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Download File A Robust Queueing Network Analyzer Based on Indices of Dispersion Wei You Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2019 c 2019 Wei You All Rights Reserved ABSTRACT A Robust Queueing Network Analyzer Based on Indices of Dispersion Wei You In post-industrial economies, modern service systems are dramatically changing the daily lives of many people. Such systems are often complicated by uncertainty: service providers usually cannot predict when a customer will arrive and how long the service will be. For- tunately, useful guidance can often be provided by exploiting stochastic models such as queueing networks. In iterating the design of service systems, decision makers usually favor analytical analysis of the models over simulation methods, due to the prohibitive compu- tation time required to obtain optimal solutions for service operation problems involving multidimensional stochastic networks. However, queueing networks that can be solved an- alytically require strong assumptions that are rarely satisfied, whereas realistic models that exhibit complicated dependence structure are prohibitively hard to analyze exactly. In this thesis, we continue the effort to develop useful analytical performance approx- imations for the single-class open queueing network with Markovian routing, unlimited waiting space and the first-come first-served service discipline. We focus on open queueing networks where the external arrival processes are not Poisson and the service times are not exponential. We develop a new non-parametric robust queueing algorithm for the performance ap- proximation in single-server queues. With robust optimization techniques, the underlying stochastic processes are replaced by samples from suitably defined uncertainty sets and the worst-case scenario is analyzed. We show that this worst-case characterization of the perfor- mance measure is asymptotically exact for approximating the mean steady-state workload in G=G=1 models in both the light-traffic and heavy-traffic limits, under mild regularity conditions. In our non-parametric Robust Queueing formulation, we focus on the customer flows, defined as the continuous-time processes counting customers in or out of the network, or flowing from one queue to another. Each flow is partially characterized by a continuous function that measures the change of stochastic variability over time. This function is called the index of dispersion for counts. The Robust Queueing algorithm converts the index of dispersion for counts into approximations of the performance measures. We show the ad- vantage of using index of dispersion for counts in queueing approximation by a renewal process characterization theorem and the ordering of the mean steady-state workload in GI=M=1 models. To develop generalized algorithm for open queueing networks, we first establish the heavy-traffic limit theorem for the stationary departure flows from a GI=GI=1 model. We show that the index of dispersion for counts function of the stationary departure flow can be approximately characterized as the convex combination of the arrival index of dispersion for counts and service index of dispersion for counts with a time-dependent weight function, revealing the non-trivial impact of the traffic intensity on the departure processes. This heavy-traffic limit theorem is further generalized into a joint heavy-traffic limit for the stationary customer flows in generalized Jackson networks, where the external arrival are characterized by independent renewal processes and the service times are independent and identically distributed random variables, independent of the external arrival processes. We show how these limiting theorems can be exploited to establish a set of linear equa- tions, whose solution serves as approximations of the index of dispersion for counts of the flows in an open queueing network. We prove that this set of equations is asymptotically exact in approximating the index of dispersion for counts of the stationary flows. With the index of dispersion for counts available, the network is decomposed into single-server queues and the Robust Queueing algorithm can be applied to obtain performance approximation. This algorithm is referred to as the Robust Queueing Network Analyzer. We perform extensive simulation study to validate the effectiveness of our algorithm. We show that our algorithm can be applied not only to models with non-exponential dis- tirbutions but also to models with more complex arrival processes than renewal processes, including those with Markovian arrival processes. Table of Contents List of Figures v List of Tables vii 1 Introduction 1 1.1 Challenges in Analyzing the Open Queueing Networks . .3 1.1.1 Jackson Networks . .3 1.1.2 Generalized Jackson Networks . .4 1.1.3 General Open Queueing Networks . .5 1.2 Approximation Algorithms . .6 1.2.1 Decomposition Approximations . .6 1.2.2 Heavy-Traffic Limit Approximations . .6 1.2.3 Robust Queueing Approximations . .8 1.2.4 Approximations Based on Non-Parametric Traffic Descriptions . .8 1.3 Main Contributions . .9 1.3.1 Non-Parametric Traffic Description for Queueing Networks . .9 1.3.2 Heavy-Traffic Limits for Stationary Network Flows . 10 1.3.3 The Robust Queueing Network Analyzer and Simulation Studies . 12 1.4 Outline . 12 2 Robust Queueing for the G=G=1 Model 14 2.1 Robust Queueing for the Steady-State Waiting Time . 15 i 2.1.1 Parametric RQ for Waiting Time in the GI=GI=1 Model . 16 2.1.2 Non-Parametric RQ for Waiting Time in the G=G=1 Model . 20 2.2 Robust Queueing for the Steady-State Workload . 21 2.2.1 The Workload Process and Its Reverse-Time Representation . 21 2.2.2 Parametric and Non-Parametric RQ for the Workload . 22 2.2.3 The Indices of Dispersion for Counts and Work . 26 2.2.4 The Indices of Dispersion and the Mean Steady-State Workload . 28 2.2.5 Robust Queueing with the IDW . 31 2.2.6 The RQ(b) Algorithm for Quantile Approximation . 32 2.2.7 Heavy-Traffic and Light-Traffic Limits . 33 2.2.8 Other Steady-State Performance Measures . 34 2.2.9 An Illustrative Simulation Example . 35 2.3 Estimating and Calculating the IDC . 37 2.3.1 The IDC's for Renewal Processes. 37 2.3.2 The IDC of the Markovian Arrival Process . 38 2.3.3 Calculation of the IDW and IDC in Some Queueing Networks . 42 2.3.4 Numerical Estimation of the IDC from Data. 43 2.4 More on Non-Parametric RQ for Waiting Times . 46 2.5 Supporting Functional Central Limit Theorems . 47 2.5.1 The Basic FCLT for the Partial Sums . 48 2.5.2 The FCLT for Other Basic Processes . 49 2.5.3 Alternative Scaling in the CLT . 51 2.5.4 The Heavy-Traffic FCLT . 52 2.5.5 The IDW and the Normalized Workload . 54 2.6 Proofs . 54 3 Robust Queueing for Queues In Series 62 3.1 Review of Stationary Point Processes . 63 3.1.1 Renewal Processes and the Laplace Transform . 63 3.1.2 Revisiting the IDC . 65 ii 3.1.3 The Palm-Khintchine Equation . 66 3.2 Heavy-Traffic Limit Theorem for GI=GI=1 Stationary Departure . 68 3.2.1 The Departure Variance in the GI=M=1 Queue . 68 3.2.2 The Departure Variance in the M=GI=1 Queue . 74 3.2.3 Heavy-Traffic Limit for the Stationary Departure Process . 76 3.2.4 Extensions . 84 3.3 Robust Queueing Network Analyzer for Queues In Series . 85 3.3.1 Approximation of the Departure IDC . 85 3.3.2 The RQNA algorithm for Queues in Series . 87 3.4 Proofs . 88 4 Heavy-Traffic Limits for Stationary Flows 98 4.1 The OQN Model . 100 4.2 Existence, Uniqueness and Convergence Via Harris Recurrence . 101 4.3 Heavy-Traffic Limit Theorems for the Stationary Processes . 106 4.3.1 Representation of the Centered Stationary Flows . 106 4.3.2 Heavy-Traffic Limit with Any Subset of Bottlenecks . 107 4.4 Heavy-Traffic Limits with One Bottleneck Queue . 111 4.4.1 A Single-Server Queue with Customer Feedback . 112 4.4.2 Networks with One Bottleneck Queue . 115 4.5 Functional Central Limit Theorem for the Stationary Flows . 118 4.6 Proofs . 120 5 Robust Queueing Network Analyzer 127 5.1 The Departure Operation . 128 5.2 The Splitting Operation . 130 5.2.1 Dependent Splitting: One Queue with Immediate Feedback . 131 5.2.2 The General Case . 132 5.3 The Superposition Operation . 136 5.3.1 Dependent Superposition: A Splitting and Recombining Example . 136 iii 5.3.2 The General Case . 139 5.4 The IDC Equation System . 143 5.5 Feedback Elimination . 145 5.5.1 Immediate Feedback Elimination . 145 5.5.2 Near-Immediate Feedback . 147 5.6 The RQNA Algorithm . 148 5.6.1 RQNA for Tree-Structured Queueing Networks . 148 5.7 Proofs . 151 6 Simulation Experiments 153 6.1 Notation and Simulation Methodology . 153 6.2 Robust Queueing for Single-Server Queues . 157 6.2.1 The GI=GI=1 Models . 157 6.2.2 A Queue with a Superposition Arrival Process . 159 6.2.3 A Ten-Queues-in-Series Example . 168 6.2.4 A MAP=MAP=1 Example . 171 6.2.5 The Queues in Series Models . 173 6.2.6 The Limitation of IDC . 175 6.3 Robust Queueing Network Analyzer for Tandem Queues . 182 6.3.1 Departure IDC Approximation in G=G=1 Models . 183 6.3.2 An Illustrative Example . 185 6.3.3 Comparisons with Previous Algorithms for Queues in Series . 188 6.3.4 RQNA Performance in Tandem Queueus . 190 6.4 Robust Queueing Network Analyzer for Open Queueing Networks .
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