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Lindley-Type Recursions THOMAS STIELTJES INSTITUTE for MATHEMATICS Lindley-type recursions THOMAS STIELTJES INSTITUTE FOR MATHEMATICS CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN c Vlasiou, Maria Lindley-type recursions / by Maria Vlasiou. – Eindhoven: Technische Universiteit Eindhoven, 2006. Proefschrift. – ISBN 90-386-0784-9 – ISBN 978-90-386-0784-9 NUR 919 Subject headings: recurrence relations / queuing theory 2000 Mathematics Subject Classification: 60K20, 60K25, 90B22 Printed by PrintPartners Ipskamp Cover design by Rik Wisman Kindly supported by a full scholarship from the legacy of L. Athanasoula Lindley-type recursions 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 25 september 2006 om 16.00 uur door Maria Vlasiou geboren te Drama, Griekenland Dit proefschrift is goedgekeurd door de promotoren: prof.dr. J. Wessels en prof.dr.ir. O.J. Boxma Copromotor: dr.ir. I.J.B.F. Adan Acknowledgements The book is finally over and I feel grateful for the faith many people have shown in me throughout all the years I have been studying. This manuscript is the result of a few years of training under the guidance of my supervisor, Ivo Adan. I thank him for his endless patience in his interaction with me, for the fact that he always made time for me and that he constantly stimulated me to keep on learning. Onno Boxma was always available to discuss anything that I would come up with, no matter if it would be the problem that I was stuck with for the past few months, or my opinion on yesterday’s tennis game. His opinion, both on maths and on tennis, will always be valued. I truly enjoyed our collaboration. Jaap Wessels has been through numerous drafts of this book and the papers that it is comprised of, and his constructive comments have led to many improvements and additions. I am blessed to be surrounded by dear friends and family. I often feel that the reason I did not quit early in this project is the constant support and advice of Krishanu Maulik. I am deeply indebted to him for his guidance and unconditional assistance all these years. His friendship is one of the most valuable gifts that I have received in this country. Ioanna Dimitriou and Fanis Matsoukas were always a phone call away, at any our of the day or night, to hear all my mathematical and personal problems. Dimitris Fotiadis and my family where always there to hear me out, even if that meant that they would have to spend several hours on the phone. A great many colleagues and staff at EURANDOM and TU/e have sustained me over the years in what must have seemed an increasingly testing endeavour. I particularly wish to thank all the support staff in EURANDOM that were always ready to answer questions ranging from peculiarities in the Dutch language to advice on how to fill in my tax form. I wish also to thank Marko Boon and Wil Kortsmit for answering any computer-related question I would come up with. They were both obliging teachers, whose guidance has directly or indirectly produced all graphs appearing in this book. Stef van Eijndhoven is the father of the intriguing differential equation (A.1), which helped me understand better the implications of my own work. I would not have initiated this project if it were not for the constant encour- agement of several professors and friends I have left back in Thessaloniki. Dr. Sotiria Harmousi did not only teach me how to handle my migraines; she also gave me valuable advice that I recall and cherish until today. I am honoured by the trust and confidence that my professors have shown in me. I would like to thank Theodora Theohari-Apostolidi, Lia Kalfa, and Michalis Marias for the constant support throughout all these years. The joy of my own family life throughout this project is altogether the creation of my partner, Bert. Even a dissertation seems a paltry offering for the daily riches of his companionship. Notation An service time of the n-th customer Bn preparation time of the n-th customer Wn waiting time of the server for the n-th customer Xn+1 Bn+1 − An A, B, W generic service, preparation, and waiting time respectively FY distribution function of the random variable Y ; e.g., FA is the service-time distribution fY density function of the random variable Y α, β, ω Laplace-Stieltjes transform of A, B, W respectively φ Laplace-Stieltjes transform of A + W λ the scale parameter of the phase-type distribution associated with the service times, whenever relevant µ the scale parameter of the phase-type distribution associated with the preparation times, whenever relevant π0 P[W = 0], i.e., the mass of the waiting-time distribution FW at the origin c(k) the covariance function cov[W1,W1+k] θ throughput Gi the Erlang distribution with i stages 1 E[Y ; E] E[Y · [E]], where Y is a random variable and E is an event 2 cY the squared coefficient of variation of the random variable Y f (i) the i-th derivative of the function f R+ the set of nonnegative real numbers =D equal in distribution Contents 1 Introduction 1 1.1 Scope of the thesis ............................ 1 1.2 Background motivation ......................... 2 1.3 The history of carousels ......................... 4 1.3.1 Storage .............................. 4 1.3.2 Picking a single order ...................... 7 1.3.3 Picking multiple orders ..................... 9 1.3.4 Design issues ........................... 10 1.3.5 Problems involving multiple carousels ............. 11 1.4 The model ................................. 13 1.5 Lindley’s recursion ............................ 16 1.6 A generalised Wiener-Hopf problem .................. 18 1.7 Overview of the thesis .......................... 23 2 General properties 25 2.1 Introduction ................................ 25 2.2 Stability .................................. 28 2.2.1 The case P[X < 0] > 0 ...................... 28 2.2.2 The case P[X < 0] = 0 ...................... 28 2.3 The recursion revisited .......................... 32 2.4 Classification of distribution functions ................. 34 2.5 Tail behaviour .............................. 36 2.6 The covariance function ......................... 40 3 Preparation times on a bounded support 45 3.1 Introduction ................................ 45 3.2 Previous results .............................. 46 3.3 Iterative approach ............................ 49 3.4 Erlang service times ........................... 51 3.4.1 Laplace transforms approach .................. 51 3.4.2 Differential equations approach ................. 55 3.5 Phase-type service times ......................... 58 3.6 Polynomial preparation times ...................... 60 3.7 Numerical results ............................. 65 3.8 Concluding remarks ........................... 66 ix x Contents 4 The G/PH model 69 4.1 Introduction ................................ 69 4.2 The alternating case ........................... 71 4.2.1 Time-dependent analysis for G/M ............... 71 4.2.2 Time-dependent analysis for G/E ................ 80 4.2.3 Steady-state distribution for G/E ................ 84 4.2.4 Steady-state distribution for G/PH ............... 88 4.3 The non-alternating case ......................... 89 4.4 Performance comparison ......................... 92 4.4.1 Stochastic ordering ........................ 92 4.4.2 Mean waiting times ....................... 94 4.5 Numerical results ............................. 97 4.6 A comparison to Lindley’s recursion .................. 98 4.6.1 The time-dependent distribution ................ 98 4.6.2 The busy cycle .......................... 101 4.6.3 The covariance function ..................... 101 5 The M/G model 103 5.1 Introduction ................................ 103 5.2 Derivation of the integral equation ................... 104 5.3 The M class ............................... 107 5.4 Steady-state distribution for M/M ................... 109 5.5 The G/M model ............................. 112 5.6 Explicit examples ............................. 113 5.7 Concluding remarks ........................... 117 6 Approximations 119 6.1 Introduction ................................ 119 6.2 Error bounds ............................... 120 6.3 Approximations of the waiting-time distribution ........... 121 6.3.1 Fitting phase-type distributions ................. 121 6.3.2 Fitting polynomial distributions ................ 122 6.3.3 Fitting distributions that have one discontinuity ....... 123 6.4 Numerical results ............................. 126 7 Dependencies 129 7.1 Introduction ................................ 129 7.2 Markov-modulated dependencies .................... 130 7.2.1 Exponential preparation times ................. 131 7.2.2 Phase-type preparation times .................. 134 7.3 Services depending on the previous preparation time ......... 135 7.4 A comparison to Lindley’s recursion .................. 140 Contents xi 8 A more general Lindley-type recursion 143 8.1 Introduction ................................ 143 8.2 Stability .................................. 145 8.3 The G/PH model ............................. 146 8.4 The M/D model ............................. 153 Final remarks 163 Minor extensions and observations ...................... 163 Further research ................................ 165 A remark on Equation (3.32) 169 Samenvatting (Summary) 171 Bibliography 173 Index 187 About the author 189 Chapter 1 Introduction 1.1 Scope of the thesis We have all had the unpleasant experience of waiting for
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