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Communicating Systems & Networks: Traffic & Performance This page intentionally left blank INNOVATIVE TECHNOLOGY SERIES INFORMATION SYSTEMS AND NETWORKS Communicating Systems & Networks: Traffic & Performance Georges Fiche & Gerard Hebuterne London and Sterling, VA First published in France in 2003 by Hermes Science entitled 'Trafic et performances des reseaux de telecoms' First published in Great Britain and the United States in 2004 by Kogan Page Science, an imprint of Kogan Page Limited Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licences issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned addresses: 120 Pentonville Road 22883 Quicksilver Drive London N1 9JN Sterling VA 20166-2012 UK USA www.koganpagescience.com © GET and Lavoisier, 2003 © Kogan Page Limited, 2004 The right of Georges Fiche and Gerard Hebuterne to be identified as the authors of this work has been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. ISBN 1 9039 9635 X British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library. Library of Congress Cataloging-in-Publication Data Fiche, Georges. [Trafic et performances des reseaux de telecoms. English] Systems and communicating networks : traffic and performance / Georges Fiche and Gerard Haebuterne. p. cm. -- (Innovative technology series. Information systems and networks) ISBN 1-903996-35-X 1. Telecommunication--Traffic. 2. Computer networks. I. Haebuterne, Gaerard. II. Title. III. Series. TK5102.5F47 2004 621.382'l--dc22 2003024763 Typeset by Kogan Page Printed and bound in Great Britain by Creative Print and Design (Wales), Ebbw Vale Contents Introduction xiii 1. Telecommunication Networks and Traffic 1 1.1. Network concepts 2 1.1.1. Network architectures 2 1.1.1.1. Network components 3 1.1.1.2. Network structures 4 1.1.2. Communication technologies: circuit and packet 7 1.1.2.1. Circuit switching 7 1.1.2.2. Packet switching 8 1.1.3. Main networks 10 1.1.3.1. Conventional fixed telephone network 11 1.1.3.2. No. 7 signalling network 12 1.1.3.3. Mobile networks 13 1.1.3.4. The internet network 17 1.1.3.5. The next generation network (NGN) 20 1.1.3.6. Private networks 23 1.2. Traffic concepts 25 1.2.1. Erlang concept 25 1.2.2. Traffic offered, traffic handled 27 1.2.3. Quality of service 28 1.2.4. Load profiles, load A and load B 28 1.2.5. Stationarity 30 1.2.6. The concept of busy hour call attempts (BHCA) 30 1.2.7. The user plane and the control plane 32 1.2.8. Characterisation of traffic 32 1.2.8.1. Characterisation of services 33 1.2.8.2. Traffic mix 34 1.2.8.3. Traffic matrix 34 1.2.8.4. "Node" level matrix 35 1.2.8.5. Network level matrix 38 vi Communicating Networks: Traffic and Performance 2. Quality of Service and Performance 41 2.1. Standardization organisms and norms 41 2.2. Quality of Service (QoS) 42 2.3. Network performance 44 2.3.1. End-to-end and intermediate performance parameters 44 2.3.1.1. Reference network configuration 45 2.3.1.2. Communication set-up reference diagram 45 2.3.1.3. Communication, transfer phase reference diagram 45 2.3.1.4. Disengagement reference diagram 46 2.3.2. Access, disengagement, and transfer phases 47 2.3.3. Parameter values 48 2.4. Traffic load conditions 49 2.4.1. Normal load, load A 50 2.4.2. High load, load B 51 2.4.3. Overload 52 2.5. Parameters and standards 55 2.5.1.Trafficability 56 2.5.1.1. Access and disengagement 56 2.5.1.2. Communication, user information transfer 60 2.5.2. Dependability parameters 63 2.5.2.1. Access 63 2.5.2.2. Transfer and disengagement 65 2.5.3. Performance during overload 66 2.5.4. Synthesis tables 67 2.5.4. l.Trafficability 68 2.5.4.2. Dependability parameters 69 3. Probability 71 3.1. Definition and properties of events 71 3.1.1. The concept of event 71 3.1.2. Complementary events 73 3.1.3. Properties of operations on events 73 3.1.3.1. Commutativity 73 3.1.3.2. Associativity 73 3.1.3.3.Distributivity 74 3.1.3.4. Difference 74 3.1.3.5. De Morgan rules 74 3.2. Probability 75 3.2.1. Definition 75 3.2.2. Basic theorems and results 75 3.2.2.1. Addition theorem 75 3.2.2.2. Conditional probability 76 3.2.2.3. Multiplication theorem 78 3.2.2.4. The a posteriori probability theorem 79 3.3. Random variable 80 Contents vii 3.3.1. Definition 80 3.3.2. Probability functions of a random variable 81 3.3.2.1. Notations 81 3.3.2.2. Distribution function or cumulated probability 81 3.3.2.3. Probability density function 81 3.3.3. Moments of a random variable 82 3.3.3.1. Moments about the origin 83 3.3.3.2. Central moments 83 3.3.3.3. Mean and variance 83 3.3.3.4. Examples of application 85 3.3.4. Pairs of random variables 86 3.3.4.1. Definition 87 3.3.4.2. Joint probability 87 3.3.4.3. Marginal probability of a pair of random variables 89 3.3.4.4. Conditional probability of a pair of random variables 89 3.3.4.5. Functions of a pair of random variables 90 3.3.4.6. Sum of independent random variables 92 3.3.4.7. Moments of the sum of independent random variables 94 3.3.4.8. Practical interest 96 3.4. Convolution 97 3.4.1. Definition 97 3.4.2. Properties of the convolution operation 98 3.4.2.1. The convolution is commutative 98 3.4.2.2. Convolution of exponential distributions 98 3.4.2.3. Convolution of normal distributions 98 3.5. Laplace transform 100 3.5.1. Definition 100 3.5.2. Properties 101 3.5.2.1. Fundamental property 101 3.5.2.2. Differentiation property 101 3.5.2.3. Integration property 102 3.5.2.4. Some common transforms 102 3.6. Characteristic function, generating function, z transform 106 3.6.1. Characteristic function 106 3.6.1.1. Definition 106 3.6.1.2. Properties 107 3.6.1.3. Inversion formula 107 3.6.1.4. Asymptotic law 110 3.6.1.5. Some usual transforms 114 3.6.2. Generating functions 115 3.6.2.1. Definition 115 3.6.2.2. Moments 116 3.6.2.3. Some usual transforms 116 3.6.2.4. Convolution 117 viii Communicating Networks: Traffic and Performance 4. Probability Laws 121 4.1. The (discrete) uniform distribution 122 4.2. The binomial law 123 4.3. The multinomial distribution 125 4.4. The geometric distribution 125 4.5. The hypergeometric distribution 127 4.6. The Poisson law 128 4.6.1. Relationship with the binomial law 130 4.7. The continuous uniform distribution 131 4.8. The normal (Gaussian) distribution 132 4.8.1. The sum of normal random variables 134 4.8.2. Statistical tables 135 4.8.3. The normal law as limiting distribution 136 4.9. The Chi-2 distribution 136 4.9.1. Limiting behaviour 137 4.10. The Student distribution 137 4.11. The lognormal distribution 138 4.12. The exponential and related distributions 139 4.12.1. The exponential distribution 139 4.12.2. The Erlang-k distribution 140 4.12.3. The hyperexponential distribution 143 4.12.4. Generalising: the Cox distribution 144 4.12.5. The Gamma distribution 144 4.13. The Weibull distribution 146 4.14. The logistic distribution 147 4.15. The Pareto distribution 148 4.16. A summary of the main results 150 4.16.1. Discrete distributions 150 4.16.2. Continuous distributions 152 5. Statistics 155 5.1. Descriptive statistics 156 5.1.1. Data representation 156 5.1.2. Statistical parameters 158 5.1.2.1.Fractiles 158 5.1.2.2. The sample mean 159 5.1.2.3. The sample variance 159 5.1.2.4. The moments 159 5.1.2.5. The mode 160 5.1.2.6. Other characterisations 160 5.2. Correlation and regression 161 5.2.1. Correlation coefficient 162 5.2.2. The regression curve 163 5.3. Sampling and estimation techniques 164