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Product-Form in Queueing Networks Product-form in queueing networks VRIJE UNIVERSITEIT Product-form in queueing networks ACADEMISCH PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Vrije Universiteit te Amsterdam, op gezag van de rector magnificus dr. C. Datema, hoogleraar aan de faculteit der letteren, in het openbaar te verdedigen ten overstaan van de promotiecommissie van de faculteit der economische wetenschappen en econometrie op donderdag 21 mei 1992 te 15.30 uur in het hoofdgebouw van de universiteit, De Boelelaan 1105 door Richardus Johannes Boucherie geboren te Oost- en West-Souburg Thesis Publishers Amsterdam 1992 Promotoren: prof.dr. N.M. van Dijk prof.dr. H.C. Tijms Referenten: prof.dr. A. Hordijk prof.dr. P. Whittle Preface This monograph studies product-form distributions for queueing networks. The celebrated product-form distribution is a closed-form expression, that is an analytical formula, for the queue-length distribution at the stations of a queueing network. Based on this product-form distribution various so- lution techniques for queueing networks can be derived. For example, ag- gregation and decomposition results for product-form queueing networks yield Norton's theorem for queueing networks, and the arrival theorem implies the validity of mean value analysis for product-form queueing net- works. This monograph aims to characterize the class of queueing net- works that possess a product-form queue-length distribution. To this end, the transient behaviour of the queue-length distribution is discussed in Chapters 3 and 4, then in Chapters 5, 6 and 7 the equilibrium behaviour of the queue-length distribution is studied under the assumption that in each transition a single customer is allowed to route among the stations only, and finally, in Chapters 8, 9 and 10 the assumption that a single cus- tomer is allowed to route in a transition only is relaxed to allow customers to route in batches. In all cases necessary and sufficient conditions for a product-form queue-length distribution are given and solution techniques for product-form queueing networks are illustrated. For the equilibrium behaviour emphasis will be on the study of the influence of state-dependent routing on the existence of product-form distributions, that is queueing networks in which customers may be blocked while routing or in which routing may be otherwise influenced by the state of the queueing network are studied. I have chosen to analyse product-form queueing networks for a number of reasons. First, as is observed above, product-forms are useful for prac- tical systems since various solution techniques (also for non product-form queueing networks) are based on product-form results for queueing net- works. Second, I have been interested in balance equations and product- forms are usually obtained when a process satisfies some notion of local v vi Preface balance. Third, because product-forms analysis of queueing networks is a method to obtain exact results, and finally because it is more fun. The basic method used to analyse queueing networks in this mono- graph is a method based on the approach used in physics. In my opinion, a physical interpretation or a physical analogue of a queueing network is very important. This can be seen from various results obtained in this monograph. For example, the dual process discussed in Chapter 5 is a direct consequence of the physical interpretation of a queueing network in terms of potential energy, Norton's theorem as obtained in Chapter 6 is a consequence of the electrical-circuit analogue of a queueing network, and the theory on queueing networks with negative and positive customers as discussed in Chapter 10 shows similarities with Dirac's theory of holes as described in quantum mechanics. Also, the interpretation of the balance equations yielding the equilibrium distribution of a queueing network is a physical interpretation of the behaviour of the queueing network. Global balance which forms the basis of the equilibrium analysis states that the flow out of a state equals the flow into a state. Although flow represents probability flow, the intuitive relation with flow of liquids is obvious. A queueing network is shown to possess a product-form equilibrium distri- bution if the flow into and out of each station of the queueing network is balanced. Therefore, the physical interpretation of a queueing network is very important. In fact, it is shown in this monograph that a queueing network possesses a product-form distribution if and only if a physical interpretation of the rate into and out of a station is possible. Although product-form distributions are the topic of all chapters of this monograph, the various chapters are independent of each other to a large extent. All chapters, except for the introductory chapters, are mainly based on publications and research reports. Therefore, all chapters are self-contained. The following table gives the papers which are the basis of the chapters of this monograph: [3] Transient product form distributions in queueing networks. (with P.G. Taylor) Research report, The University of Western Australia, Department of Mathematics, August 1990, submitted. [4] A note on the transient behaviour of the Engset loss model. To ap- pear: Stochastic Models [5] A dual process associated with the evolution of the state of a queue- ing network at its jumps. Research Memorandum 1991-39, Vrije Uni- Preface vii versiteit, Faculteit der Economische Wetenschappen en Econometrie, Amsterdam, submitted. [6] A generalization of Norton's theorem. (with N.M. van Dijk) Research Memorandum 1991-61, Vrije Universiteit, Faculteit der Economische Wetenschappen en Econometrie, Amsterdam, submitted. [7] Aggregation of Markov chains. To appear: Stoch. Proc. Appl. [8] Spatial birth-death processes with multiple changes and applications to batch service networks and clustering processes. (with N.M. van Dijk) Adv. Appl. Prob. 22, 433-455 (1990). [9] Product forms for queueing networks with state dependent multiple job transitions. (with N.M. van Dijk) Adv. Appl. Prob. 23, 152-187 (1991). [10] Local balance in queueing networks with positive and negative cus- tomers. (with N.M. van Dijk) Research Memorandum 1992-1, Vrije Universiteit, Faculteit der Economische Wetenschappen en Econome- trie, Amsterdam, submitted. Sections are numbered within a chapter. A reference to Section 3 is to Section 3 of the current chapter and a reference to Section 2.3 is to Sec- tion 3 of Chapter 2. Formulas are numbered within a section. A reference to formula (5.7) is to formula (5.7) appearing in Section 5 of the current chapter, and a reference to formula (3.5.7) is to formula (5.7) appearing in Section 5 of Chapter 3. A similar convention is used for theorems and figures. I should like to take this opportunity of thanking the National Operations Research Network in the Netherlands (LNMB { Landelijk Netwerk Mathe- matische Besliskunde) for providing an excellent training programme for Ph.D.-students. In particular, I wish to thank the LNMB for providing funds allowing me to visit The University of Western Australia, Perth, Australia, from July 1, 1990 until August 12, 1990. With respect to this visit my thanks are also due to dr. P.G. Taylor for the fruitful discussions and joint research during this visit. I should like to thank my colleagues at the Department of Econometrics of the Vrije Universiteit for stimulating discussions about this, that and the other during coffee, lunch and tea breaks. viii Contents I am also very grateful to prof.dr. H.C. Tijms and prof.dr. N.M. van Dijk for their encouragement, comments and advice during the past years in which I was employed at the Vrije Universiteit, and to R.D. Nobel for carefully reading parts of this monograph. Finally, I owe a special debt of gratitude to Carla for her constant support during the past years. Amsterdam, February 1992 Richard Boucherie Contents 1 Introduction 1 1 Motivation . 1 2 Outline . 4 2 Preliminaries 11 1 Basic results on Markov chains . 11 2 Queueing network model . 19 3 Balance equations . 25 4 Product-form distributions; literature . 29 5 Review of assumptions . 38 3 Transient product-form distributions 39 1 Introduction . 39 2 Model and canonical representation . 40 2.1 Canonical form for the open network . 42 2.2 Canonical form for the closed network . 44 3 Sufficient conditions . 46 4 Necessary conditions for the open network . 50 5 Necessary conditions for the closed network . 56 6 Discussion and general remarks . 59 4 Transient behaviour of the Engset loss model 63 1 Introduction . 63 2 Engset loss model . 64 3 Transient queue-length distribution . 66 5 Dual processes 75 1 Introduction . 75 2 The primal process . 77 3 Definition of the dual process . 83 ix x CONTENTS 4 The dual routing function . 88 4.1 Similar transitions . 90 4.2 Similar subtransitions . 93 4.3 Similar transitions; reversed potential . 94 4.4 Similar subtransitions; reversed potential . 95 5 Relation between the primal and the dual process . 96 5.1 Equal probability flow . 97 5.2 Palm probabilities . 100 6 Examples . 105 6.1 Complementary slackness relations . 105 6.2 Blocking examples . 106 6.3 Alternative transition rates; dual states . 109 6.4 Alternative transition rates; traffic equations . 110 6.5 Self-dual process . 111 6.6 Customer-vacancy duality . 112 6 Norton's theorem 115 1 Introduction . 115 2 Electrical circuit theory . 117 3 Queueing network model . 120 4 Decomposition into clusters . 124 4.1 Routing: Conditions . 124 4.2 Routing: Examples . 129 4.3 Service: Conditions . 134 4.4 Service: Examples . 135 5 Norton's theorems . 138 6 Practical applications of Norton's theorems . 146 7 Examples . 149 7.1 Global throughput determines local behaviour; work- load balancing . 149 7.2 Internal blocking . 151 7.3 Nested aggregation .
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