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Aggregate Matrix-Analytic Techniques and Their Applications W&M ScholarWorks Dissertations, Theses, and Masters Projects Theses, Dissertations, & Master Projects 2002 Aggregate matrix-analytic techniques and their applications Alma Riska College of William & Mary - Arts & Sciences Follow this and additional works at: https://scholarworks.wm.edu/etd Part of the Computer Sciences Commons Recommended Citation Riska, Alma, "Aggregate matrix-analytic techniques and their applications" (2002). Dissertations, Theses, and Masters Projects. Paper 1539623412. https://dx.doi.org/doi:10.21220/s2-ky81-f151 This Dissertation is brought to you for free and open access by the Theses, Dissertations, & Master Projects at W&M ScholarWorks. It has been accepted for inclusion in Dissertations, Theses, and Masters Projects by an authorized administrator of W&M ScholarWorks. For more information, please contact [email protected]. U MI MICROFILMED 2003 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with with permission permission of the of copyright the copyright owner. owner.Further reproductionFurther reproduction prohibited without prohibited permission. without permission. Aggregate matrix-analytic techniques and their applications A Dissertation Presented to The Faculty of the Department of Computer Science The College of William k Mary in Virginia In Partial Fulfillment Of the Requirements for the Degree of Doctor of Philosophy by Alma Riska 2002 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPROVAL SHEET This dissertation is submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy A. Riska Approved, November 2002 wgema Smirm Thesis Advisor Gianfranco Ciardo /KdJT VVeizhen Mao Xiaodong Zhang / !/ / Mark SquiMante lM T.J. Watson Research Center ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission To my father and the memory of my mother iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table of Contents Acknowledgments xi List of Tables xiii List of Figures xx Abstract xxi 1 Introduction 2 1.1 Contributions ........................................................................................... 4 1.2 Organization ........................................................................................... 6 2 Background 8 2.1 Basic notation ........................................................................................... 9 2.1.1 Kendall N otation .......................................................................... 10 2.2 Random variables ..................................................................................... 12 2.2.1 Exponential distribution .............................................................. 13 2.2.2 Variations of exponential distribution ........................................ 14 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.3 Heavy tailed distributions ...................................................................... 15 2.4 Markov processes .................................................................................... 18 2.4.1 Discrete time Markov chains (D TM Cs) ...................................... 18 2.4.2 Continuous time Markov chains (C TM C s) ................................ 21 2.5 Markov chains with repetitive structure ................................................ 22 2.5.1 Quasi birth-death processes ........................................................ 23 2.5.2 GI/M/l-type processes ................................................................. 26 2.5.3 M/G/1-type processes ................................................................. 28 2.5.4 GI/G/l-type processes ................................................................. 29 2.6 Phase-type distributions ......................................................................... 30 2.6.1 Fitting algorithms for PH distributions ...................................... 33 2.6.1.1 EM A lgorithm .............................................................. 33 2.6.1.2 Feldmann-YVhitt (FW) Algorithm ................................. 35 2.7 Markovian Arrival Process ...................................................................... 36 2.7.1 Fitting techniques for M A Ps ........................................................ 38 2.8 Aggregation techniques ............................................................................ 38 2.8.1 Exact aggregation and decomposition ........................................ 40 2.8.2 Stochastic complementation ........................................................ 42 2.9 Chapter summary .................................................................................... 46 3 Matrix-Analytic Methods 48 3.1 Matrix geometric solutions for GI/M/1-type and QBD processes . 50 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.1.1 Additional measures of in te re s t .................................................. 52 3.2 Why does a geometric relation hold for QBD processes? ........................ 54 3.3 Generalization: geometric solution for the GI/M/1 processes ............... 58 3.4 Why M /G /l processes are more difficu lt .............................................. 61 3.5 Example: a BMAP1 /C0 X2 /I q u eu e....................................................... 61 3.6 Generalization: derivation of Ramaswami’s recursive form ula ................ 65 3.7 General solution of M/G/1-type processes ........................................... 68 3.7.1 Ramaswami’s formula .................................................................. 69 3.7.2 Explicit computation of G ........................................................... 71 3.7.3 Fast FFT Ramaswami's formula .................................................. 71 3.8 Explicit computation of R for Q B D s ..................................................... 75 3.9 Conditions for stability ............................................................................. 77 3.10 Chapter summary ...................................................................................... 79 4 Data Fitting Algorithms 81 4.1 Long-tailed behavior ................................................................................ 82 4.2 D&C E M .................................................................................................. 86 4.2.1 D&C EM experimental re s u lts .................................................. 88 4.2.1.1 TVaces............................................................................ 89 4.2.1.2 D&C EM statistical analysis .................................... 90 4.2.2 D&C EM queueing system analysis ......................................... 94 4.2.3 Computational efficiency of D&C E M ...................................... 96 vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.2.4 Refined D&C EM fitting algorithm ...................................... 97 4.2.4.1 Results with the refined D&C E M ............................. 100 4.3 D&C M M ................................................................................................. 102 4.4 MAP fitting algorithm ............................................................................ 105 4.4.1 Hidden Markov model for parameter estim ate .................... 107 4.4.2 Generation of MAP from HMM o u tp u t ................................ I ll 4.4.3 Experimental re su lts ............................................................... 115 4.4.3.1 Traces............................................................................. 115 4.4.3.2 Experimental settin g ..................................................... 116 4.4.3.3 Peak TYaffic Period (Trace A ) ...................................... 117 4.4.3.4 Off-Peak TVaffic Period (TVace B ) ................................ 122 4.4.3.5 Strong Long-Range Dependence (Trace C ) ................ 124 4.5 Chapter summary ..................................................................................... 127 5 ETAQA Methodology 128 5.1 E taqa - M / G / 1 ................................................................................................ 130 5.1.1 Computing measures of interest for M/G/ 1-type processes . 137 5.1.2 Time and storage complexity .................................................... 142 5.2 E taqa -G I/M /1 ................................................................................................ 146 5.2.1 Computing measures of interest for GI/M/1-type processes . 149 5.2.2 Time and storage complexity .................................................... 150 5.3 E taqa -QBD ................................................................................................... 154 vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.3.1 Computing measures of interest for QBD processes ............... 155 5.3.2 Time and storage complexity ................................................... 157 5.4 Computational efficiency ........................................................................ 160 5.5 Numerical stability of the m ethod ......................................................... 163 5.6 MAMSolver ............................................................................................. 170 5.6.1 MAMSolver input and o u
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