Queueing Networks and Markov Chains Modeling and Performance Evaluation with Computer Science Applications Second Edition Gunter Bolch Stefan Greiner Hermann de Meer Kishor S Trivedi WILEYINTERSCIENCE A JOHN WILEY & SONS, INC., PUBLICATION Queueing Networks and Markov Chains This Page Intentionally Left Blank Queueing Networks and Markov Chains Modeling and Performance Evaluation with Computer Science Applications Second Edition Gunter Bolch Stefan Greiner Hermann de Meer Kishor S Trivedi WILEYINTERSCIENCE A JOHN WILEY & SONS, INC., PUBLICATION Copyright 02006 by John Wiley & Sons, Inc All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 ofthe 1976 United States Copyright Act, without either 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bibliographical references and index ISBN- I3 978-0-47 1-56525-3 (acid-free paper) ISBN- I0 0-47 1-56525-3 (acid-free paper) I Markov processes Queuing theory I Bolch, Gunter QA76.9E94Q48 2006 004.2’401519233-dc22 Printed in the United States of America 200506965 Contents Preface to the Second Edition Xlll Preface to the First Edition Introduction 1.1 Motivation 1.2 Methodological Background 1.2.1 Problem Formulation 1.2.2 The Modeling Process 1.2.3 Evaluation 1.2.4 Summary 1.3 Basics of Probability and Statistics 1.3.1 Random Variables 1.3.2 Multiple Random Variables 1.3.3 Transforms 1.3.4 Parameter Estimation 1.3.5 Order Statistics 1.3.6 Distribution of Sums xv 1 10 12 15 15 30 36 38 46 46 V vi CONTENTS Markov Chains 51 Markov Processes 2.1.1 Stochastic and Markov Processes 2.1.2 Markov Chains 2.2 Performance Measures 2.2.1 A Simple Example 2.2.2 Markov Reward Models 2.2.3 A Casestudy 2.3 Generation Methods 2.3.1 Petri Nets 2.3.2 Generalized Stochastic Petri Nets 2.3.3 Stochastic Reward Nets 2.3.4 GSPN/SRN Analysis 2.1 2.3.5 2.3.6 2.3.7 2.3.8 51 51 53 71 71 75 80 90 94 96 97 101 A Larger Exanlple 108 Stochastic Petri Net Extensions 113 Non-Markoviarl Models 115 Symbolic State Space Storage Techniques 120 Steady-State Solutions of Markov Chains 123 Solution for a Birth Death Process 125 Matrix-Geometric Method: Quasi-Birth-Death Process 127 3.2.1 The Concept 127 3.2.2 Example: The QBD Process 128 3.3 Hessenberg Matrix: Non-Markovian Queues 140 3.3.1 Nonexporlential Servicc Times 141 3.3.2 Server with Vacations 146 3.4 Numerical Solution: Direct Methods 151 3.4.1 Gaussian Elimination 152 3.4.2 The Grassmanrl Algorithm 158 3.5 Numerical Solution: Iterative Methods 165 3.5.1 Convergence of Iterative Methods 165 3.5.2 Power Method 166 3.5.3 Jacobi's Method 169 3.5.4 Gauss-Seidel Method 172 3.1 3.2 3.6 3.5.5 The Method of Successive Over-Relaxation 173 Comparison of Numerical Solution Methods 177 3.6.1 Case Studies 179 Steady-State Aggregation/Disaggregation Methods 4.1 4.2 Courtois’ Approximate Method 185 4.1.1 Decomposition 186 192 4.1.2 Applicability 4.1.3 Analysis of the Substructures 194 4.1.4 Aggregation and Unconditioning 195 4.1.5 The Algorithm 197 Takahashi’s Iterative Method 198 4.2.1 The Fundamental Equations 199 4.2.2 Applicability 201 4.2.3 The Algorithm 202 4.2.4 Application 202 4.2.5 Final Remarks 206 Transient Solution of Markov Chains 5.1 5.2 Transient Analysis Using Exact Methods 5.1.1 A Pure Birth Process 5.1.2 A Two-State CTMC 5.1.3 Solution Using Laplace Transforms 5.1.4 Numerical Solution Using Uniformization 5.1.5 Other Numerical Methods Aggregation of Stiff Markov Chains 5.2.1 Outline arid Basic Definitions 5.2.2 5.2.3 5.2.4 5.2.5 5.2.6 5.2.7 6.2 209 210 210 213 216 216 221 222 223 224 227 Aggregation of Fast Recurretit Subset.s Aggregation of Fast Transient Subsets Aggregation of Initial State Probabilities 228 Disaggregations 229 The Algorithm 230 An Example: Server Breakdown arid Repair 232 Single Station Queueing Systems 6.1 185 241 Notation 242 6.1.1 Kendall’s Notation 242 6.1.2 Performance Measures 244 Markovian Queues 246 6.2.1 The M/M/l Queue 246 viii CONTENTS 6.2.2 The M/M/ca Queue 249 6.2.3 The M/M/m Queue 250 6.2.4 The M / M / l / K Finite Capacity Queue 251 6.2.5 Machine Repairman Model 252 6.2.6 Closed Tandem Network 253 6.3 Non-Markovian Queues 255 6.3.1 The M / G / Queue 255 6.3.2 The GI/M/l Queue 261 6.3.3 The GI/M/m Queue 265 6.3.4 The GI/G/1 Queue 265 6.3.5 The M/G/m Queue 267 6.3.6 The GI/G/m Queue Priority Queues 6.4.1 Queue without Preemption 6.4.2 Conservation Laws 6.4.3 Queur: with Preemption 6.4.4 Queue with Time-Dependent Priorities 6.5 Asymmetric Queues 6.5.1 Approximate Analysis 6.5.2 Exact Analysis 6.6 Queues with Batch Service and Batch Arrivals 6.6.1 Batch Service 6.6.2 Batch Arrivals 6.7 Retrial Queues 6.7.1 M / M / Ret.rial Queue 6.4 269 272 272 278 279 280 283 284 286 295 295 296 299 300 6.7.2 M / G / l Retrial Queue 301 6.8 Special Classes of' Point Arrival 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Approximation algorithms, 659 Allen-Cunneen, 266, 271 Bard-Schweitzer, 422 Cosmetatos, 268, 271, 277 Courtois’ method, 185 Crommelin, 269 diffusion, 463 heavy traffic, 282 Kimura, 267, 272 Kingman-Kollerstrom, 270 Kulbatzki, 266 Martin’s, 267 networks, 421 869 870 /ND€X product-form, 469 APU, 535, 728 Arc multiplicity, 97 marking-dependent, 99 Arcs, 94 Arrival overall rate, 324 process, 246 rate, 324 theorem, 384, 401, 517 Associate processing unit, 535 Associated processor model, 728 Asymmetric MVA, 579 nodes, 577 queueing systems, 286 SCAT, 580 SUM, 578 system, 283, 286 GI/G/m, 285 M/M/m, 285 ASYM-MVA, 579 ASYM-SCAT 580 ASYM-SUM, 578 ATM, 48 networks, 135, 775 B Bard-Schweitzer approximation, 422 Batch arrival, 296 full, 295 GREEDY, 295 minimum, 295 service, 295 size, 295 BCMP theorem, 357-358, 371 Version 1, 360 Version 2, 361 Binomial coefficient, 355 Birth rate, 125 Birth-death process, 125 250, 252 BJB, 456 Blocking after service, 592 before service, 592 factor, 565 network, 592, 594 probabilities, 597 rejection, 593 repetitive, 593 types, 592 BMAP, 309 BOTT, 447, 578 Bottapprox method, 505 Bottapprox method, 447, 578 mrilticlass, 450 Bottleneck analysis, 447, 467-468 Bounds analysis, 452 asymptotic, 452 453 balanced job, 452, 456 lower bound, 269, 421 optimistic, 452 pessimistic, 452 upper bound, 265, 269, 421 Boxnia-Cohen-Huffels formula, 272 Brumelle, 269 C Cache, 704 cohcrence protocol 796, 798 write-once, 798 CPU, 611 hit ratio, 641, 643 miss, 642, 704 strategies, 795 Web, 639, 641 CCNC algorithm, 370 CDF, 18-19, 23, 37, 52 conditional, 52 Cell loss probability, 775 Central limit theorem, 35 Central-server, 321 model, 321, 389, 555 Chain, 52 Chapman-Kolmogorov equation, 55, 66 Cheapernet, 710 Class switching, 353, 383, 395, 525, 725, 727 mixed priorities, 526 Client server system, 707, 763 Closed network single class, 371 tandem, 253 Closed network 332 Closing method, 507 Coefficient of variation, 19-20, 23-25, 31, 295 Communication systems, 709 Corriposite node, 492 Composition method, 620 Concept of chains, 353, 727 Conditional probability, 33 Conditioning method, 637 Confidencr interval, 42, 631 Congestion avoidance, 648 Conservation law, 279 Constant bit rate, 628 Contention time, 545 INDEX Continiious parameter process, 52 Continuous random variable, 18, 46 Control variates, 637 Convolution, 48 algorithm, 369, 371, 659 method, 375, (il9 operator, 371 Correction factor, 266 Correlation coefficient, 35 Cosrrretatos approximation, 268, 271, 277 Counting process, 303 Courtois’ approximate method, 185 Covariance, 34 Coverage factor, 71 Cox-2, 494 Cox-m, 494 CPH, 118 Cromrnelin approximation, 269 CSMA/CD network, 708 CSPL, 669 CTMC, 65, 289, 332, 334, 337, 369, 657 ergotlic, 69 state transition diagram, 286 time-homogeneous, 65 Cumulative average, 635 distribution function, 18 measure, 210 D DA, 463 DAC algorithm, 370 Death rate, 125 Decomposition method, 479 of Cliylla, 481 of Gelenbe, 481 of Kiihn, 482 of Piijolle, 481 of Whitt, 481 Degree of multiprogramming, 321 Density majorizing, 622 DES, 2, 607, 610, 659 DiffServ, 720 Diffusion approximation, 463 Direct transformation method, 621 Disaggregation step, 200 Discipline dynamic priorities, 243 FCFS, 243 IS, 243 LBPS, 364 LCFS, 243 preemption, 243 PS, 243 queueing discipline, 278 871 RR, 243 SIRO, 243 static priorities, 243 WEIRDP, 364 Discrete random variable, 15, 31, 36 Discrete-event simulation, 607, 657, 659 Discrete-parameter process, 52 Discrete-time Markov chain, 54 Distribution arbitrary normal, 19 Bernoulli, 617, 633 branching Erlang, 26 Cox, 26, 242 deterministic, 242 discrete, 618 empirical, 614 Erlang, 242, 619 phases, 334 generalized, 26 k phases, 23 m phases, 405 T phases, 48 exponential, 20, 242, 248, 256, 614, 618 function, 18 gamma, 25 general, 242 general independent, 242 generalized Erlang, 26 geometric, 619 hyperexponential, 22, 242, 620 hypergeometric, 623 hypoexponential, 24, 620 k phases, 25 r phases, 49 log- logistic, 17- 18 negative binomial, 620, 624 non-exponential, 463 normal, 19, 621, 632 of sums, 46 Pareto, 29, 614, 616, 618 Poisson, 623 Rayleigh, 617 618, 621 Student’s t , 632 triangular, 617-618 uniform, 614, 616, 618 waiting time, 249, 262 Weibull, 28, 614-615, 618 DPH, 118 DSPN, 113 DTMC, 54, 353, 657 E EBOTT, 505 Embedding techniques, 140, 259 Ensemble average, 634 Entropy, 471 872 lNDEX function, 471 Equilibrium, 332 probability vector, 68 Equivalent network, 595 ERG, 96, 110 algorithm, 102 Ergodic, 63, 70, 332, 342 chains, 384 CTMC, 69 Markov chain, 63 ESCAF, 502 Estimate, 39 Estimator, 39 ESUM, 505 Ethernet, 708 Expectation, 31 Expected instantaneous reward rate, 79 Expected value, 16, 18 Extended BOTT, 505 Extended MVA 514 Extended reachability graph, 96 Extended SUM, 505 F Fast state, 223 FB, 295 FDDI, 709 FES method, 410, 542 multiple nodes, 414 node, 410, 542 Fire 95 Firing rate marking-dependent, 101 Firing time, 96 Fission-fusion, 562 Fixed-point iteration, 445 Flexible production systems, 745 Flow, 480 Flow-equivalent server method, 369, 410, 542 Fokker-Planck equation, 463 Fork-join systems, 561 FSPN 114 Full batch policy, 295 G Gaussian elimination, 177 Gauss-Seidel iteration, 173 Generator matrix, 335 G I / G / l queue, 265 GI/G/m queue, 269 GI/M/I queue, 261 GI/M/m queue, 265 Global balance, 332 equations, 69, 332, 335, 337, 347 Gordon/Newell theorem, 346, 349 Grassmann algorithm, 152 GREEDY policy, 295 GSPN, 96 Guards, 99 H Heavy traffic approximation, 282 Heavy-tailed, 29 Hessenberg matrix, 142 Heterogeneous multiple servers, 283 Hierarchical models, 657, 795 I IBP, 628 Immediate transition, 96 Importance sampling, 610, 637-638 Importance splitting, 610 Impulse reward, 76 Infinitesimal generator matrix, 68 Inhibitor arcs, 98 Initial marking, 95 Initial probability vector, 56 Initialization bias, 634 Input arc, 94 Input place, 94 Instantaneous measure, 210 Instantaneous reward rate 77 Intelligent initialization, 634 Internal concurrency, 550 Interval estimates, 630 IntServ 720 Inverse transform, 614 / subsystems, 544 IPF‘, 627 Irreducible, 62, 70 ISDN channel, 767 J Jackson’s method, 511, 581 Jackson’s theorem, 342 Jacobi method, 170 Jensen’s method, 216 Joint cumulative distribution function, 52 Joint distribution function, 31 Joint probability density function, 52 Joint probability mass function, 31 K Kendall’s notation, 242-243 Kimura, 483 approximation, 267, 272 Kirigman, 269 Kingman-Kollerstrom approximation, 270 Kolmogorov backward equation, 67 Kolmogorov forward equation, 67 INDEX KramerlLmgenbach-Belz, 271, 483 formula, 266 kth-order statistic, 46 Kulbatzki, 271 approximation, 266 formula, 271 L LAN, 708 709 Laplace transform 37-38 358 Laplace-Stieltjes transform, 37, 248, 256 261 Latency time, 545 LBANC, 369 LBPS, 364 Likelihood function, 41 ratio, 43 Linear regression, 40 Little’s theorem, 245, 247, 249, 253, 276, 288, 290, 329, 384 Load-dependent, 358 Local balance, 335-336 conditions, 358 equations, 336-337, 341 property, 339, 341 Long-rangc dependency, 30 Long-tailetl, 29 Loss system, 286, 292 LST, 248 Lurnpable, 192 approxiillate, 201 pairwise, 201 M Machine rcpairrnan model, 252, 321 Macro state, 199 Majorizing, 622 M A M , 25!1, 311, 314 MAP, 306, 311 Marchal, 269, 483 Marginal probabilities, 330, 431 Marie’s method, 491 Marking, 94 dependrnt arc mtiltiplicity, 99 firing rate, 101 tangible, 96 vanishing, 96 Markov chain, 5:i ergodic, 63 nrodulal.ed Bernoulli process, 627 fluid source, 629 Poisson process, 303, 306, 626, 772 873 process, 52 property, 52, 65 renewal process, 303 reward model, 8, 75 Martin’s approximation, 267 Master slave model, 727 Matrix equation, 332 Matrix-analytic method, 259, 311, 314 Matrix-geometric method, 127, 263 Maximum entropy method, 470 closed networks, 473 open networks, 471 Maximum-likelihood estimation, 41 MB, 295 Mean, 29 Mean number of jobs, 250, 328, 331 of visits, 324, 326 Mean queue length, 251, 262, 329 Mean recurrence time, 63, 69 Mean remaining service time, 274, 279 Mean residual life, 255 Mean response time, 251 252, 262, 320, 375 Mean time t o absorption, 76 to failure, 71 t o repair, 72 Mean value, 16, 18 analysis, 369, 384, 659 Mean waiting time, 251, 255, 329 MEM, 470 hlemory constraints, 541, 547 Memoryless property, 53, 256 hlemory requirements, 424 Merging, 480 Method accept ance-reject ion, 622 bottapprox, 447, 578 coniposition, 620 conditioning, 637 convolution, 375, 619 Courtois’ approximate, 185 decomposition, 479 direct transformation, 621 FES, 410, 414, 542 How-equivalent server, 542 flow-equivalent server method, 410 Jackson’s, 511, 581 Jacobi, 170 Jensen’s 216 Marie’s 491 matrix-analytic, 259, 311, 314 matrix-geometric, 127, 263 maximum entropy, 470 874 lNDEX closed networks, 473 opcm networks, 471 mixing, 620 of batch means, 636 of independent replications, 634-635 of moments, 39 of shadow server, 522 of simultaneous displacement, 170 power, 166 Pujolle, 511 SUM 440, 505, 578 Takahashi’s, 198 M / G / queue, 255 M G M , 127, 263 M/G/ni queue, 267 Minimum batch policy, 295 Mixing method, 620 M / M / queue, 246 M / M / l / K queue, 251 MMBP, 627 MMFS, 629 M / M / c c queue, 249 M / M / m queue, 250 M M P P , 303, 306, 626, 772, 775 Moment, 37, 263 n t h , 16, 19 central, 17, 19 second central, Monoprocessor model, 725 Monotonicity 440 MRM, 8, 75 instantaneous reward rate, 77 MRSPN, 113 MTBDD, 120 MTTA, 76 M T T F , 71 M T T R , 72 Multiclass closed networks, 378 Multilevel models, 657 Multiple j o b classes, 325, 330 Miiltiple servers 328 Multiprocessor system, 704 loosely coupled, 705 tightly coupled, 704, 795 Multiprograniming system, 321 MVA, 369, 384, 421, 447, 579 N Network approximation, 421 BCMP, 353 closed, 253, 321, 324, 346, 355, 406 cyclic, 253 load-dependent service, 405 mixed, 325, 401 multiclass, 325, 330 closed, 378 open, :324, 355 product-form, 339, 364 421 separable, 339 single class, 323, 326 tandem, 253 with blocking, 592 Newton-Raphson method 446 N H P P , 625 Node, 321 Node types -/G/1-FCFS HOL, 538 - / G / l - F C F S PRS, 540 -/G/m-FCFS HOL, 539 -/G/m-FCFS PRS, 540 -/M/I-FCFS HOL, 514 537 - / M / l - F C F S P R S , 514, 539 -/M/l-LBPS 364 -/M/l-SIRO, 364 -/M/l-WEIILDP, 364 -/M/nl-FCFS HOL, 538 -/M/ni-FCFS PRS,540 product-form, 340 Non-exponential distribution, 463 Non-lossy system, 288 asymmetric, 288 Non-product-form networks, 461 Normalization condition, 291, 327, 330, 336, 338 constant, 339, 347 359, 369, 371 -372, 378 Normalized blocking factor, 565 Norrnalized speedup 565 Normalized standard deviation 17 Normalized synchronization overhead, 565 Norton’s theorem, 410 NPFQN, n-step rerurrrtnce probability, 62 transition probabilities, 55 Number of jobs, 244 -245, 359 ODE, 221 One-limited service, 714 One-step transition probabilities, 54 On-Off, 625 Ordinary differential equation, 221 O u t p u t analysis: 629, 639 O u t p u t arc, 94 Overall throughput, 328, 331 P PADS, 633 Pairwise lumpable, 201 Parallel processing, 551 with asynchronous tasks, 551 INDEX PASTA theorem, 401 Pdf, 18-20, 23-25, 29, 48, 52 PEPSY, 658 Percentiles, 83 Performability, 76-77, 804 Performance evaluation, 724 measures, 244, 326, 335 Periodic, 63 Peripheral devices, 321 Petri net, 94, 96 states, 94 PFQN, 3, 331 Place, 94 Pmf, 15 PN, 94, 96 states, 94 Point estimates, 630 Point process, 302 Poisson arrival streams, 358 process, 21, 246 Pollaczek-K hintchine formula, 256, 266 transform equation, 256 Polling, 146 asymmetric limited service, 147 exhaustive-service systems, 147 gated-service systems, 147 single-service systems, 147 symmetric limited services, 147 Population vector, 326, 357 Positive recurrent, 63 Power method 166 Preemption, 279 Priority, 98, 725 class, 272 fixed, 281 function, 280-282 networks, 514 queueing, 272 system, 272, 281-282 static, 282 strategy mixed, 727 time dependent, 280 Private cache blocks, 797 Probability density function, 18 generating function, 36 marginal, 327, 330, 373 mass function, 15 geometric, 246 n-step recurrence, 62 of loss, 286, 293 one-step transition, 54 875 routing, 323-326, 337 steady-state, 246, 250, 253, 262, 326 transition, 65 unconditional state, 67 vector, 244 equilibrium, 68 steady-state, 68 waiting, 250, 267 Process Bernoulli, 624 interrupted, 628 Markov modulated, 627 switched, 628 Markov, 52 renewal, 303 On-Off, 625, 645 point process, 302 Poisson, 21, 246, 624, 643 interrupted, 627 Markov modulated, 303, 306, 626, 772 non-homogeneous, 625 quasi-birth-death, 127 sem i-Mar kov, 101 stochastic, 52 Product-form, 340 approximation, 469 expressiort, 347 node types, 340 queueing network, 336, 369, 421, 440 multiclass, 441 single, 440 solution, 341, 369 Property local balance, 341 M M, 341 Markov, 52, 65 memoryless, 53, 248 stat ion-balance, 34 Pujolle method, 51 + Q QBD, 127 Quasi-birth-death process, 127 Queue length, 245 Queueing discipline, 242-243, 358 network, 321 closed, 336 closed tandem, 321 non-product-form, 657 product-form, 657 system asymmetric, 286 elementary, 242 symmetric, 286 876 lNDfX Quota nodes, 533 RS, 95 R S Race model, 96 Random early detection, 648 Random selection, 286 Random variable Bernoulli, 16: 620 binomial, 16, 37, 620 continuous, 33 discrete, 614 Erlang, 31, 38 exponential, 31, 38 gamma, 31, 38 geometric, 16, 37 hyperexponential, 31, 38 hypoexponential, 31, 38 majorizing, 622 normal, 621 Pascal, 620 Poisson, 16, 37 statistically independent, 31 sum, 34 Random variate, 614 Randomization, 124 Rare event, 609 Reachability graph, 110 Reachability set, 95 Reachable, 62, 70 RECAL algorithm, 369 Recurrent non-null, 63 null, 63 positive, 63 state, 62 Relative arrival rate, 324 Relative utilization, 324, 453 Relaxation parameter, 173 Remaining service time, 267 Renewal process Markov, 303 Repetitive blocking, 593 Residual state holding time, 70 Response time, 245, 248, 251-252 RESQ, 561 RESTART, 610; 638 Reward rate, 76 expected instantaneous, 79 specification, 101 RG, 102 Robustness, 487 Routing matrix, 353, 355 probability, 323 load-dependent, 364, 384 Sample mran, 635 SBP, 628 SCAT, 427, 502 algorithm, 502, 580 core algorithm, 428 extended algorithm, 435 multiple server node algorithm, 431 single server node algorithm, 427 Second moment, 17 Seek time, 545 Self-correcting approximation technique, 427 Self-similar, 29 Semi-Markov proccss, 101 Serialization, 549 Service blocking, 593 rate, 324 load-dependent, 384, 410 station, 321 single, 321 time distribution, 263 generally distributed, 353 remaining, 255 Shadow equation, 535 model, 522 network, 523 node, 522 service times, 523 technique, 522, 535 extended, 727 Shared cache blocks, 798 SHARPE, 543, 551, 687 Short circuit, 411, 492 Simulation, 607 algorithm, 607 application, 639 computer, 607 continuous, 610 deterministic, 610 discrete, 610 discrete event, 607 distributed, 611 distribution driven, 611 event oriented, 611 initialization, 634 language, 609, 612 model, 608-610 packages, 612 parallel and distributed, 610, 638 process oriented, 611 INDEX program, 607 regenerative, 610 runs, 609, 614 sequential, 611 software libraries, 612 steady-state, 610, 634 stochastic, 610 symbolic, 611 synthetic, 61 terminating, 610, 631 tools, 609, 612 trace driven, 611 Simultaneous displacement, 170 resource possession, 541, 544 Single class closed network, 371 Single server node, 327 Single station queueing system, 241 Single-buffer polling system, 108 SIRO, 364 Slow state, 223 SMP, 101 Sojourn time, 64 Solution technique numerical, 335 SOR, 173 Speedup, 565, 636 Split-merge system, 562 Splitting, 481 SPNP, 666 Squared coefficient of variation, 27-29 Standard deviation, 17 State diagram, 54 probabilities, 286-288, 291, 335 space, 52, 353 transition diagram, 54, 286, 335, 337 variables, 607 Stationary, 57 Stationary probability vector, 57 Steady-state probability, 289, 341 vector, 68, 332-333 solution, 336 Stochastic process, 52 transition matrix, 54 Strategy last batch processor sharing (LBPS), 364 WEIRDP, 364 Substitute network, 492 Successive over-relaxation, 173 SUM, 440, 447, 505, 578 Summation method, 440, 505, 578 Surrogate delays, 547 Symmetric system 877 GI/G/m, 285 M / M / m , 285 Synchronization overhead, 565 System availability, 73 equation, 421 gracefully degrading, performance, 73 reliability, 73 T Takahashi’s method, 198 Tangible marking, 96 Task completion, 74 precedence graph, 561 Terminal system, 323, 557 Terminals, 323 Theorem arrival, 384, 401 BCMP, 357-358, 371 central limit, 35 Gordon/Newell, 349 Jackson’s, 342, 747 Little’s, 245, 247, 249, 253, 276, 288, 290, 329, 384 Norton’s, 410 PASTA, 401 Throughput, 245, 324, 328, 330 Time-average accumulated reward, 77 Timed transition, 96 Time-homogeneous, 53 Token, 94 Token rotation time, 715 T P M , 59 Traffic equations, 324, 344, 472, 714 Transfer blocking, 592 time, 545 Transient solution, 209 state, 62 probabilities, 56 uniformization, 216 Transition, 94 enabled, 95 fire, 95 immediate, 96 probability, 65 rates, 66 timed, 96 Tree-convolution, 384 U Uniformization, 124 Unit vector, 69 878 INDEX UNIX, 724 Upper time limit, 282 Utilization, 244, 253, 327, 330 V Vanishing marking, 96 Variance, 17, 19-20, 23 -25, 27-29, 31, 246, 249 reduction, 610, 636 Visit ratio, 324, 347, 355 W Wafer production system, 756 Waiting time, 245 WAN, 709 Z z-transform, 36, 38 ... WILEYINTERSCIENCE A JOHN WILEY & SONS, INC., PUBLICATION Queueing Networks and Markov Chains This Page Intentionally Left Blank Queueing Networks and Markov Chains Modeling and Performance Evaluation with. . .Queueing Networks and Markov Chains Modeling and Performance Evaluation with Computer Science Applications Second Edition Gunter Bolch Stefan Greiner Hermann de Meer Kishor S Trivedi WILEYINTERSCIENCE... web site at www .wiley. com Library of Congress Cataloging-in-Publication Data: Queueing networks and Markov chains : modeling and performance evaluation with computer science applications / Gunter