# Centrum Wiskunde & Informatica · PDF file ACKNOWLEDGEMENTS Conducting the research that has led to this monograph, has been a great pleasure for me. Now that this phase is reaching

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• Polling Models From Theory to Traffic Intersections

• ii

• Polling Models From Theory to Traffic Intersections

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de rector magnificus, prof.dr.ir. C.J. van Duijn, voor een

commissie aangewezen door het College voor Promoties in het openbaar te verdedigen op maandag 4 april 2011 om 16.00 uur

door

Marcus Aloysius Antonius Boon

geboren te Heerlen

• Dit proefschrift is goedgekeurd door de promotoren:

prof.dr.ir. O.J. Boxma

en

A catalogue record is available from the Eindhoven University of Technology Library. ISBN: 978-90-386-2449-5

• ACKNOWLEDGEMENTS

Conducting the research that has led to this monograph, has been a great pleasure for me. Now that this phase is reaching its end, it is just as great a pleasure for me to express my gratitude to everybody who has helped to make this thesis possible.

First and foremost, I am greatly indebted to my supervisors Onno Boxma and Ivo Adan for encouraging me to start writing a Ph.D. thesis. Their never-ending enthusiasm has given me all the motivation that was necessary to finish the thesis successfully. Onno, you have given me a great start by pushing me in the right direction, but you also had the patience to let me wander into different directions that I saw fit. I greatly admire the way that you always manage to reserve time for anybody needing it. Ivo, you are one of the most enthusiastic persons I know. After a session in front of your blackboard I always left your room full of spirit and ideas about how to tackle the problem.

Secondly, I would like to thank the other members of my doctorate committee. I am thankful to the core committee members, Sem Borst, Rob van der Mei, and Richard Boucherie, for their valuable remarks, suggestions, and for the discussions at various occasions, including - or perhaps especially - those that were not related to research. Moreover, I am very honoured to have Uri Yechiali, a foremost expert in the field of polling systems, as a member of the doctorate committee. The last member of my doc- torate committee, Jacques Resing, deserves a special thanks for all the time he has spent explaining the ins and outs of branching-type service disciplines, and for the valuable discussions about the mixed gated/exhaustive service discipline during the ValueTools 2008 conference in Athens.

Furthermore, I want to express my gratitude to the co-authors of the papers on which this thesis is based. In alphabetical order: Doug Down, Rob van der Mei, Sandra van Wijk, and Erik Winands. Although being mentioned last in this list, Erik Winands has been one of the most influential people on my research, and may certainly be considered as a third, unofficial, supervisor.

Finally, I would like to thank all the people that contributed (directly or indirectly) to the realisation of this research: the board of the department of mathematics and computer science, for creating the possibility for me to be a Ph.D. student for three days per week; all of my colleagues who consequently had to take over some of my tasks; all of my colleagues at EURANDOM and the Stochastics section, for creating a pleasant atmosphere to work in; my family and friends, for making sure that I enjoy the time outside working hours at least as much as the time during working hours; and finally Nicole and Erik, for creating the loving atmosphere at home.

Marko Boon February 2011

• vi ACKNOWLEDGEMENTS

• CONTENTS

Acknowledgements v

1 Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Polling models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Literature review 7 2.1 Applications of polling models . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Analysis of polling models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

I Customer behaviour 41

Introduction to Part I 43

3 Smart customers 45 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 Model description and notation . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.3 Queue length distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.4 Waiting time distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.5 Cycle times, visit times and intervisit times . . . . . . . . . . . . . . . . . . . 56 3.6 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4 Reneging at polling instants 63 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.2 Model description and notation . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.3 Cycle times, (inter)visit times and waiting times . . . . . . . . . . . . . . . . 65 4.4 Queue length distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.5 Vacation system with exhaustive service . . . . . . . . . . . . . . . . . . . . . 70 4.6 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

II System behaviour 79

Introduction to Part II 81

• viii CONTENTS

5 Multiple priority levels 83 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.2 Model description and notation . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.3 Joint queue length distribution at polling epochs . . . . . . . . . . . . . . . . 84 5.4 Marginal queue lengths and waiting times . . . . . . . . . . . . . . . . . . . . 85 5.5 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6 Priority based mixed gated/exhaustive service 99 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.2 Model description and notation . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.3 Joint queue length distribution at polling epochs . . . . . . . . . . . . . . . . 101 6.4 Cycle times, visit times and intervisit times . . . . . . . . . . . . . . . . . . . 102 6.5 Waiting times and marginal queue lengths . . . . . . . . . . . . . . . . . . . . 103 6.6 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.7 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

III Signalised intersections 113

Introduction to Part III 115

7 Closed-form waiting time approximations 117 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 7.2 Model description and main result . . . . . . . . . . . . . . . . . . . . . . . . . 118 7.3 Derivation of the approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 120 7.4 Numerical study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

8 Signalised intersections with exhaustive traffic control 135 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 8.2 Model description and notation . . . . . . . . . . . . . . . . . . . . . . . . . . 137 8.3 Heavy traffic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 8.4 Light traffic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 8.5 Interpolations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 8.6 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 8.A Input settings for Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

9 Signalised intersections with conflicts 163 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 9.2 Model description and notation . . . . . . . . . . . . . . . . . . . . . . . . . . 163 9.3 Heavy traffic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 9.4 Light traffic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 9.5 Interpolations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 9.6 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

Bibliography 173

Summary 189

Curriculum Vitae 191

• 1 INTRODUCTION

1.1 Motivation

Waiting in a queue is an unavoidable nuisance in everyday life. Queues may be visible, like people in supermarkets, cars stuck in a traffic jam, patients waiting in a hospital, or invisible, like data packets in computer networks or jobs in a printer queue. Neverthe- less, they are always a source of annoyance, impatience and loss of valuable time and money. For these obvious reasons it is of much practical relevance to gain insight into the processes that cause queues to develop and disappear again. Driven by a rapidly growing number of applications, the mathematical study of waiting lines has become an important mathematical discipline by itself, known as queueing theory.

Nowadays, more than 100 years after the first queueing model was introduced by A. K. Erlang in 1909 to model delays in telephone conversations [83], it is impossible to count the various queueing models that have appeared in the literature. This the- sis focusses on one particular type of model, the so-called polling model. Few models have received as much attention in the abundan

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