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VU Research Portal VU Research Portal Fluid Limit Approximations of Stochastic Networks Remerova, M. 2014 document version Publisher's PDF, also known as Version of record Link to publication in VU Research Portal citation for published version (APA) Remerova, M. (2014). Fluid Limit Approximations of Stochastic Networks. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal ? Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. E-mail address: [email protected] Download date: 27. Sep. 2021 Fluid Limit Approximations of Stochastic Networks ISBN: 978-90-6464-775-8 c 2014 M. Remerova Cover design by Maria Remerova. The cover visualises the general idea of fluid limits: they provide a “helicopter view” of the system trajectories which strips away unessen- tial details and allows to see some key features. Print: GVO Drukkers & Vormgevers B.V. | Ponsen & Looijen, Ede VRIJE UNIVERSITEIT Fluid Limit Approximations of Stochastic Networks ACADEMISCH PROEFSCHRIFT ter verkrijging van de graad Doctor aan de Vrije Universiteit Amsterdam, op gezag van de rector magnificus prof.dr. F.A. van der Duyn Schouten, in het openbaar te verdedigen ten overstaan van de promotiecommissie van de Faculteit der Exacte Wetenschappen op dinsdag 20 mei 2014 om 13.45 uur in de aula van de universiteit, De Boelelaan 1105 door Maria Remerova geboren te Chita, Rusland promotoren: prof.dr. A.P. Zwart prof.dr. S.G. Foss To my parents, Vladimir Frolkov and Svetlana Frolkova Acknowledgements This thesis would not have been possible without the help and encouragement of many people. I take this opportunity to express my gratitude to them. First and foremost, I want to thank my advisors Bert Zwart and Sergey Foss, who have always so generously and enthusiastically shared their knowledge with me and kept me motivated. Bert and Sergey Georgievich, I am deeply grateful for your devoted guid- ance, your patience with my style of working, and your genuine caring attitude. Sergey Georgievich, thank you for converting me from a careless writer to a self-critical one. Bert and Maria, you helped me a great deal to settle in this country, I really appreciate it. Also thank you for the advice to try teaching at the TU/e which I am very much enjoying! The results described in this dissertation were obtained in collaboration with Bert, Sergey and Josh Reed. Josh, it was truly a pleasure working with you. Thank you for coming all the way from the US to serve on my defence committee. I also thank the other com- mittee members Rob van den Berg, Sem Borst, Michel Mandjes and Floske Spieksma for the careful reading of my manuscript and valuable comments, and for cooperation in setting the defence date. I gratefully acknowledge the financial support for my Ph.D. project provided by The Netherlands Organization for Scientific Research (NWO). I thank my institute, the CWI Amsterdam, for making my relocation to the Netherlands a fuss-free and joyful experi- ence, and for the many opportunities I have had there as a Ph.D. student: from various career trainings to language courses to ping-pong tables. I thank the Dutch Network on the Mathematics of Operations Research (LNMB) for the excellent courses in queueing theory and OR. In particular, I enjoyed the convex analysis course by Erik Balder, and I apply the techniques I learned there in my dissertation. A lot of excitement in my Ph.D. life has been due to conferences and working visits since I love travelling so much. I will cherish the memories of the cozy Cambridge and the holy Jerusalem and many others. I am happy to have met many new friends during these years: Sihan, Demeter, Kerem, Jan-Pieter, Joost, Arnoud, Natalia (the CWI crowd so far), Melania (VU), Lera (RUG), Tanya (UvA), the list being far from complete. To you, guys, and to my very special old friend Lena (Twente), thank you (check all that apply ©) for our collaborative achieve- ments in the LNMB courses and research, for the whole lot of your help on all sorts of i ii occasions including production of this thesis, for the fun we had together, and for you being there to lend me your ear and give me good advice. To Mitya, my husband, from the bottom of my heart, thank you for being with me all the way, for loving me and never reproaching me. I wish I had half of your kindness. Finally, I want to thank my parents and brother. Although me moving so far away to do my studies was not easy for them, they chose to trust and support my aspirations. Mum, Dad, Yegor, thank you so much! I am where I am now because of you, and every compliment and congratulation I get on my work is equally yours! I hope this book will be some joy to you. Masha March 2014 Contents 1 Introduction 1 1.1 Stochastic networks . .1 1.2 Fluid limits . .2 1.3 Motivation 1: fluid limits for stability . .6 1.4 Motivation 2: fluid limits as approximations . .8 1.5 Methods of proving convergence to fluid limits . 12 1.6 Overview of the thesis . 15 1.7 Notation . 17 2 An ALOHA-type Model with Impatient Customers 19 2.1 Introduction . 19 2.2 Stochastic model . 20 2.3 Fluid model . 22 2.4 Fluid limit theorem . 25 2.5 Proof of Theorem 2.1 . 26 2.5.1 Non-zero initial state . 26 2.5.2 Zero initial state . 28 2.6 Proof of Theorem 2.2 . 33 2.7 Proof of Theorem 2.3 . 34 2.7.1 A representation of the population process . 35 2.7.2 C-tightness and limiting equations . 37 2.7.3 Proof of Lemma 2.3 . 39 2.7.4 Proof of Lemma 2.4 . 47 2.8 Proofs of auxiliary results . 50 3 Bandwidth-Sharing Networks with Rate Constraints 51 3.1 Introduction . 51 3.2 Stochastic model . 53 3.3 Fluid model . 56 3.4 Fluid limit theorem . 63 3.5 Fixed-point approximations for the stationary distribution . 64 3.6 Proof of fluid model properties . 67 3.6.1 Proof of Theorem 3.1 . 67 3.6.2 Proof of Theorem 3.2 . 69 iii iv CONTENTS 3.6.3 Proof of Theorem 3.3 . 72 3.7 Proof of Theorem 3.5 . 75 3.7.1 Load process . 76 3.7.2 Compact containment . 76 3.7.3 Asymptotic regularity . 77 3.7.4 Oscillation control . 82 3.7.5 Fluid limits are bounded away from zero . 83 3.7.6 Fluid limits as FMS’s . 85 3.8 Proof of Theorem 3.6 . 88 3.9 Proofs of auxiliary results . 90 4 Random Fluid Limit of an Overloaded Polling Model 95 4.1 Introduction . 95 4.2 Stochastic model . 99 4.3 Connection with MTBP’s . 101 4.4 Fluid limit theorem . 103 4.5 Proofs for Section 4.3 . 106 4.5.1 Proof of Lemma 4.1 . 106 4.5.2 Proof of Lemma 4.3 . 107 4.6 Proofs for Section 4.4 . 111 4.6.1 Additional notation . 111 4.6.2 Preliminary results . 112 4.6.3 Proof of Thereom 4.1 . 117 4.6.4 Proof of Theorem 4.2 . 118 4.7 Proofs of auxiliary results . 121 5 PS-queue with Multistage Service 123 5.1 Introduction . 123 5.2 Stochastic model . 125 5.3 Fluid model . 127 5.4 Fluid limit theorem . 130 5.5 Equivalence of the two fluid model descriptions . 131 5.6 Proof of Theorem 5.3 . 133 5.7 Proof of Theorem 5.4 . 134 5.8 Candidate Lyapunov functions for PS . 137 Bibliography 145 Nederlandse samenvatting 153 Chapter 1 Introduction 1.1 Stochastic networks This thesis belongs to queueing theory — a domain of mathematics that specialises in modeling real-life service systems such as mobile, computer and manufacturing net- works, and pursues the goal of explaining, predicting and controlling the behavior of these systems. Modeling and analysis tools of queuing theory come mainly from prob- ability theory and operations research. In a service system, some participants (servers) deliver a certain type of service to others (customers) according to some rule (service discipline). Depending on a particular appli- cation, these notions can read differently. For example, in a computer network, servers are links, customers are packets of data, and the service discipline is called a transmis- sion protocol. There are simple models that involve a single waiting line, or a queue. More complicated models that consist of a number of queues interacting through the service discipline and/or customer routing are called stochastic networks, where “sto- chastic” refers to the random components of the model. A probabilistic approach to modeling of real-life service systems is very natural. Indeed, when we deal with human behavior, there is always room for uncertainty. For example, call centers do not know when exactly in the course of the day they will receive calls and how long answering them will take.
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