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Extremes and Fluid Queues UvA-DARE (Digital Academic Repository) Extremes and fluid queues Dieker, A.B. Publication date 2006 Document Version Final published version Link to publication Citation for published version (APA): Dieker, A. B. (2006). Extremes and fluid queues. Ponsen & Looijen. General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl) Download date:30 Sep 2021 Extremes and uid queues Extremes and uid queues / Antonius Bernardus Dieker, 2006 Proefschrift Universiteit van Amsterdam Met lit. opg. { Met samenvatting in het Nederlands Omslagontwerp door Tobias Baanders Gedrukt door Ponsen & Looijen BV ISBN 90-5776-151-3 Dit onderzoek kwam tot stand met steun van NWO Extremes and uid queues ACADEMISCH PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magni¯cus prof. mr. P. F. van der Heijden ten overstaan van een door het College voor Promoties ingestelde commissie, in het openbaar te verdedigen in de Aula der Universiteit op donderdag 9 maart 2006, te 10.00 uur door Antonius Bernardus Dieker geboren te Amsterdam Promotiecommissie: Promotor: prof. dr. M. Mandjes Overige leden: dr. A. A. Balkema prof. dr. ir. O. J. Boxma prof. dr. C. A. J. Klaassen prof. dr. I. Norros prof. dr. T. Rolski dr. ir. W. R. W. Scheinhardt dr. P. J. C. Spreij Faculteit der Natuurwetenschappen, Wiskunde en Informatica Dankwoord Dit boekje is het resultaat van vier jaar zweten. Toen ik aan dit project begon kon ik me nauwelijks voorstellen wat het schrijven ervan allemaal zou behelzen, maar het uiteindelijke resultaat vervult mij met trots. Hoewel alleen mijn naam op de omslag staat, had ik het niet kunnen schrijven zonder de hulp van velen. Graag richt ik het woord tot enkele personen in het bijzonder, en haal enkele speciale herinneringen naar boven. Michel, jij bent zonder twijfel de persoon die de meeste invloed heeft gehad op mijn werk; de hoofdstukken 8, 9, 10 en 14 hebben wij samen geschreven. Niet alleen deze samenwerking heb ik als zeer prettig ervaren, maar ook het vertrouwen dat jij altijd in mij gesteld hebt. De deur stond en staat altijd open voor een uitgebreide discussie over uiteenlopende onderwerpen. Veel dank hiervoor. Dit proefschrift heeft ook erg gepro¯teerd van vele discussies met collega's uit binnen- en buitenland, waarvan ik hier slechts enkelen kan noemen. Hoofdstuk 13 is gezamenlijk werk met Krzy¶s en Tomasz; jullie ben ik in het bijzonder erkentelijk. Bert, jij hebt mij altijd van goede adviezen voorzien. Misja, bedankt voor je ongezouten kritiek op een deel van het manuscript. Verder gaat mijn dank uit naar Thyra en Richard voor alle Twentse inspanningen en het instituut EURANDOM voor de Brabantse gastvrijheid. Ik kan niet aan de afgelopen jaren denken zonder dat mooie herinneringen naar boven komen aan de plaatsen waar dit werk mij heeft gebracht. Ik denk daarbij niet alleen aan de grote conferenties in BraziliÄe en Canada, maar ook aan de werkbezoeken aan Polen en Ierland (waarvoor dank, Krzy¶s, Tomasz en Neil). Een langere tijd heb ik in Parijs doorgebracht, op deze mogelijkheid geattendeerd door Peter. In Parijs heb ik de basis gelegd voor een artikel met Marc [109]. Ren¶e, van alle kamergenoten neem jij een speciale plaats in. Vier jaar lang wist jij het uit te houden tussen mijn ongelooijke troep, waarvoor respect op z'n plaats is. Na ruim twee jaar werd onze kamer aangevuld met Regina; enigszins verrast door de saaie werkomgeving van twee mannen leukte jij de kamer al snel op met een paar planten. Loijc, jou wil ik bedanken voor jouw geduld als ik Frans probeerde te praten. Mijn familie heeft mij van heel dichtbij zien worstelen aan dingen die ik hen nooit precies heb kunnen uitleggen. Pap, mam, Jos, Miranda en Marjon, heel erg bedankt voor alle steun. Jasper, jij weet altijd een lach op m'n gezicht te toveren. Jullie weten dat de afgelopen vier jaar niet altijd zo makkelijk was als dit gelikte boekje doet vermoeden. De wiskundige stellingen en talloze formules uit dit proefschrift verbleken bij mijn grootste ontdekking van de afgelopen vier jaar: Cyndi. Lieverd, ik weet niet hoeveel punten ik je moet geven voor alles wat je voor me hebt gedaan! Ik ben blij dat we samen van Ierland kunnen gaan genieten. Amsterdam, januari 2006 Ton Contents 1 Introduction 1 1.1 The uid queue . 3 1.2 Fluid networks . 6 1.3 Approximations for the input process . 8 1.4 Outline of the thesis . 10 2 Techniques 13 2.1 Regular variation . 13 2.2 Weak convergence . 15 2.3 Large deviations . 17 2.4 Tail asymptotics . 20 A Gaussian queues 23 3 Background on Gaussian processes 25 3.1 Gaussian measures and large-deviation principles . 26 3.2 Two fundamental inequalities . 28 3.3 The double sum method . 29 3.4 Outline of Part A . 32 4 Conditional limit theorems 33 4.1 Introduction . 33 4.2 Queueing results . 36 4.3 Preliminaries . 38 4.4 Weak convergence results . 41 4.5 Proofs for Section 4.2 . 47 4.6 Proof of Proposition 4.5 . 48 5 Extremes of Gaussian processes 51 5.1 Introduction . 51 5.2 Description of the results and assumptions . 53 5.3 Special cases: stationary increments and self-similar processes . 57 5.4 A variant of Pickands' lemma . 64 5.5 Four cases . 67 5.6 Upper bounds . 73 5.7 Lower bounds . 77 iv Contents 6 Reduced-load equivalence 85 6.1 Introduction . 85 6.2 Description of the result . 87 6.3 Examples . 88 6.4 Proof of Theorem 6.2 . 89 B Simulation 93 7 Background on simulation 95 7.1 Importance sampling and asymptotic e±ciency . 96 7.2 Examples: random walks . 98 7.3 Outline of Part B . 101 8 Conditions for asymptotic e±ciency 103 8.1 Introduction . 103 8.2 The Varadhan conditions for e±ciency of exponential twisting . 104 8.3 Comparison with Sadowsky's conditions . 107 8.4 Uniqueness of the exponential twist . 110 8.5 An example . 111 9 Random walks exceeding a nonlinear boundary 113 9.1 Introduction . 113 9.2 Sample-path large deviations and exceedance probabilities . 114 9.3 Path-level twisting . 115 9.4 Step-level twisting . 120 9.5 A comparison . 124 10 Simulation of a Gaussian queue 127 10.1 Introduction . 127 10.2 The bu®er-content probability . 129 10.3 Simulation methods . 133 10.4 Evaluation . 139 10.5 Concluding remarks . 144 10.A Appendix: proofs . 145 C L¶evy-driven uid queues and networks 149 11 Background on L¶evy processes 151 11.1 Random walks . 151 11.2 L¶evy processes . 155 11.3 Outline of Part C . 157 12 Factorization embeddings 159 12.1 Introduction . 159 12.2 On factorization identities . 161 12.3 L¶evy processes with phase-type upward jumps . 164 12.4 Perturbed risk models . 166 12.5 Asymptotics of the maximum . 173 Contents v 13 Quasi-product forms 179 13.1 Introduction . 179 13.2 Splitting times . 181 13.3 The P-distribution of (X; F ) . 184 P 13.4 The k#-distribution of H . 186 13.5 Multidimensional Skorokhod problems . 189 13.6 Tandem networks and priority systems . 194 13.A Appendix: a compound Poisson process with negative drift . 200 14 Extremes of Markov-additive processes 205 14.1 Introduction . 205 14.2 A discrete-time process and its extremes . 207 14.3 Markov-additive processes and their extremes . 223 14.4 The uid queue: theory . 233 14.5 The single queue: examples . 237 14.6 Tandem networks with Markov-additive input . 242 14.7 Concluding remarks . 244 Bibliography 247 Author Index 261 Samenvatting (Summary) 265 About the author 269 CHAPTER 1 Introduction In queueing theory, the main entity is a station where service is provided; examples include counters, call centers, elevators, and tra±c lights. Customers seeking this service arrive at the system, where they wait if the service facility cannot immediately allocate the required amount of service. They depart after being served. In most of these applications, it is intrinsically uncertain at what time customers arrive and how long they need to be served. This explains why probability theory plays an important role in the analysis of queueing systems. In order to design these systems optimally, it is desirable to know how this randomness in- uences the performance of the system. Therefore, it is crucial to study system characteristics that reect this performance. For instance, one may wish to analyze the probability distribu- tions of the queue length or the waiting time of a customer, as a function of the random arrival process and service requirements. The investigation of such a.
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