This item is the archived peer-reviewed author-version of: Analyzing queues with customer differentiation using matrix analytic methods Reference: van Velthoven Jeroen.- Analyzing queues with customer differentiation using matrix analytic methods Antwerpen, Universiteit Antwerpen. Faculteit Wetenschappen. Departement Wiskunde & Informatica, 2008, 179 p. Handle: http://hdl.handle.net/10067/1109350151162165141 Institutional repository IRUA Faculteit Wetenschappen Departement Wiskunde & Informatica Analyzing Queues with Customer Differentiation using Matrix Analytic Methods Een analyse van wachtrijsystemen met differentiatie in klanten gebruikmakend van matrix analytische methodes Proefschrift voorgelegd tot het behalen van de graad van doctor in de Wetenschappen aan de Universiteit Antwerpen, te verdedigen door Jeroen VAN VELTHOVEN Promotor: Prof. dr. Benny Van Houdt Copromotor: Prof. dr. Chris Blondia Antwerpen, 2008 Contents Preface xi Acknowledgements xiii Notations and Acronyms xv Summary and Organization xvii I Introduction1 1 Queueing Theory and Markov Chains3 1.1 Markov Chains...................................4 1.2 Queueing Theory..................................6 1.2.1 Arrival Process...............................6 1.2.2 Scheduling Discipline............................7 1.2.3 Service Process...............................8 1.2.4 Some Basic Queueing Systems......................9 1.2.5 Customer Differentiation..........................9 2 Matrix Analytic Methods 11 2.1 Structured Processes................................ 12 2.1.1 M/G/1 and GI/M/1 Type Markov Chains................ 12 2.1.2 Quasi-Birth-Death Markov Chains.................... 12 2.1.3 Tree-Like Processes............................. 14 2.2 The D-MAP/PH/1 Queue............................. 16 2.2.1 The Discrete-Time Markovian Arrival Process.............. 16 2.2.2 Phase-Type Service Time......................... 17 2.2.3 Modeling the D-MAP/PH/1 Queue................... 18 II Steady State Analysis 19 3 Impatient Customers 21 3.1 Related Work.................................... 22 3.2 General Customer Impatience in the D-MAP/PH/1 Queue.......... 22 3.2.1 The Discrete-Time D-MAP/PH/1+GI Queue.............. 23 i Contents 3.2.2 Impatient Customers in the System................... 24 3.2.3 Service Time Aware Impatient Customers................ 28 3.2.4 Impatient Customers in the Waiting Room............... 30 3.2.5 Quasi-Birth-Death Reduction....................... 33 3.2.6 Numerical Examples............................ 37 3.3 Minimizing the Abandonment Probability.................... 40 3.3.1 The Geo/Geo/1 Queue with a Limited Sojourn Time......... 40 3.3.2 The Geo/Geo/1 Queue with a Limited Waiting Time......... 47 3.3.3 Queues with Impatient Customers and Geometric Service Times... 49 4 Priority Scheduling 51 4.1 Related Work.................................... 52 4.2 The Impact of Finite Buffers........................... 52 4.2.1 System Characteristics........................... 53 4.2.2 Finite High Priority Buffer........................ 54 4.2.3 Finite Low Priority Buffer......................... 58 4.2.4 Two Finite Buffers............................. 63 4.2.5 Two Infinite Buffers............................ 64 4.2.6 Numerical Examples............................ 67 4.2.7 Conclusions................................. 70 5 A Tree-Like Process Approach to Priority Modeling 71 5.1 Related Work.................................... 72 5.2 The Queueing Model................................ 72 5.3 A QBD Formulation................................ 73 5.4 A Tree-Like Process Approach.......................... 73 5.4.1 Definition of a Markov Process on a (K − η + 1)-ary Tree....... 74 5.4.2 The Infinitesimal Generator........................ 77 5.4.3 Constructing a Tree-Like QBD Process................. 79 5.5 K-Class Priority Queue as a Tree-Like Process................. 82 5.5.1 Characterization of the Tree-Like Process................ 84 5.5.2 Computing the V -Matrix......................... 87 5.5.3 Solving the Boundary Condition..................... 95 5.6 Loss Rate Computation.............................. 99 6 Multilevel Feedback Queues 105 6.1 Related Work.................................... 106 6.2 Introduction..................................... 106 6.3 QBDs on Binomial-Like Trees........................... 107 6.4 Steady State Analysis............................... 109 6.5 The Multilevel Feedback Queue.......................... 111 6.5.1 Markov Model............................... 112 6.5.2 Transition Matrices............................ 114 6.5.3 Performance Measures........................... 118 ii Contents III Transient Performance Measures 123 7 Quasi-Birth-Death Markov Chains 125 7.1 Related Work.................................... 126 7.2 Introduction..................................... 126 7.3 QBDs with Marked Time Epochs: a Review................... 127 7.4 QBD Reduction................................... 130 7.4.1 Computing the R-Matrices........................ 132 7.4.2 Computing the Invariant Probabilities.................. 136 7.4.3 Algorithmic Overview........................... 138 7.5 Numerical Examples................................ 141 8 Tree-Like Processes 147 8.1 Related Work.................................... 148 8.2 Introduction..................................... 148 8.3 Tree-Like Process with Marked Time Epochs.................. 149 8.3.1 Process Definition............................. 149 8.3.2 Discrete Erlangization and Reset Markov Chains............ 150 8.3.3 Computing the V -Matrix......................... 153 m 8.3.4 Calculating the Probability Vectors πn (J)................ 156 8.4 Transient Analysis of the CTM Protocol with Free Access........... 157 8.4.1 Protocol Definition............................. 157 8.4.2 Analytical Model.............................. 159 8.4.3 Performing a Transient Analysis..................... 160 Bibliography 172 Related Publications 173 Nederlandse Samenvatting 175 iii List of Figures 1.1 A basic queueing system..............................6 2.1 A Quasi-Birth-Death Markov chain........................ 13 2.2 Structure of the first three levels of a tree-structured Markov chain...... 14 3.1 Response time distribution of successful impatient customers......... 37 3.2 Response time distributions of successful service time (un)aware customers that are impatient in the entire system or only in the waiting room......... 38 3.3 Expected number of lost customers........................ 39 3.4 Expected response time of a successful impatient customer........... 39 + 4.1 Illustration of the functions z and ξ(A(z)), for z 2 R ............. 58 4.2 Comparison of the loss rate of low priority packets for each of the six approaches with λ1 = 0:4;H = 25;L = 20 and a varying rate λ2 .............. 68 4.3 Comparison of the low priority loss rate obtained by each of the six approaches for λ1 = 0:4; λ2 = 0:2;H = 25 and a varying capacity L ............ 69 4.4 Comparison of the coefficient for the tail approaches, λ1 = 0:4;H = 25;L = 20 70 5.1 Illustrative example of the stochastic process capturing the behavior of the 5-class priority queueing system with η = 3................... 76 5.2 Block sizes required for the QBD approach and the tree-structured process. 77 5.3 Block sizes required for the QBD approach and the tree-like process..... 80 5.4 Illustrative example from Figure 5.1( η = 3) revisited after the transformation of the Markov process into a tree-like QBD process............... 81 5.5 Dimensioning the high priority buffers...................... 101 5.6 Influence of the low priority service rates on the type-5 loss probabilities... 101 5.7 Influence of the low priority service rates on the type-5 loss probabilities under equal low priority arrival rates.......................... 102 5.8 Influence of the low priority arrival rates on the type-5 loss probability.... 103 5.9 Influence of the high priority traffic on the low priority loss rates....... 103 6.1 Structure of the first three levels of a binomial-like tree Markov chain and the matrices characterizing its transitions...................... 108 6.2 Average queue lengths in a multilevel feedback queue with 10 priorities and different arrival characteristics........................... 118 6.3 Average queue lengths in a multilevel feedback queue with 10 priorities under varying processing time distributions....................... 119 v List of Figures 6.4 Average queue length in the 10-class multilevel feedback queue........ 120 6.5 Average response time of type t customers in a multilevel feedback queue under varying arrival characteristics and with different processing time distributions 121 7.1 Queue length distributions of the D-MAP/PH/1 queue at time t = 50.... 141 7.2 Queue length distributions at different time epochs............... 142 7.3 Average queue length of the D-MAP/PH/1 queue............... 143 7.4 Execution times (s) to get the transient queue length distribution for r ≤ 50. 143 7.5 Waiting time distribution of the n-th customer in a D-MAP/PH/1 queue.. 144 7.6 Average waiting time of the n-th customer in a D-MAP/PH/1 queue..... 145 8.1 The CTM protocol for random multiple access communication........ 158 8.2 Throughput in slot n under Poisson arrivals................... 160 8.3 Initial throughput in the n-th slot under Poisson arrivals............ 161 8.4 Average time until the n-th event under Poisson input............. 162 8.5 Mean time until the i-th success of a size n conflict............... 163 8.6 Throughput in the n-th slot under MMPP