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Stability in Adversarial Queueing Theory Panagiot Is Tsaparas Stability in Adversarial Queueing Theory Panagiot is Tsaparas ,A thesis submitted in conformity with the requirements for the degree of Master of Science Graduate Department of Corn puter Science University of Toronto @ Copyright by Panagiotis Tsaparas 1997 National Library Bibliothèque nationale 1+m of Canada du Canada Acquisitions and Acquisitions et Bibliographie Services services bibliographiques 395 Wellington Street 395, rue Wellington OttawaON KlAON4 Ottawa ON KIA ON4 Canada Canada The author has granted a non- L'auteur a accordé une licence non exclusive licence allowing the exclusive permettant à la National Library of Canada to Bibliothèque nationale du Canada de reproduce, loan, distribute or sell reproduire, prêter, distribuer ou copies of this thesis in microform, vendre des copies de cette thèse sous paper or electronic formats. La forme de microfiche/nlm, de reproduction sur papier ou sur format électronique. The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts fiom it Ni la thèse ni des extraits substantiels may be printed or othenirise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation. Abstract -4 queueing network is a set of interconnected servers. Customers are injected continu- ously in the system. inducing some workload for each semer. A fundamental issue that arises in this context is that of stability: will the total workload in the sÿstem remain bounded over tirne:' in t his thesis, we investigate the question of stability within the model defined in [BKRf 961. where workload is induced by an adversarÿ. Prior work in the adversarial model considers only packet routing networks. a speciai case of queueing networks, where ail custoniers require unit service tirne. We estend this mode1 to include more general queueing networks, where the customers may require different service times at different servers of the network. We show that the queueing poiicies proven to be universally stable for packet rou t- ing networks, are universally stable in the new adversarial mode1 as well. Llnlike the packet routing networks. we prove that in the new model. the unidirectional ring is not universally stable. Furtherrnore. we define four new queueing policies for the new model. which depend upon the service times of the customers. and we esarnine their stability. Finally. following the outline in [BEiSWSG]. we provide a detailed proof that the Nearest-To-Go queueing policy can become unstable when workload is injected at arbitrarily small constant rate. Acknowledgements This thesis would not have been possible without the help of my supervisor .-\Ilan Borodin, who supported me in every possible way. His ent husiasrn was alwaÿs a source of motivation. while his humor and class made working with him an enjoyable esperience. Mÿ greatest thanks also to my second reader Charlie Rackoff. who devoted a lot of tirne and care in reading my thesis. and whose comments improved substantially its final form. 1 would also like to thank Jon Kieinberg and Madhu Sudan who generously offered me help, advice and information in the early stages of my thesis and Demetris .khlioptas for many helpful suggestions and discussions. Lydia Kavraki had always been a mentor throughout my undergraduate and graduate years. 1 owe a lot to my friends here in Toronto who helped me keep my life in perspective. In no particular order. Theodoulos Garefalakis. Slichalis and Petros Faloutsos, Vasilis Tzcrpos, Demetris Achlioptas, Mimi Vista, Stergios Anastasiades, Nikos Iioudas, Themis Palpanas. 'iick Zahariades. Vaso Bartzoka. Florine Tseu. Melanie Balko. and Lucia Moura. They al1 added something new to my life. -4 special thanks to Kathy fin. the best graduate secretary ever. 1 would also like to thank rny friends from Greece. Lefteris, Aris. -4ntwnhs Tzo, -4ntwnhs Fou. Atalanti, Mitsos. k'iannis Spanakis. Although miles away. I feel that ttiey somehow t rave1 wit h me. There are too many things 1 should thank Thalia for. to sum them up here: she always managed to give me ent husiasm to seize life. Last but certainly not least. 1 would like to thank my family which has alwaÿs been a strong reference point in rny life, and always will be. Contents 1 Introduction 1 1.1 TheProblem ................................... 1 1.2 Adversarial Queueing Theo- .......................... 2 1.3 Ourresults .................................... 4 2 Previous Work 6 2.1 Classical Queueing Theory ............................ 6 . 2.1.1 Single-ciass networks ........................... I 2-12 ,L iulticlass networks ........................... S 2.1.3 Instabilities in multiclass networks ................... 10 2.2 .Applications to packet routing ......................... 12 3 The New Adversarial Mode1 15 3.1 Mode1 definition ................................. 1.5 3.2 Universal stability of queueing policies ..................... 18 3.2.1 SISisuniversallystable ......................... 19 3.2 LIS is universally stable ......................... 'LO 3.2.3 FTG is universally stable ........................ 22 3.3 Instability of the ring ............................... 24 3.3.1 Instability of NTG . LTTG and MSD on the ring ........... 25 3.3.2 InstabiIity of FIFO on the ring ..................... 21 3.1 The MTTG queueing policy ........................... 30 3.4.1 Stability of MTTG ............................ 31 3.4.2 Instability of MTTG ........................... 33 4 Instability of NTG for constant rate injections 36 4.1 Mode1 description ................................. 36 1.2 Definitions ..................................... 38 4.3 System properties ................................. 41 4.4 The Proof of Instability ............................. 47 4.5 InstabiIity of packet routing networks ..................... -19 5 Conclusions 53 List of Figures 41 The4x4network.t'. .............................. 37 4.2 The 4 x 4 network -1;' .............................. 51 Chapter 1 Introduction The Problem Qiieueing networks are important models of compiex systems. such as packet routing net- works. rnanufacturing systems. and time shared cornputer systems. In this nork we analyze the behavior of queueing networks where customers are continuously injected into the sys- tem. Each customer induces some workload in the network. A crucial question arising in this contest is that of stability - will the total workload in the sÿstem remain bounded as the systern runs for an arbitrarily long period of time*? The answer to this question typi- cally depends on the rate at which tvorkload is injected into the system and the queueing policy that is used at the nodes of the network. A queueing network is a set of iriterconnected servers. We assume tliat eacli server is associated with a single queue of infinite size. and each queue is serviced by a sirigte server. Customers arrive at the servers of the network. When the service of some customer at some server is completed the custorner rnoves to another server or it esits the system. IF more than one customer wishes to be serviced at some server S. the queueing policy chooses one of the customers to be serviced nest: the rernainder of these customers wait in the queue of server S. ..\ queueing policy is work-conserving if a server is never idle whenever there is at least one customer in the queue of the server. A queueing policy is called non-preemptive if the service of some customer at some server is never interrupted until it is completed. In general, a queueing network is characterized by the following parameters. 0 The process that injects the custorners in the network. 0 The routing of the customers in the network. 0 The service times of the custorners at the servers. The queueing policy. It is obvious that if the rate at which workload arrives at some server is greater than the rate at which the workload is serviced at that server. then the system becomes unstable. .A natural condition for stability, referred to as the rate condition. is that the rate at which workload is induced at any server is strictly Iess than the service rate of the server. Intuitively, it seems that the rate condition should be sufficient for stabiIity of any queueing network. This is true for man. systems [liIei5. Iie179], but it is not true in al1 cases [LI<91, RS92. DW96. Bra94a. Bra9lb. SeiS-l]. The question of necessary and sufficient conditions for stability is one of the rnost interesting questions in the area of queueing networks. Queueing networks are the object of study of queueing theory. Typical assumptions in queueing theory are that the customers are injected according to a time invariant dis- tribution (often a Poisson process), and that the service time of a custonier, or customer type at some server is a random variable following a tirne invariant distribution ( often an esponential distribution). In this thesis we work within a model proposed by Borodin et al. [BIiR+96]. in which probabilistic assumptions are replaced by worst-case in pu ts. The underlying goal is to determine whether it is feasible to prove stability results even when customers are injected by an adversary, rather than a randomized process: the adversary deterrnines the route of the customers in the system and the service tirne they receive at each server. For systems that are not stable, we seek to establish instability within the weakest possible adversarial model. 1.2 Adversarial
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