Queueing Networks and Markov Chains.pdf

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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 WILEY- INTERSCIENCE A JOHN WILEY & SONS, INC., PUBLICATION

Queueing Networks and Markov Chains

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Copyright 0 2006 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 the prior written permission of the Publisher, or authorization through payment o f the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 1 1 1 River Street, Hoboken, NJ 07030, (201) 748-601 I , fax (201) 748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and spccifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic format. For information about Wiley products, visit our 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 Bolch . , . [et al.].-2nd rcv. and enlarged ed. p. cm. “A Wiley-lnterscience publication.” Includes 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. 2. Queuing theory. I. Bolch, Gunter QA76.9E94Q48 2006 004.2’4015 19233-dc22 Printed in the United States of America. 1 0 9 8 7 6 5 4 3 2 1 200506965

Contents Preface to the Second Edition Preface to the First Edition 1 Introduction ... X l l l xv 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Motivation 1 . . . . . . . . . . . . . . . . . . . 5 1.2 Methodological Background . . . . . . . . . . . . . . . . . . . 6 1.2.1 Problem Formulation 1.2.2 The Modeling Process . . . . . . . . . . . . . . . . . . . 8 . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.3 Evaluation 1.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 12 . . . . . . . . . . . . . . . 15 1.3.1 Random Variables 15 1.3.2 Multiple Random Variables . . . . . . . . . . . . . . . . 30 36 1.3.3 Transforms 1.3.4 Parameter Estimation . . . . . . . . . . . . . . . . . . . 38 1.3.5 Order Statistics 46 . . . . . . . . . . . . . . . . . . . 46 1.3.6 Distribution of Sums 1.3 Basics of Probability and Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V

vi CONTENTS 2 Markov Chains 2.1 Markov Processes 51 . . . . . . . . . . . . . . . . . . . . . . . . . 51 . . . . . . . . . . . . 51 53 2.2 Performance Measures 2.1.1 Stochastic and Markov Processes 2.1.2 Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.2.1 A Simple Example . . . . . . . . . . . . . . . . . . . . . 71 2.2.2 Markov Reward Models . . . . . . . . . . . . . . . . . . 75 2.2.3 A Casestudy 80 2.3 Generation Methods 90 94 2.3.1 Petri Nets . . . . . . . . . . . . 96 2.3.2 Generalized Stochastic Petri Nets 2.3.3 Stochastic Reward Nets . . . . . . . . . . . . . . . . . . 97 . . . . . . . . . . . . . . . . . . . 101 2.3.4 GSPN/SRN Analysis 2.3.5 108 . . . . . . . . . . . . . 113 2.3.6 Stochastic Petri Net Extensions . . . . . . . . . . . . . . . . . . 115 2.3.7 Non-Markoviarl Models 2.3.8 Symbolic State Space Storage Techniques . . . . . . . . 120 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Larger Exanlple . . . . . . . . . . . . . . . . . . . . . 3 Steady-State Solutions of Markov Chains 123 . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Hessenberg Matrix: Non-Markovian Queues 3.2.1 The Concept 3.2.2 Example: The QBD Process . . . . . . . . . . . . . . . 125 3.1 Solution for a Birth Death Process 3.2 Matrix-Geometric Method: Quasi-Birth-Death Process . . . . 127 127 . . . . . . . . . . . . . . . 128 . . . . . . . . . . 140 . . . . . . . . . . . . . . 141 3.3.1 Nonexporlential Servicc Times 3.3.2 Server with Vacations . . . . . . . . . . . . . . . . . . . 146 . . . . . . . . . . . . . . 151 . . . . . . . . . . . . . . . . . . . 152 . . . . . . . . . . . . . . . . 158 . . . . . . . . . . . . . 165 . . . . . . . . . . . . 165 166 169 . . . . . . . . . . . . . . . . . . . 172 . . . . . . . 173 . . . . . . . . . . 177 3.6.1 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . 179 3.5.1 Convergence of Iterative Methods 3.5.2 Power Method 3.5.3 Jacobi's Method 3.5.4 Gauss-Seidel Method 3.5.5 The Method of Successive Over-Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Gaussian Elimination 3.4.2 The Grassmanrl Algorithm 3.6 Comparison of Numerical Solution Methods 3.4 Numerical Solution: Direct Methods 3.5 Numerical Solution: Iterative Methods

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