Markov-Modulated and Feedback Fluid Queues

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Markov-Modulated and Feedback Fluid Queues Markov-modulated and feedback fluid queues Werner Scheinhardt Faculty of Mathematical Sciences University of Twente P.O. Box 217 7500 AE Enschede The Netherlands ISBN 90-3651248-4 MARKOV-MODULATED AND FEEDBACK FLUID QUEUES PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Universiteit Twente, op gezag van de rector magnificus, prof. dr. F.A. van Vught, volgens besluit van het College voor Promoties in het openbaar te verdedigen op vrijdag 4 december 1998 te 15.00 uur. door Willem Richard Werner Scheinhardt geboren op 24 februari 1969 te Santiago Dit proefschrift is goedgekeurd door de promotor en de assistent promotor, prof. dr. ir. J.H.A. de Smit dr. ir. E.A. van Doorn Voorwoord Aan het eind van dit proefschrift gekomen, rest nog het schrijven van het begin ervan, het voorwoord. Gebruikelijk is om daarin allen te bedanken die op ´e´en of andere wijze aan de totstandkoming van het proefschrift hebben bijgedragen. Graag houd ik deze traditie in ere, en wel omdat de volgende personen dit ten volle verdienen. Uiteraard wil ik beginnen met mijn dagelijks begeleider Erik van Doorn hartelijk te bedanken voor zijn inzet en enthousiasme. Van v´o´or het eerste sollicitatiegesprek tot na het laatste 2-teken was hij intensief betrokken bij mijn doen en laten. Een belangrijk deel van het onderzoek in dit proefschrift is in samenwerking met hem tot stand gekomen. Ook de samenwerking met Dick Kroese was aangenaam en productief. Hij wist vaak antwoorden die ik zelf niet zou hebben gevonden, en leerde mij omgaan met het symbolisch manipulatie-pakket \Mathematica", dat heel wat \tiresome analysis" heeft uitgevoerd. De overige twee personen die direct betrokken waren bij het hier beschreven onderzoek zijn Jacques Resing en Ivo Adan van de Technische Universiteit Eindhoven; ook met hen was het een eer en genoegen om samen te werken. Vervolgens gaat natuurlijk mijn dank uit naar de leden van de promotiecommissie, met op de eerste plaats mijn promotor, Jos de Smit. Hoewel hij het vaak druk had, toonde hij zich altijd ge¨ınteresseerd en betrokken. Ook de andere commissieleden (Hans van den Berg, Onno Boxma, Wim Nawijn, Ignas Niemegeers en Henk Zijm) ben ik dank verschuldigd, zowel voor de uitvoering van de hen toevertrouwde formele taken als voor het aangename en nuttige contact daarbuiten. De nog niet genoemde (oud-)collega's van de afdeling DOS worden allen hartelijk be- dankt voor hun interesse en gezelschap. Helaas is het niet mogelijk om hen allen hier bij naam te noemen, dus wil ik het laten bij deze twee mensen: Ineke van Eijkern-Moraal, onze secretaresse die op praktisch gebied altijd de beste oplossing, het juiste formulier, en niet te vergeten het nieuwste nieuwtje kon bieden. En natuurlijk Rieske Hadianti, met wie ik bijna 3 jaar lang de kamer met plezier heb gedeeld, dus: terima kasih sekali, Rieske! Tijs Huisman, Sindo N´u~nez Queija en Simone Sassen wil ik bedanken voor de stimule- rende gesprekken en leuke vakanties in binnen- en buitenland. John Hoff dank ik voor de illustratie op de voorkant van dit proefschrift. Ten slotte bedank ik hier graag van harte de vier mensen die deze vier jaar zonder twijfel het meest hebben meegeleefd, namelijk mijn ouders, Karen, en natuurlijk Sonja. Werner Scheinhardt Enschede, november 1998 v vi Contents Voorwoord v Contents vii 1 Introduction 1 1.1Introduction................................... 1 1.1.1 Fluidmodelsforqueueingsystems.................. 1 1.1.2 Outlineofthischapter......................... 2 1.2ThetraditionalMarkov-modulatedfluidmodel................ 2 1.3LiteratureonMarkov-modulatedfluidmodels................ 4 1.3.1 Introduction............................... 4 1.3.2 Justificationfortheuseoffluidmodels................ 5 1.3.3 Findingthestationarydistribution.................. 6 1.3.4 Extensionsandotheraspects..................... 8 1.4Connectionwithothertypesoffluidmodels................. 11 1.5Contributionsinthisthesis........................... 13 1.5.1 Feedback................................. 13 1.5.2 Infinite state space N ......................... 15 1.5.3 Overview................................ 16 2 Some birth-death fluid models 19 2.1Introduction................................... 19 2.2Preliminaries.................................. 20 2.3Finitestatespace................................ 22 2.4 Infinite state space with N+ < 1 ....................... 24 2.4.1 Analysis................................. 24 2.4.2 Example................................. 26 2.5 Infinite state space with N− < 1 ....................... 28 2.5.1 Analysis................................. 28 2.5.2 Example................................. 34 2.5.3 CompletionofproofofTheorem2.9.................. 37 vii 3 A birth-death fluid model with feedback 43 3.1Introduction................................... 43 3.2 Finite waiting room and infinite credit reservoir ............... 45 3.2.1 Preliminaries.............................. 45 3.2.2 Stationaryjointdistribution...................... 46 3.2.3 Examples................................ 48 3.3 Infinite waiting room and infinite credit reservoir .............. 49 3.3.1 Preliminaries.............................. 49 3.3.2 Stationaryjointdistribution...................... 50 3.3.3 Stationarysojournandwaitingtimedistributions.......... 52 3.4Alternativeanalysisviadiscretization..................... 52 3.5 Infinite waiting room and finite credit reservoir ............... 55 3.5.1 Preliminaries.............................. 55 3.5.2 Analysisoftheapproximativemodel................. 56 3.5.3 Validationoftheapproximativemodel................ 59 3.6Two-leveltrafficshaper............................. 60 3.6.1 Introduction............................... 60 3.6.2 Numericalresults............................ 62 4 Some two-buffer fluid models 65 4.1Introduction................................... 65 4.2Tandemmodel................................. 66 4.3 Tandem model: stability ............................ 68 4.4Tandemmodel:stationarymarginaldistributions.............. 71 4.5Tandemmodel:stationaryjointdistribution................. 75 4.6Dualmodel................................... 82 4.7 Dual model: stability . ............................ 83 4.8Dualmodel:stationaryjointdistribution................... 84 4.9Dualmodel:spectralanalysis......................... 88 4.9.1 Introduction............................... 88 4.9.2 Anapproximativemodel........................ 88 4.9.3 Takingthelimit............................. 92 5 A two-buffer fluid model with feedback 97 5.1Introduction................................... 97 5.2Model...................................... 99 5.2.1 Generalmodel.............................. 99 5.2.2 Modelling a two-level traffic shaper . ................. 100 5.2.3 Modelling a tandem queue with finite second buffer . ...... 101 5.3 Stability ..................................... 101 viii 5.4 Stationary marginal distribution of (Ct).................... 102 5.4.1 Introduction............................... 102 5.4.2 Anembeddedprocess.......................... 103 5.4.3 Twospecialcases............................ 105 5.4.4 Generalcase............................... 108 5.5Stationaryjointdistribution.......................... 111 5.6 Special case: infinite credit buffer ....................... 124 5.7Generalisation:finitedatabuffer....................... 127 Bibliography 131 Index 139 Samenvatting 141 Summary 143 Over de auteur 145 About the author 145 ix x Chapter 1 Introduction 1.1 Introduction 1.1.1 Fluid models for queueing systems In traditional queueing theory the object of interest is usually a system of one or more servers at which customers arrive, who want to receive some kind of service. Since there is uncertainty about the actual arrival times of the customers and/or their service re- quirements, they may have to wait for service in a queue. In the course of this century many interesting and insightful results have been found for a wide range of variants of this model. The basic assumptions in any of these models can be summarized in the no- tation A=B=n=m − S, which is an extension of Kendall's characterisation of a queueing system. Here, A and B indicate the distributions of interarrival times and service times (that may or may not be independent), n is the number of servers, m the maximum num- ber of customers that can be accommodated in the system, and S specifies the service discipline. In their search for more advanced ways to model practical queueing situations, researchers have added a wide variety of features to this basic model, such as various types of feedback, multi-class customers, priorities with or without preemption, server vacations, polling systems, state-dependent arrivals and services, impatient customers, batch arrivals, batch services, and many more. Indeed, the only thing that seemed to stand the test of time was that in all of these models individual customers arrive to receive some kind of service, reflecting the discrete nature of the modeled phenomena. In the last ten to fifteen years, this last remaining pillar has been torn down as well, allowing for models in which some continuous entity, referred to as fluid, takes the role of the individual customers. In these models, fluid flows into a fluid reservoir according to some stochastic process. The server may be thought of as a tap at the bottom of the reservoir, allowing fluid to flow out. The rate at which this happens is often constant, but may also be stochastic, possibly including zero. Since the fluid reservoir takes the role of the traditional customer queue, it is often referred to as fluid
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