Performance of Computer Communication Systems: A Model-Based Approach Boudewijn R Haverkort Copyright © 1998 John Wiley & Sons, Ltd Print ISBN 0-471-197228-2 Electronic ISBN 0-470-84192-3 PERFO~CI~ 0E; CoMPUTER COMMUNICA~ON SYSTEMS PERFORMAN~E~F COMPUTER COMMUNICATION SYSTEMS A Model-Based Approach BOUDEWIJN R HAVERKORT Rheinisch- Westfalische Technische Hochschule Aachen, Germany John Wiley & Sons, Ltd Chichester l New York l Weinheim l Brisbane l Singapore Toronto l Copyright 1998 by John Wiley & Sons Ltd, Baffins Lane, Chichester, West SussexPO 19 1UD, England National 1243 779777 International (+44) 1243 779777 e-mail (for orders and customer service enquiries): cs-books@wiley.co.uk Visit our Home Page on http://www.wiley.co.uk or http://www.wiley.com Reprinted June 1999 All Rights Reserved.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 under the terms of the Copyright, Designs and PatentsAct 1988 or under the terms of a licence issued by the Copyright Licensing Agency, 90 Tottenham Court Road, London W 1P 9HE, UK, without the permission in writing of the Publisher Neither the author nor John Wiley & Sons Ltd accept any responsibility or liability for loss or damage occasioned to any person or property through using the material, instructions, methods or ideas contained herein, or acting or refraining from acting as a result of such use The author and Publisher expressly disclaim all implied warranties, including merchantability or fitness for any particular purpose There will be no duty on the author or Publisher to correct any errors or defects in the software Designations used by companies to distinguish their products are often claimed as trademarks In all instanceswhere John Wiley & Sons is aware of a claim, the product names appear in initial capital or all capital letters Readers,however, should contact the appropriate companies for more complete information regarding trademarks and registration Other Wiley Editorial Oflees New York Weinheim Brisbane Singapore Toronto l l l l Library of Congress Cataloguing in Publication Data Haverkort, Boudewijn R Performance of computer communication systems : a model-based approach / Boudewijn R Haverkort p cm Includes bibliographical references and index ISBN O-471-97228-2 (alk paper) Computer networks-Evaluation Electronic digital computers-Evaluation Queuing theory Telecommunication systems-Evaluation, Stochastic processes I Title TK5 105.5.H375 1998 004.6-dc2 98-27222 CIP British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 471 97228 Produced from Postcript files supplied by the author Printed and bound in Great Britain by Biddles Ltd, Guildford and King’s Lynn This book is printed on acid-free paper responsibly manufactured from sustainable forestry, in which at least two trees are planted for each one used in paper production Johannes In memory Hermannus of my father Hendrikus Haverkort Contents xvii Preface I Performance modelling Model 1.3 Stochastic 1.4 Queueing 1.4.1 1.4.2 1.5 1.6 1.7 evaluation: processes techniques models models The principle of queueing Single queues: the Kendall Model construction 1.5.2 Model solution 11 notation 15 18 19 19 reading Exercises law 2.1.1 Understanding 2.1.2 Proof of Little’s The simplest 2.3 Further 2.4 Exercises 21 queue 21 Little’s 21 24 law law Little’s queueing model: the MI M 11 queue reading of stochastic 25 28 28 31 processes Overview 13 15 law and the MIMI1 11 Further Stochastic 1.5.1 2.2 3.1 aim and approach solution Tool support Little’s 2.1 stochastic Introduction 1.1 Performance 1.2 with processes 31 Contents Vlll 3.2 3.3 3.4 3.5 Renewal processes Discrete-time Markov chains Convergence properties of Markov chains Continuous-time Markov chains 3.6 3.7 3.8 3.9 3.5.1 From DTMC to CTMC 3.5.2 Evaluating the steady-state and transient behaviour Semi-Markov chains The birth-death process The Poisson process Renewal processes as arrival processes 3.9.1 Phase-type distributions 3.9.2 Phase-type renewal processes 3.10 Summary of Markov chains 3.11 Further reading 3.12 Exercises II Single-server MIMI1 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 queueing queueing models models General solution of the MIMI1 queue The MIMI1 queue with constant rates The PASTA property Response time distribution in the MIMI1 queue The MIMI m multi-server queue The MIMI 00 infinite-server queue Job allocation in heterogeneous multi-processors The MIMI11 m single-server queue with bounded buffer The MIMI m Im multi-server queue without buffer The MIM(l((K queue or the terminal model Mean values for the terminal model Further reading Exercises MIGjl-FCFS queueing models 5.1 The M/Gil result 5.2 An intuitive proof of the M/G/l result 34 37 41 43 43 45 51 53 53 54 55 58 62 63 63 67 69 70 73 74 76 77 79 80 83 85 86 88 92 92 95 95 99 iX Contents 5.2.1 5.2,2 5.3 5.4 5.5 5.5.2 5.7 103 A formal proof of the MlGll result The MlGll model with batch arrivals MI G/ queueing systems with server breakdowns 5.5.1 5.6 100 Residual lifetime Intuitive proof Single arrivals Batch arrivals 104 107 109 110 111 112 Further reading Exercises 112 M/G/ queueing models with various scheduling disciplines 6.1 Non-preemptive priority scheduling 6.2 Preemptive priority scheduling 115 115 121 123 6.4 Shortest job next scheduling Round robin scheduling 6.5 Processor sharing scheduling 128 6.6 129 6.7 Scheduling based on elapsed processing time Further reading 6.8 Exercises 131 6.3 G)Mll-FCFS and GIGIl-FCFS queueing 7.1 The GlMll queue 7.2 The GIG11 queue 7.3 7.4 7.5 results for the G(GI queue Further reading Exercises Approximate PHIPHIl 8.1 8.2 models The MIMI1 queue The PHIPHIl queue 8.2.1 8.2.2 8.2.3 queueing A structured description of the CTMC Matrix-geometric solution Stability issues Performance measures Numerical aspects 8.3.1 Solving the boundary equations 8.2.4 8.3 models 126 131 133 133 139 143 143 144 145 145 148 148 152 153 154 155 155 Contents X 8.4 8.3.2 A successive substitution 8.3.3 The logarithmic 8.4.1 The M/PHI1 q ueue: an explicit 8.4.2 The PHlMlm 8.6 Other matrix-geometric 8.7 Further models with 162 163 164 167 solution 169 169 Exercises 173 models Characterisation of polling 9.1.1 Basic terminology 9.1.2 The visit order 9.1.3 The scheduling Cyclic polling: 9.3 Count-based 9.5 for R reading 9.2 9.4 expression queue The caudal curve 9.1 162 A few special cases Polling 158 algorithm 8.5 8.8 III reduction 156 algorithm Count-based 174 174 symmetric cyclic polling Exhaustive service: 9.4.2 Some approximate evaluation 9.5.1 Timed-token 9.5.2 Approximating 9.5.3 The influence models exact analysis of the IBM token ring the timed-token 9.7 Further 9.8 Exercises timer models 10 Open queueing 10.2 Feed-forward 187 189 191 194 197 models 199 networks 10.1 Basic terminology 186 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Other Wiley Editorial Oflees New York Weinheim Brisbane Singapore Toronto l l l l Library of Congress Cataloguing in Publication Data Haverkort, Boudewijn R Performance of computer communication systems. .. (Electronic) Part Performance stochastic I modelling processes with Performance of Computer Communication Systems: A Model-Based Approach Boudewijn R Haverkort Copyright © 1998 John Wiley & Sons Ltd ISBNs:... conserving, 98 working set, 287 x2-test, 422 Performance of Computer Communication Systems: A Model-Based Approach Boudewijn R Haverkort Copyright © 1998 John Wiley & Sons Ltd ISBNs: 0-471-97228-2 (Hardback);