An Application of Matrix Analytic Methods to Queueing Models with Polling by Kevin Granville A thesis presented to the University of Waterloo in fulfilment of the Doctor of Philosophy in Statistics Waterloo, Ontario, Canada, 2019 c Kevin Granville 2019 Examining Committee Membership The following served on the Examining Committee for this thesis. The decision of the Exam- ining Committee is by majority vote. External Examiner Dr. Yiqiang Zhao Associate Dean (Research and Graduate Studies), Faculty of Science, Carleton University Supervisor Dr. Steve Drekic Professor, Statistics and Actuarial Science Internal Members Dr. Gordon Willmot Professor, Statistics and Actuarial Science Dr. Yi Shen Assistant Professor, Statistics and Actuarial Science Internal-external Member Dr. Qi-Ming He Professor, Management Sciences ii Author's Declaration I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. iii Abstract We review what it means to model a queueing system, and highlight several components of interest which govern the behaviour of customers, as well as the server(s) who tend to them. Our primary focus is on polling systems, which involve one or more servers who must serve multiple queues of customers according to their service policy, which is made up of an overall polling order, and a service discipline defined at each queue. The most common polling orders and service disciplines are discussed, and some examples are given to demonstrate their use. Classic matrix analytic method theory is built up and illustrated on models of increasing complexity, to provide context for the analyses of later chapters. The original research contained within this thesis is divided into two halves, finite population maintenance models and infinite population cyclic polling models. In the first half, we investigate a 2-class maintenance system with a single server, expressed as a polling model. In Chapter 2, the model we study considers a total of C machines which are at risk of failing when working. Depending on the failure that a machine experiences, it is sorted into either the class-1 or class-2 queue where it awaits service among other machines suffering from similar failures. The possible service policies that are considered include exhaustive, non- preemptive priority, and preemptive resume priority. In Chapter 3, this model is generalized to allow for a maintenance float of f spare machines that can be turned on to replace a failed machine. Additionally, the possible server behaviours are greatly generalized. In both chapters, among other topics, we discuss the optimization of server behaviour as well as the limiting number of working machines as we let C . As these are systems with a finite population ! 1 (for a given C and f), their steady-state distributions can be solved for using the algorithm for level-dependent quasi-birth-and-death processes without loss of accuracy. When a class of customers are impatient, the algorithms covered in this thesis require their queue length to be truncated in order for us to approximate the steady-state distribution for all but the simplest model. In Chapter 4, we model a 2-queue polling system with impatient customers and ki-limited service disciplines. Finite buffers are assumed for both queues, such that if a customer arrives to find their queue full then they are blocked and lost forever. Finite buffers are a way to interpret a necessary truncation level, since we can simply assume that it is impossible to observe the removed states. However, if we are interested in approximating an infinite buffer system, this inconsistency will bias the steady-state probabilities if blocking probabilities are not negligible. In Chapter 5, we introduce the Unobserved Waiting Customer approximation as a way to reduce this natural biasing that is incurred when approximating an infinite buffer system. Among the queues considered within this chapter is a N-queue system with exhaustive service and customers who may or may not be impatient. In Chapter 6, we extend this approximation to allow for reneging rates that depend on a customer's place in their queue. This is applied to a N-queue polling system which generalizes the model of Chapter 4. iv Acknowledgements I would like to begin by thanking my PhD supervisor, Dr. Steve Drekic. Over the past five years he has been a mentor, colleague, and friend, who has continually supported me as I devel- oped my skills both as a researcher and as an educator. Ever since introducing me to the world of queueing theory, he has always encouraged me to follow my instincts and tackle research problems of my own design, as well as providing me many opportunities to present my work to peers in our field. I am extremely grateful to have had the privilege to work with him, as well as for the time and effort that he has spent helping me get to where I am today. I would also like to thank my Master's supervisor Dr. Adam Kolkiewicz and Bachelor's supervisor Dr. Zhaozhi Fan for helping guide me through these degrees, for pushing me to pursue further work in academics, as well as helping set the foundations of my experience with research as a younger student. I am grateful to the members of my thesis committee, Dr. Yiqiang Zhao, Dr. Qi-Ming He, Dr. Gordon Willmot, and Dr. Yi Shen. In addition to helping with this process, I am thankful to Dr. He for inviting me to participate in a series of workshops, as well as to Dr. Willmot and Dr. Shen for having taught me courses on ruin theory and stochastic processes, respectively. I am thankful for the assistance of Mary Lou Dufton who was my first contact in the De- partment of Statistics and Actuarial Science. In addition to administrative support, she helped in my decision to attend the University of Waterloo and in the selection of both of my graduate programs, while also being the one who introduced me to Dr. Drekic. I would like to express my gratitude to my friends and family. To the friends who I have known most of my life, as well as those who I met in Waterloo and helped make my time as a graduate student enjoyable. To my sister, my brother, and my grandparents. Finally, I would like to thank my parents, whose unwavering love and support have allowed me to pursue my dreams and find my place in life. Thank you, everyone. v Dedication I dedicate this thesis to my loving parents, Scott and Marilyn. vi Contents List of Figures x List of Tables xii List of Abbreviations xiii 1 Introduction 1 1.1 What's in a Queueing Model? . .1 1.2 Using Matrix Analytic Methods . .3 1.2.1 Generating the Generator . .3 1.2.2 Example 1: Analyzing the M=M=1 Queueing Model, a Birth-and-Death Process . .8 1.2.3 Continuous Phase-Type Distributions . 12 1.2.4 Example 2: Analyzing the M=P H=1 Queueing Model, a Level-Independent QBD....................................... 18 1.2.5 Customer Behaviour: Reneging . 25 1.2.6 Example 3: Analyzing the M=P H=1 + M Queueing Model, a Level- Dependent QBD . 26 1.2.7 Polling Systems . 32 1.2.8 Discussion of Polling Model Literature . 42 1.3 Main Contributions . 46 Part I : Finite Population Maintenance Models 49 2 A 2-Class Maintenance Model with a Finite Population and Competing Ex- ponential Failure Rates 50 2.1 Discussion of Literature . 50 2.2 Model Assumptions . 52 2.3 Exhaustive and Non-preemptive Priority Service Models . 54 2.4 Preemptive Resume Priority Models . 63 2.4.1 Case 1: =1.................................. 63 I 2.4.2 Case 2: =2.................................. 69 I 2.5 Numerical Examples . 75 2.5.1 Setup . 75 2.5.2 Simultaneous Optimization of E[NW] and E[ ]............... 76 S 2.5.3 Limiting Behaviour . 78 2.5.4 Optimization Problem . 82 vii 3 A 2-Class Maintenance Model with Dynamic Server Behaviour 86 3.1 Discussion of Literature . 86 3.2 Model Assumptions . 88 3.3 Model Construction and Analysis . 91 3.3.1 State Space and State-Dependent Decision Probabilities . 91 3.3.2 Select Service Policies and Their Decision Probability Matrices . 92 3.3.3 Steady-State Probabilities . 94 3.3.4 Sojourn Time Distribution . 98 3.4 Results Concerning the Expected Number of Working Machines . 102 3.4.1 Limit Theorems . 102 3.4.2 Connection to Mean Sojourn Times . 109 3.5 Numerical Examples . 114 3.5.1 (a; b) Threshold Optimization . 114 3.5.2 Class-1 Sojourn Time Densities for (a; a) Threshold Policies . 117 3.5.3 Smart Bernoulli Optimization . 122 Part II : Infinite Population Cyclic Polling Models 128 4 A 2-Class Polling Model with Class-Dependent Reneging, Switchover Times, and Phase-Type Service 129 4.1 Discussion of Literature . 129 4.2 Model Assumptions . 130 4.3 Determination of the Steady-State Probabilities . 132 4.4 Determination of the Waiting Time Distribution . 137 4.5 Numerical Examples . 143 4.5.1 A Cost Optimization Problem Concerning Expected Time Spent Waiting in System . 143 4.5.2 Examining Actual Waiting Times and the Reduction Effect . 149 5 The Unobserved Waiting Customer Approximation 154 5.1 Discussion of Literature and Introduction to UWC . 154 5.2 M=M=1 Queue . 156 5.3 M=M=1 + M Queue . 158 5.4 M=M= Queue . 160 1 5.5 M=P H=1 Queue . 160 5.6 M=P H=1 + M Queue .