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Performance Analysis of Working Vacation Queueing Models in Communication Systems

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Performance Analysis of Working Vacation Queueing Models in Communication Systems PERFORMANCE ANALYSIS OF WORKING VACATION QUEUEING MODELS IN COMMUNICATION SYSTEMS A Thesis Submitted for the Award of the Degree of DOCTOR OF PHILOSOPHY by COSMIKA GOSWAMI (Roll Number: 05612304) to the DEPARTMENT OF MATHEMATICS INDIAN INSTITUTE OF TECHNOLOGY GUWAHATI GUWAHATI 781039, INDIA October 2010 TH-954_05612304 CERTIFICATE It is certi¯ed that the work contained in the thesis entitled \Performance Analysis of Working Vacation Queueing Models in Communication Systems" by Cosmika Goswami (Roll Number: 05612304), a student in the Department of Mathematics, Indian Institute of Technology Guwahati, for the award of the degree of Doctor of Philosophy, has been carried out under my supervision and that this work has not been submitted elsewhere for a degree. Guwahati 781039 Dr. N. Selvaraju October 2010 Thesis Supervisor iii TH-954_05612304 iv TH-954_05612304 Dedicated to My Husband and My Parents v TH-954_05612304 vi TH-954_05612304 ACKNOWLEDGEMENT First, I would like to thank my supervisor, Dr. N. Selvaraju, without whose guidance and suggestions this work would not have seen the light of day. I am very grateful to him for giving me many patient hearings and correcting me when I have gone astray. Discipline, scienti¯c method and mathematical rigor are some of the virtues that he has repeatedly imbibed in me. I am thankful to the members of my Doctoral committee, Dr. Rajen Kumar Sinha, Dr. Natesan Srinivasan and Dr. Sriparna Bandopadhyay for their valuable suggestions given at crucial junctures during the period of the work. I would like to thank Prof. Jyotiprasad Medhi (Gauhati University), Prof. Krishna B. Athreya (Iowa State University, USA), Dr. Aloke Goswami (ISI, Kolkata) and Dr. T. Venkatesh (IIT Guwahati) for their invaluable conceptual guidance. I would like to thank my many friends, who cannot be named here individually, for helping me throughout. I am thankful to the Indian Institute of Technology Guwahati for being my home for the last ¯ve years and providing me with a healthy academic environment for pursuing research. Special thanks to Larry and Sergey for leaving their Ph.Ds at Stanford and setting up Google. I could never hope to imitate them. I would like to thank my parents, in-laws and dear brother(in-law)s and sisters for having ¯rm belief in me and providing me crucial moral support. I would like to thank my husband for understanding me and for being with me throughout. Above all, I would like to thank God. Cosmika vii TH-954_05612304 viii TH-954_05612304 ABSTRACT This thesis studies, both analytically and through numerical experiments, the performance of queueing models with a `working vacation' policy arising naturally in communication systems, especially in wavelength division multiplexing (WDM) networks. In a queueing system with this vacation policy, the server switches between vacation and non-vacation periods. Unlike in a classical vacation framework, this system serves customers even during vacation periods but at a lower service rate. We study some working vacation queueing models incorporating di®erent features of system characteristics with a view to assisting optimization and guiding the design of new generation systems. First, we consider a single-server queueing model with working vacations and corre- lated arrivals as network tra±c is seldom similar and this often leads to system congestion and packet loss. The model is studied in discrete-time scale by constructing a quasi-birth- death (QBD) process. Since the matrix-geometric method gives an e±cient way to solve homogeneous QBDs, we use this method to analyze and study the performance of the correlated model. The analogous continuous-time model is also outlined. Next, we con- sider a ¯nite-bu®er model with this working vacation policy and correlated arrivals to lay the emphasis on the role of correlation in arrivals on customer loss probabilities. A multi- server model is presented next, where the servers obey asynchronous multiple working vacation policy and the formulated non-homogeneous QBD process is analyzed using the ¯nite truncation method of approximation. A working vacation model with di®erent priority classes of customers is studied as priority based tra±c can enhance network e±ciency and ensure quality of service (QoS). Explicit expressions for system performance measures are obtained and also comparisons are made for di®erent classes of customers. Another important model considered is with retrial or repeated attempts of customers. This model, mostly seen in mobile networks, is analyzed to obtain closed-form solutions. Finally, a queue with impatient customers is studied with two types of working vacation policies, multiple and single working vacation policy, and comparisons are made to determine the most e®ective policy. ix TH-954_05612304 x TH-954_05612304 Contents Abbreviations xv List of Figures xvi List of Tables xxi 1 Introduction 1 1.1 Queueing theory . 1 1.2 Queues in communication systems . 10 1.3 WDM networks and working vacation queues . 13 1.4 Phase-type distribution . 18 1.5 Markovian arrival process . 27 1.6 Quasi-Birth-Death process . 31 1.7 Solution methods of queueing models . 37 1.7.1 Transform methods . 37 1.7.2 Matrix-Geometric Method . 39 1.8 Outline of the thesis . 40 2 A Queue with Correlated Arrivals 43 2.1 The discrete-time MAP/PH/1/WV model . 44 xi TH-954_05612304 2.1.1 Stability condition . 49 2.1.2 The rate matrix R . 51 2.1.3 Stationary distribution . 51 2.2 Waiting time distribution . 53 2.3 Regular busy period and busy cycle . 57 2.4 Numerical examples . 60 2.4.1 Comparison of MMBP and DPH arrival models . 62 2.4.2 Autocorrelation and DMAP/Geo/1/WV(Geo) model . 64 2.5 The continuous-time MAP/PH/1/WV model . 72 3 A Finite-bu®er Queue with Correlated Arrivals 77 3.1 Model description . 78 3.1.1 Stationary distribution at arbitrary epochs . 80 3.1.2 Stationary distribution at pre-arrival epochs . 81 3.2 Performance measures . 82 3.3 MMPP arrival process model: A special case . 83 3.4 Numerical examples . 85 3.4.1 E®ect of bu®er size . 86 3.4.2 E®ect of burstiness . 88 3.4.3 E®ect of correlation . 88 4 A Multi-server Queue with Impatience 95 4.1 Model description . 96 4.2 Stationary distribution . 99 4.3 The ¯nite truncation method . 101 xii TH-954_05612304 4.4 Performance measures . 105 4.5 Numerical examples . 106 4.5.1 E®ect on cut-o® value Kf ....................... 107 4.5.2 E®ect on mean queue length . 108 4.5.3 E®ect on blocking probability . 109 4.5.4 Average number of servers busy in non-vacation . 110 4.5.5 Average customer loss . 110 5 A Priority Queue with Vacation Interruptions 125 5.1 Model description . 126 5.2 Length of busy period . 129 5.3 Busy cycle . 130 5.4 Queue length . 134 5.5 Waiting time . 137 5.6 Response time . 138 5.7 Numerical examples . 139 6 A Retrial Queue 145 6.1 Model description . 146 6.2 Stationary distribution . 148 6.3 System e±ciency . 156 6.3.1 The availability of the server . 156 6.3.2 The blocking probability . 157 6.3.3 The distribution of number of customers in the orbit . 157 6.3.4 The distribution of number of customers in the system . 158 xiii TH-954_05612304 6.4 Stochastic decomposition . 159 6.5 Numerical results . 160 7 A Queue with Impatient Customers 167 7.1 Multiple working vacation model . 168 7.1.1 Stationary distribution . 169 7.1.2 Performance measures . 176 7.1.3 Stochastic decomposition in MWV model . 178 7.2 Single working vacation model . 180 7.2.1 Performance measures . 182 7.2.2 Stochastic decomposition in SWV model . 183 7.3 Comparison of the models . 184 8 Conclusions 187 References 191 List of Publications based on the Thesis 208 xiv TH-954_05612304 Abbreviations AMWV Asynchronous Multiple Working Vacation ATM Asynchronous Transfer Mode DMAP Discrete-time Markovian Arrival Process DPH Discrete-time Phase-type EAS Early Arrival System FCFS First Come First Served LAN Local Area Network LAS Late Arrival System LDQBD Level-Dependent Quasi-Birth-Death process LST Laplace-Steiltjes Transform MAC Medium Access Control MANET Mobile Ad hoc Network MAP Markovian Arrival Process MMBP Markov-modulated Bernoulli Process MMPP Markov-modulated Poisson Process MWV Multiple Working Vacation OBS Optical Burst Switching p.d.f. Probability density function p.m.f. Probability mass function PGF Probability Generating Function PH Phase-type QBD Quasi-Birth-Death process QoS Quality of Service SWV Single Working Vacation VI Vacation Interruption WDM Wavelength Division Multiplexing WV Working Vacation xv TH-954_05612304 xvi TH-954_05612304 List of Figures 2.1 Autocorrelation vs ¸1, with ¸2 = 0:1...................... 66 2 2.2 Mean queue length vs ¹v, with ¹b = 0:7, c = 2, θ = 0:1. 66 2 2.3 Mean queue length vs ¹v, with ¹b = 0:7, c = 2, ¸ = 0:2. 67 2 2.4 Mean queue length vs θ, with ¹b = 0:7, ¹v = 0:5, c = 2. 67 2.5 Mean queue length vs θ, with ¹b = 0:7; ¹v = 0:5, ¸ = 0:3. 68 2 2.6 Mean queue length vs c , with ¹b = 0:7; ¹v = 0:3; θ = 0:4, ¸ = 0:3. 68 2 2.7 Mean queue length vs ¹v, with ¹b = 0:7; θ = 0:8; ¸ = 0:4, c = 0:5. 69 2.8 Mean queue length vs Ã1, with ½ = 0:65. 70 2.9 Mean queue length vs Ã1, with ½ = 0:9..................... 70 2.10 Mean queue length vs ¹v of M/M/1/WV(M) model, with ¹b = 2; ¸ = 1:25.
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