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An Application of Martingales to Queueing Theory

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An Application of Martingales to Queueing Theory t+, g'cl+ University of Adelaide Department of Applied Mathematics PhD Thesis An Application of Martingales to Queueing Theory Matthew Roughan B.Sc. (Ma)Hons. 20th December, 1993 ;'ijwun ,J *ul iq* ü Dedication This thesis is dedicated to Syivia Daina Luks who has tolerated me these many years. ll Acknowledgement I would like to thank my supervisor Charles Pearce for his valuable suggestions and advice. I would further like to thank the numerous people who have at some stage in this thesis provided heip or encouragement such as the staff and students of the Teletraffic Research Centre and the Applied Mathematics Department at the University of Adelaide. I would also like to thank Bong Dai Choi and Nick Duffield who on their brief visits supplied motivation for continuing. Specifically I would like to thank Andrew Coyle for proof reading this thesis I would like also to acknowledge the help and support of my family This thesis ïyas funded by a Australian Postgraduate Priority Research Award and a supplementary scholarship from the Teletraffic Research Centre at Adelaide University. lrl Statutory Declaration I, Matthew Roughan, state that this work contains no material which has been accepted for the award of any other degree or diploma in any university or other tertiary institution and, to the best of my knowledge and belief, contains no material previously published or written by another person, except where due reference has been made in the text. I give consent to this copy of my thesis, when deposited in the University Library, being available for loan and for photocopying. SIGNED: DATE: ?0[ tur'K lv Contents 1 Introduction 1 2 Renewal Theory 7 2.L Introduction l 2.2 Simple renewal theory I 2.3 Basic Markov renewal theory L2 2.3.1 Delayed Markov renewal processes 15 2.3.2 Generalised Markov renewal processes 16 2.3.3 Discrete time 17 2.4 Multi-phase Markov renewal process 18 2.4.L Definitions 18 2.4.2 Results 23 2.5 Multi-phase Markov renewal process with an infinite number of phases, 28 3 The Multi-phase M/G/t Queue 31 3.1 Introduction . 31 3.1.1 Themodel ... 34 3.7.2 Probabilistic elements 36 3.2 The martingale 37 3.2.L Stability and recurrence 38 3.2.2 Regularity of the stopping times 40 3.2'.3 Use of the Optional Sampling Theorem 45 3.2.4 Martingale arguments for stability 48 3.3 Relationship with the MRP 50 v 3.3.1 Discussion 51 3.4 Generalisations of server behaviour 53 3.4.1 Blocking versus zero service time 56 3.4.2 Later modifi.cations 57 3.5 Generalisations of the phases 58 3.6 Infinitely-many phases 60 .],t A single-phase example 61 4 Two-phase examples 62 4.0,1 Motivation 63 4.1 Results 65 4.2 Fixed upward threshold 7t 4.2.L Some limiting cases (( 4.2.2 Modifications 78 4.3 The geometrically-distributed random-time threshold 80 4.3.1 Some limiting cases 85 4.3.2 Modifications 86 4.4 Fixed-Time Threshold 87 4.4.L Some limiting cases 92 4.4.2 Modifications 92 4.5 Other random thresholds 93 4.6 The probability of a given phase 94 4.6.I The length of the phases 95 4.7 Summary 97 5 Three-phase examples 100 5.1 Two fixed upward thresholds r02 5.2 Downward thresholds 108 6 A breakdown/repair model 111 7 A two-threshold, infinite-phase example. LL7 7.I Motivation . I20 vl 7.2 Results L2L 7.3 Numerica^l examples I27 7.4 The probability of a given regime r37 7.5 The M/G/1/N+1 queue 139 7.5.I A check of the blocking probabilities L4T I Conclusion t43 8.1 Further work 143 8.2 Block-matrix geometric techniques L44 8.3 Dénoûement . r46 A Probability and Martingales t47 A'.1 Elementary probability theory and notation r47 4.1.1 Random variables 148 A.L.2 Independence L49 4.1.3 Expectation 150 4.1.4 Convergence 153 4.1.5 Conditional expectation 154 4.2 Martingale theory 156 B Vectors and rnatrices 159 8.1 Eigenvalues and eigenvectors . 160 8.2 Norms 161 8,3 Non-negative matrices 162 8.4 Matrices and probability 163 C Queueing theory 165 C.1 The Poisson process. 165 C.2 Queues 166 C.2.I Queue discipline 167 c.3 The embedded process r67 C,4 Useful theorems 168 C.5 Waiting times 169 vll List of Figures 2.L The multi-phase Markov renewal process. 19 2.2 A generalised Markov renewal process. 28 2.3 The multi-phase renewal process with an infinite number of phases. 30 4.7 The two-phase Markov rene\Mal process. 63 5.1 The three-phase Markov renewal process . 101 5.2 The MRP for one upwards and one downwards threshold . 109 5.3 The three-phase MRP for one upwards and one downwards threshoid. 110 7.I A generalised Markov renewal process. 118 7.2 The multi-phase renewal process with an infinite nurnber of phases. 119 7.3 The two-regime process with K:7, L:4 and deterministic service times . 128 7.4 The two-regime process with K:3, L-l and negative exp. service times . I29 7.5 The two-regime process with K:3, L-3 and negative exp. service times . 130 7.6 The two-regime process with K:5, L:5 and negative exp. service times . 131 7.7 The two-regime process with K:6, L-3 and negative exp. service times . L32 7.8 The two-regime process with K:7,L:4 a,nd negative exp. service times . 133 7.9 The two-regime process with K:7rL:7 and negative exp. service times . I34 7.10 The two-regime process with K-7,L:7 and Erlang order 2 service times 135 7.11 The two-regime process with K:5,L-4 and Erlang order 4 service times 136 vlll Summary Systems such as theMlGll queue are of great interest in queueing theory. Techniques such as Neuts's block matrix methodology have traditionally been used on the more complicated generalisations of this type of quene. In this thesis I develop an alternative method which uses martingale theory and some renewal theory to find solutions for a class of M lG lI type clueues. The theory, originally applied by Baccelli and Makowski to simple queueing problems, derives its key result from Doob's Optional Sampling Theorem. To make use of this result some renewal theoretic arguments a e necessary, This allows one to find the probability generating function for the equilibrium distribution of customers in the system. Chapter 2 develops the renewal theoretic concepts necessary for the later parts of the thesis. This involves using the key renewal theorem on a modified type of Markov renewal process to obtain results pertaining to forward recurrence tifnes. Chapter 3 contains the martingale theory and the main results. The type of processes that can be dealt with are described in detail. Briefly these consist of processes where the busy period is broken into a series of phases. The transitions between phases can be controlled in a number of ways as long as they obey certain rules. Some examples are: phases ending when there are more than a certain number of customers in the system or when-the busy period has continued for a certain number of services. The behaviour of the server can be different in each phase. For instance, the service-time distribution or the service discipline may change between phases. The main result uses Doob's Optional Sampling Theorem and so we must establish a number of conditions on the martingale used. We establish a simpie criterion for these conditions to hold. Finally in this chapter we examine the simplest case, the M/G/1 queue. The following chapters contain a number of examples. Standard probabilistic arguments are used to obtain the necessary conditions and results to use the theorems of Chapter 3. The examples considered include cases with two, three, four and an infinite number of phases. The theoretical results are supported by a number of simulations in the latter case. Finally we have some suggestions for possibie future work and the conclusion. tx , Chapter 1 Introduction Aim The aim of this thesis is to apply a martingale technique to queueing theory. Martingales have been used successfully in financial modelling and elsewhere and there is a well developed theory built around them. One of probability's great achievements is in its use in the study of queues and so it is surprising that such a powerful area of probability such as martingale theory has only been slowly taken up in queueing theory. In order to redress this we have demonstrated how a technique suggested by F. Baccelli and A.M. Makowski can be extended to solve some queueing problems of interest. The Technique The amount published on the use of martingales in queueing theory, compared to the total amount of literature on queueing theory, is minimal. Where martingaies have been used they tend to be deployed on a particular part of a larger problem and not as a general technique for solving problems. Some examples where this is not true are Rosenkrantz (1983), Kella and Witt (1992), Ferrandiz (1993), Baccelli (t046) and Baccelli and Makowski (1985,1986,1989,1991). In these papers the approach is to solve a problem or problems using martingales as the fundamental means.
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