lie05,Nac96a]does not carry over to the c-NSC problem. The reasonfor this is that the proof makes use of the discontinuity points in the objective of theT-PESP; in the c-NSC problem we have a continuous objective function.

1.3 A Revised MIP Formulation for the NSC problem

In the previous sections of this chapter, we reviewed existing approaches for thecoordination of traffic signals in networks, and spent some effort in appropriatelydefining the Network Signal Coordination (NSC) problem. Now, in this section, a

mixed-integer linear programming approach for the NSC problem is presented.In this approach, which is based on an approach by Gartner et al. [GLG76],also [GLG75a] and [GLG75b], we minimize the total delay occurring in a trafficnetwork by adjusting optimal offsets. In fact, in our MIP model, we also allowa variable phase sequencing and a non-uniform cycle length at the signals of thenetwork. Therefore, the MIP approach that we develop in this section actuallysolves a more general problem than the c-NSC.

The section is organized as follows: first, in 1.3.1 we introduce the traffic sce-nario that we consider and discuss the assumptions that we make. Moreover, weintroduce the mathematical notation that we need. Note that we adopt most of thenotation from Gartner et al., see [GLG75a]. Thereafter, in Sections1.3.2to1.3.6we


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develop the MIP, which we finally state and discuss in Section1.3.7. Notice that in

Section1.3.2we also motivate the major model technique of using link-wise definedoffsets instead of node-wise defined ones.

1.3.1 Introduction

We assume the following scenario. We consider an (inner-city) traffic network withfixed-time signals at the junctions. Through that network, vehicles move on pre-scribed paths. Notice that we only consider individual traffic. Further, non-uniformcycle lengths at the signals are explicitly permitted. Moreover, the network is as-sumed to operate at near-saturated condition. That means that we mainly face hightraffic volumes on the links. Thus, as already motivated by Wormleighton [Wor65],

it is justified to model the traffic flow through the network macroscopically, i.e., wedo not consider each vehicle individually, but rather consider groups of vehicles, so-called platoons. Doing so, we further assume the traffic volumes to be given link-wise.In the model, all considerations, e.g., the calculation of delay, are done separatelyfor each link. So, we set up a mathematical model that minimizes the sum of delayson all links in the network with offsets between the signals and, what we call splitmode or phase sequencing, as decision variables.

The traffic network is modeled by a directed multi-graph G = (V, A), where theverticesvVrepresent the signalized junctions and the edges aA stand for thetraffic flow between the junctions. We will also use the terms node and link or arc,respectively. For an edgea = (u, v) Awe call the node u upstreamnode of edge a,whereas v is referred to as downstreamnode ofa. The reason for allowing paralleledges is that we distinguish the traffic flow on a link depending on the signal groupsof the upstream and the downstream junctions. Moreover, letL denote the set ofcircuits of the underlying undirected graph ofG. In the following, we use the termscircuit and cycle synonymously.

In the remainder, whenever it is obvious from the context we omit indices fordifferent copies of a link. In Fig. 1.3 all relevant data for one signal and theirnotation, respectively, is depicted.

1

23

r

g

v

c

signal groups

Figure 1.3: A signal timing plan of Signal v is shown. Exemplarily, the length gof the green phase for signal group 2, and the length r of the red phase for signalgroup 1 are depicted.

A first remark is that we do not consider amber phases at the signals. Instead,we assume the signal timing plan to only include green and red phases for all signal


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groups at a signal. This is no limitation, since it is accepted to transform a usual

timing planwith the amber phaseinto one without an amber phase, using theconcept of an effective green. This is carried out in more detail in[GLG75a].

During the following sections, we successively increase the level of detail thatwe use for explaining our approach. Therefore, for example, the non-uniform cyclelengths are introduced not until Section1.3.6. Thus, assume for now that the signalsin the network operate with the same cycle length.

Let c denote the cycle length at the fixed-time controlled signals in thenetwork. Further, the length of a red phase is denoted byr, whereasg stands for thelength of a green phase. The indexing for these quantities works as follows: becauseall considerations within the model are done edge-wise, we index g andr edge-wise,too. That means, for an edge a A, ga denotes the length of the green phase atthe downstream signal of a. The length of a red phase ra is defined analogously.Moreover, let a denote the travel time on link a, i.e., the time which needs theplatoon to move between the two signals that are incident to arc a.

Observe, the following notationally simplification: during the next sections, weoften refer to an edge via the identifier a A. Sometimes, it will be necessary torefer to the nodes incident to a. Then, we use, e.g., the notationa = (u, v) A.Since we allow parallel edges in the graph, an edge is not sufficiently characterizedby its incident nodes. Rather, one actually would have to include the respectivesignalgroups at the signalsu and v, which then uniquely characterize the traffic flow,represented by a. For clarity reasons, and when there is no confusion possible, wethus omit the identifiers for the signalgroups at u andv .

To summarize this short introduction: first, we assume the vehicles in the networkto move in platoons, and, second, we conduct all modeling steps link-wise.

1.3.2 The Offsets

This section is intended to treat two important tasks. First, we formally introducethe term offset, in both of its possible interpretations. Second, we motivate, thedecision for the link-wise defined offsets for our mixed-integer linear program.

We begin the development of our mixed-integer linear program with the mostimportant variables, the offsets. Although in the problem formulation for the NSC

one is interested in offsets for the signals, there are actually two possibilities to definethe offset: node-wise or link-wise. First, we illustrate the difference between the twoviewpoints and, second, we give formal definitions.

In Figure1.4the two possibilities to interpret the term offset are shown. First, inFig.1.4(a), we illustrate node-wise defined offsets. There, the term offsets refers tothe time span between the zero point in the signals timing plan and the networksglobal zero point. These two reference points can be set arbitrary, however, theyhave to be fixed beforehand. Second, in Fig. 1.4(b), we illustrate a link-wise definedoffset. There, the term offset refers to the time span between the beginnings of thegreen phases of the upstream and downstream signal of the link. Of course, in bothcases the offset value is only unique modulo the cycle lengthc.


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1

1

2

2

3

3

u

v

network zero pointsignal zeropoint

(a) node offset

1

1

2

2

3

3

u

v

(b) link offset

Figure 1.4: The two possible interpretations of the term offset are shown. In(a),the offset is defined node-wise, whereas in(b)it is defined for each edge aA. Inthe example, the linka represents traffic flow between signal group 2 of signal u andsignal group 3 of signal v .

For a node v V we denote with v the node offset at junction v. Further, werequirev [0, c). For an arc a, the link offset variable is denoted by a. Later onin the section, we discuss bounds for the offsets a.

Now, before we formally show the connection between the v and the a, wehave to introduce the intra-node offsets denoted by . For a node vV, letRv bethe set of its signalgroups. Then, the intra-node offset pv refers to the time spanbetween the beginning of the green phase of signal group p

Rv and the signal

reference point to which the node offsets are linked. See Figure1.7on page25forexamples. For now, we assume the intra-node offsets to be an input parameter.Later, in Section1.3.4they become variables, though.

At this point, we can formalize the connection between the node offsets and thelink offsets . Consider the edgea= (u, v) A. Assume thata[ra+ac, ra+a].This will be motivated later. Then, let os(a) denote the signal group at nodeu, theorigin of a. Further, ds(a) denotes the signal group at the destination of a, thenode v. Here, os(a) Ru and ds(a) Rv. Then, there exists an integer pa, suchthat

u +

os(a)

u v + ds(a)v + pa c = a. (1.2)

The connection stated in Eq. (1.2) is rather easy to see. We illustrate an examplein Figure1.5.

Hence, the following can be observed. A model that exclusively contains nodeoffsets can be considered equivalent to a model that exclusively contains link off-sets. Therefore see Equation (1.2). On the one hand, with fixed values ofuand vthere always exists a uniquepa such that a a witha[ra+ a c, ra+ a] exists.On the other hand, one can uniquely determine the vector out of the link offsets.However, therefore one node offset has to fixed, which is, though, not a limitation.This is, because two solutions , V can be considered equivalent if one is just


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1

1

2

2

3

3u

v

u

v

a3u

2v

Figure 1.5: The connection between node-wise defined offsets and link-wise definedones allows for a straightforward translation between the two terms.

a translation of the other, namely, if there exists a constant q, such that

a= (a+ q) mod c, aA. (1.3)

However, then, with one v , vV, fixed, one simply uses (1.2) to uniquely deter-

mine all v propagating the a, e.g., along a spanning tree of the underlying graph.That this propagation works out is ensured by so-called cycle constraints, which arediscussed in detail in Section1.3.3.

Now that we know that there exist several possibilities to model the quantityoffset for our signal coordination approach, we discuss why we decided to exclusivelyinclude the link offsets into our model.

The main reason, why we build up the model with the link offsets, is the objectivefunction. Using only the node offsets v for the model, we would have faced thefollowing problem. As we already mentioned, the objective function is separable.That means, we have an objective function for each arc a A measuring the linkperformance. These functions have to be defined in dependence of the differencevu, for a (u, v) A. However, as it will become clear in Section 1.3.5, where wediscuss the objective function in detail, we need to model the objective, such that itis convex on an interval of length c, which we know in advance. Using exclusivelythe vector, ensuring such a property, in fact, costs a modulo operation per link,i.e., number of edges many integer variables. As we will see in Section1.3.3,buildingup the model with the link offsets only causes additional constraint and integervariables, whereis the cyclomatic number of the graph and= |A| |V|+1. Still,we will remark on the link offset versus node offset question again in Sections 1.5and4.4.

So, we decided to use the link-wise defined offset for our MIP approach. However,we introduce a further variablethe platoons arrival time which is, though,linear dependent from the offset .

On link a = (u, v)A, the arrival time of the head of the platoon at junction vis denoted by a. The relation between the arrival time a and the links offset a,is given by

a a + ra = a, (1.4)


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with a being the travel time of the platoon and ra denoting the length of the red

phase at the downstream signalgroup of the link. In addition, we require the arrivaltime a to lie in the interval [0, c). This will be useful when we use the a for theobjective function, i.e., to evaluate the delay on link a. Note that the bounds for aimply that for the according link offset, it holds that

a[ra+ a c, ra+ a]. (1.5)In Figure 1.6 we illustrate that connection between the arrival time and the linkoffset.

r

c

Figure 1.6: The figure illustrates the simple linear dependency between the arrivaltime of a platoon, , and the link offset .

1.3.3 The Cycle Equations

Consider the setLof cycles ofG. Note that since we are considering a multigraph,L includes cycles that contain only two parallel or anti-parallel edges.

For a particular circuit L that is traversed clockwise,F() is the set of forwardedges andR() the set of reverse edges, respectively. With c denoting the networkscycle length we require the variables to fulfill

aF()

a aR()

a= n c, (1.6)

for all cycles L and with n . Then the following connection between (1.2)and (1.6) can observed:

Lemma 1.4. For a link offset vector A, the following two statements areequivalent:

(i) there exists a vector V s.t. (1.2) holds for all edges a A,(ii) for all L, (1.6) holds.

Proof. For simplicity, we assume that all signals have exactly one signal group.Thus, (1.2), reads as a = v u +pa c. Then, assuming (i), summing up


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offsets a along a cycle Lresults in a telescope sum and (ii) holds. On the otherhand, with (ii), one can easily find a

V

fulfilling (1.2): first, fix one v, fora v V, to an arbitrary value. Then, second, propagate the link offsets along theedges of a spanning tree simply using (1.2). Then, the node offsets are well definedbecause (ii) holds.

So, the periodicities expressed in Equations (1.6) are physical constraints thatare necessary and sufficient to equivalently model the NSC problem with link offsets,instead of node offsets. For example when there is a cycle for which (1.6) does nothold, the transformation of the offsets is not well defined. That means, propagatingtheausing Equations (1.2) brings a contradiction. See Fig.1.7for a small example.Note that the intra-node offsets have to be included as well, since the link offsets

A

B

C

A= 0AB= 35

B= 35

BC= 55

C= 10

CA= 55?

A= 65

Figure 1.7: Consider this small example network with a cycle length of c = 80s.Assume that the node offset at vertex A was set to 0. Then propagating the linkoffset along the cycle reveals node offsets of 35 and 10 for the vertices B and C,respectively. However, if then the link offset on link (C, A) is not equal to 70 + 80

but instead, e.g., equal to 55, the node offset at A ceases to be well defined.

do not respect changes of signal groups at a junction. This means the following.Suppose, we are considering a circuit that contains two links that share a node v.Further, assume these edges have the same orientation with respect to the clockwisetraversal of the circuit. If then the edges do not share the same signal group atnode v, the shifting between the two signal groups has to be taken into account.Notice that for the intra-node offsets, too, the traversal direction has to be minded.For a more evolved example, see Fig.1.8. There, all relevant time gaps are depictedthrough arrows, i.e., directed arcs. Consider now, for example the point in time when

the green phase of the first signal group of Signal 1 begins. Starting from that point,following the link offset arcs and the intra-node arcs one can traverse a cycle. Thus,for this example, the sum

21 + 13 23 + 13 23 + 22 + 24 24 + 34 14 31 (1.7)must equal an integral multiple of the cycle length, since otherwise no consistentsetting of the node offset at Signal 1 is possible. So, Equation (1.6) in fact turns outto amount toaF()

a

aR()a+

aF()

os(a)u ds(a)v

aR()

os(a)u ds(a)v

= n c,

(1.8)


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11

1

1

12

2

2

2

2

3

3

33

3

413

23

24

14

21

22

31

13

23

34

24

Figure 1.8: An example network G = ({1, 2, 3, 4}, {(1, 3), (2, 3), (2, 4), (1, 4)}). Inbluish colors, the link offsets are sketched whereas the intra-node offsets are depictedin red.

whereos(a) and ds(a) again denote the signal groups at the origin and destinationof the link a = (u, v).

An important observation is that, although the cycle equations (1.8) have to holdfor each cycle L, they do not have to be defined explicitly for all cycles. Alreadyin 1966, Little [Lit66]observed that it suffices to define the cycle equations for the

cycles that are induced by the chords of a spanning tree. Later, in 2002 Liebchen andPeeters [LP02]introduce the class of integral cycle basis. Moreover, they show thatit suffices to require Equation (1.8) for the elements of an integral cycle basis [LP02]in order to ensure that it holds for all cycles of the graph. Note that the class ofintegral cycle bases is a proper superset of the class of strictly fundamental cyclebases that are defined by spanning trees [LR07].

A second important observation related to the cycle equations (1.8) concerns theinteger variable n, for L. For computational issues it is important to providebounds for them. However, with the help of the bounds on the link offsets, (1.5), onecan easily define an upper bound n and a lower bound n on n. Namely, we get

n=

1c

aF()(ra+ a)

aR()

(ra+ a c) +

(1.9)

and

n=

1

c

aF()(ra+ a c)

aR()

(ra+ a) +

(1.10)

as trivial bounds. Note that the terms

are a shortcut form the extended formas in (1.8). With these bounds on the integer variables at hand, one can consider


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the product that was also denotedwidthof a cycle basis in [Lie06],B

n n + 1 (1.11)

as quality measure of a basis B for the computation of the MIP. Hence, as we canchoose any integral cycle basis for the MIP formulation, it seems promising to takethe smallest one with respect to (1.11). However, we are only aware of one singlecontribution, see Liebchen [Lie06,LPW05], that ever tested the actual influence ofa cycle basis on the computation time of the MIP.

In Sect. 4.1we provide a second computational study that illuminates the cor-relation between the quality of a cycle basis and the computational performance ofthe according mixed-integer linear program.

1.3.4 A Variable Phase Sequencing

In contrast to the approach by Gartner et al. [GLG76], our model offers the possibilityto choose between different red-green split modes, i.e., we allow a variable phasesequencing. In the previous sections we assumed a fixed signal timing plan at thesignals for clarity reasons. In this section we formally introduce the variable phasesequencing. This, however, may not be confused with an entirely free choice oflengths of red and green phases.

In detail, thevariable phase sequencingworks as follows. For a vertex vV,Rvdenotes the set of signal groups at v . Then, we consider v different predetermined

modes of operation at the signal v,i.e.,v different signal timing plans. Nevertheless,these different signal timing plans must have some properties. First, the signal groupshave to stay the same, i.e., throughout all modes, the turns at a junction that arecontrolled by a signalgroup p Rv stay the same. With this, the possibility of anassignment of the traffic to signalgroups at the upstream and at the downstreamsignal is kept.

A second property is that the lengths of the green phases and red phases arethe same for all modes. Thus, the variable phase sequencing offers the function-ality of choosing between different modes each giving different intra-node offsets tothe signalgroups. In particular, the parameter

v,pm denotes the intra-node offset at

junction v of signal groupp

Rv in modem, where m = 1, . . . , v. Then, we intro-

duce binary variablesdv,m for all nodesv and modes m = 1, . . . , v. These variablesmanage the selection of a mode. Namely, with Equations (1.12), the actually chosenintra-node offset of signalgroupp at node v is allocated to the variable v,p:

vm=1

dv,m v,pm = v,p vV, p Rv. (1.12)

Of course, one has to ensure that exactly one mode is adjusted. This is done inEquations (1.13).

v

m=1dv,m = 1 vV. (1.13)


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With Figure1.9 we give an impression of how such different modes or the according

signal timing plans could look like.

2,14

2,43

2,22

2,31

1

11

1

2

22

23

33

3 v

vv

v

4

44

4

Figure 1.9: An example of different predetermined signal timing plans, i.e., modes,is shown. Exemplarily, some of the are indicated.

Observe that the introduction of variable timing plans, and thereby changing thestatus of the intra-node offsets from parameters, as introduced before, to variables,is well compatible with the cycle equations proposed in Section1.3.3. In particular,

in (1.8) one only has to replace the parameter os(a)u and ds(a)v by the accordingvariables u,p and v,p. Note that here it is important to still be able to assign anedge to itssignal group. However, when considering the definitions (1.9) and (1.10)of the bounds for the integer variables for the cycles, we observe that these boundsbecome much weaker.

1.3.5 The Objective Function

In this section we introduce the objective function. Our mathematical model is builtto minimize the total traffic network delay that is due to missing or poor coordination

within the network. Now, we will specify how the delay is actually determined byevaluating the arrival pattern of the platoons.First, remember that due to the assumption of the vehicles moving in platoons

through the network, we have a separable objective function. Thus, the delayon a link does only depend on the offset of that link. Therefore, consider thelink a = (u, v)A and assume certain signalgroups at the origin of the link and atthe destination. Now, the traffic, i.e., the platoon, leaves the junctionu. Namely,throughout the green phase of signalgroup os(a) at the origin signal u, the platoonis released. Then, the platoon is moving on the link towards signalv. The transittime on that link is determined by the head, i.e., the first vehicle, of the platoon. SeeFigure 1.10(a) for an illustration. Finally, the platoon arrives at signalv. Observe


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that the length of the platoon then may have changed. This means that although the

platoon was loaded for a timespan with length equal to the length of its green phaseat u, the length of the platoon can have changed while traversing the link. This isdue to dispersion effects on the link that appear especially on links that representlong streets. However, note that the actual necessary information for calculating the

1

1

2

2

3

3u

v

a

(a)

2

a

v

(b)

Figure 1.10: In Figure(a)we schematically depict the platoon moving from Signal-group 2 at Signal u to Signalgroup 2 at Signal v. In (b)we focussed on the detailthat is considered when evaluating this links delay. Notice that all informationabout intra-node offsets are left out. Only the arrival time, , of the platoon has tobe known.

delay are the arrival time a and the length of the platoon at its arrival at signal v .

The effects that change the platoon shape in between are not pursued.Further, for the calculation of the delay on link a, we slightly change the per-

spective, see Figure1.10(b). Namely, we consider the timespan of length c not as itis given from the signal timing plan, but beginning with the red phase and followedby the green phase, as indicated in Figure 1.10(b). This changed viewpoint is alsorecognizable in Fig.1.6.

Then, the situation is as follows. The platoon can pass the signal as long as itshows green. If the signal shows red, the vehicles have to stop and a waiting queue isformed. When the signal turns green again, the queue is released. For an illustrationof this queueing see Figure1.11(a). Now, as we want to quantify the delay, we haveto introduce some parameters. As an input of the model, we are given in advance

the traffic volume on the links. So, with fa indicating the number of vehicles onlink a = (u, v)A, which we measure in vehicles per hour, it is straightforward tocalculate a vehicle arrival ratef pa within the platoon. Namely, it is defined as

f pa = c fa3600 pa . (1.14)

Notice that we assume a uniform rate with the platoon. So, this rate f pa is the ratewith which the waiting queue is built up, when there is a red at signalv. Then, whenthe signal turns green, the queue is released with a saturation rate ofsa. However, ifat the beginning of the green phase there are still vehicles arriving, see the situationthat is depicted in Figure1.11(b),the net-saturation rate is sa f pa.


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A1 A2 A3

(a)

A4

(b) (c)

Figure 1.11: Depending on the arrival time of the platoon, the vehicles queue at thesignal. For example, if the platoon arrives at the destination signal at the beginningof the green phase, (c), and additionally the platoon is not longer than the greenphase, then no delay arises.

Applying these straightforward ideas of how the waiting queue behaves, makes acalculation of the occurring delay easy. When interpreting this process geometrically,determining the delay is nothing else but calculating the size of the area that visual-izes the queue over time in function of the arrival of the platoon relative to start ofred phase. See for an example Figure 1.11(a)where the arrival time of the platoonis a= 0. We denote with Z() (z()) the function that measures the total (average)delay of that link in dependence of the arrival time. Then, we observe the followingtotal delay for that particular example,7

Z(0) =A1+ A2+ A3=p2a f pa

2

+ (ra

pa)

pa

f pa+

p2a f p2a2sa

. (1.15)

Then, of course, this quantity has to be normalized with the factor pa f pa in orderto obtain the delay per vehicle on link a witha= 0. At this point, we abstain fromproviding all possible cases, i.e., explain how Z() is calculated for other values ofand with different assumptions on how, e.g.,pa is related to ra and ga. Nevertheless,an important assumption is the following: we require that the waiting queue musthave dissipated completely when the signal turns red. Without such an assumption,considering a time span of length c would not be sufficient to determine the delay.

Now, how do we use the function z() to obtain an objective for our mixed-integerprogram? A first observation is that we cannot use the above described functionz()as objective function, because it is not linear in the arrival time . In fact, the

function z () is piece-quadratic inwhich can be seen from Figure 1.11(b). In sucha situation, i.e., the arrival time a lies for example in the interval [ra pa2 , ra+ pa2],the size of the area depicted with A4 is quadratic ina.

As a consequence, we will use a piece-wise linear approximation ofz() as follows.We evaluate in advance the function z() for particular iand use the points (i, z(i))for a linear approximation. Note first thatz(0) = z (c). Further, it is obvious thatfor = 0 the periodic function z() attains its maximum. On the other hand,for =ra, z() is minimal. Thus, at the best, the platoon arrives exactly when thesignal turns green.

7assuming pa ra


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1.3 A Revised MIP Formulation for the

thesis wu en sch 2008

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