stochastic modeling and analysis of telecoms networks

Stochastic Modeling and Analysis of Telecom Networks www.it-ebooks.info Stochastic Modeling and Analysis of Telecom Networks Laurent Decreusefond Pascal Moyal Series Editor Nikolaos Limnios www.it-ebooks.info First published 2012 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. 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 licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd John Wiley & Sons, Inc. 27-37 St George’s Road 111 River Street London SW19 4EU Hoboken, NJ 07030 UK USA www.iste.co.uk www.wiley.com © ISTE Ltd 2012 The rights of Laurent Decreusefond and Pascal Moyal to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. ____________________________________________________________________________________ Library of Congress Cataloging-in-Publication Data Decreusefond, Laurent, 1966- Stochastic modeling and analysis of telecom networks / Laurent Decreusefond, Pascal Moyal. p. cm. Includes bibliographical references and index. ISBN 978-1-84821-238-1 1. Mobile communication systems. 2. System analysis Mathematical models. I. Moyal, Pascal, 1978- II. Title. TK5103.2.D43 2011 621.3845'6051922 dc23 2011036865 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN: 978-1-84821-238-1 Printed and bound in Great Britain by CPI Group (UK) Ltd., Croydon, Surrey CR0 4YY www.it-ebooks.info Table of Contents Preface ix Chapter 1. Introduction 1 1.1. Traffic, load, Erlang, etc. 1 1.2. Notations and nomenclature 7 1.3. Lindley and Beneˇs 10 1.4. Notes and comments 18 P ART 1: DISCRETE-TIME MODELING 21 Chapter 2. Stochastic Recursive Sequences 23 2.1. Canonical space 24 2.2. Loynes’s scheme 30 2.3. Coupling 34 2.4. Comparison of stochastic recursive sequences 40 2.5. Notes and comments 43 Chapter 3. Markov Chains 45 3.1. Definition and examples 45 3.2. Strong Markov property 49 3.3. Classification of states 52 3.4. Invariant measures and invariant probability 60 3.5. Effective calculation of the invariant probability 75 3.6. Problems 77 3.7. Notes and comments 80 Chapter 4. Stationary Queues 83 4.1. Single server queues 84 4.2. Processor sharing queue 104 www.it-ebooks.info vi Networks Modeling and Analysis 4.3. Parallel queues 106 4.4. The queue with S servers 117 4.5. Infinite servers queue 124 4.6. Queues with impatient customers 127 4.7. Notes and comments 146 Chapter 5. The M/GI/1 Queue 149 5.1. The number of customers in the queue 149 5.2. Pollacek-Khinchin formulas 153 5.3. Sojourn time 156 5.4. Tail distribution of the waiting time 158 5.5. Busy periods 160 P ART 2: CONTINUOUS-TIME MODELING 167 Chapter 6. Poisson Process 169 6.1. Definitions 170 6.2. Properties 176 6.3. Discrete analog: the Bernoulli process 181 6.4. Simulation of the Poisson process 183 6.5. Non-homogeneous Poisson process 185 6.6. Cox processes 189 6.7. Problems 189 6.8. Notes and comments 191 Chapter 7. Markov Process 193 7.1. Preliminaries 193 7.2. Pathwise construction 195 7.3. Markovian semi-group and infinitesimal generator 199 7.4. Martingale problem 215 7.5. Reversibility and applications 220 7.6. Markov Modulated Poisson Processes 226 7.7. Problems 232 7.8. Notes and comments 234 Chapter 8. Systems with Delay 237 8.1. Little’s formula 237 8.2. Single server queue 241 8.3. Multiple server queue 245 8.4. Processor sharing queue 252 8.5. The M/M/∞ queue 253 8.6. The departure process 254 www.it-ebooks.info Table of Contents vii 8.7. Queuing networks 255 8.8. Problems 265 8.9. Notes and comments 268 Chapter 9. Loss Systems 271 9.1. General 271 9.2. Erlang model 274 9.3. The M/M/1/1 + C queue 276 9.4. The “trunk” effect 279 9.5. Engset model 280 9.6. IPP/M/S/S queue 281 9.7. Generalized Erlang models 285 9.8. Hierarchical networks 289 9.9. A model with balking 294 9.10. A call center with impatient customers 301 9.11. Problems 303 9.12. Notes and comments 304 P ART 3: SPATIAL MODELING 307 Chapter 10. Spatial Point Processes 309 10.1. Preliminary 309 10.2. Stochastic geometry 310 10.3. Poisson process 311 10.4. Stochastic analysis 326 10.5. Problems 336 10.6. Notes and comments 337 Appendix A. Mathematical Toolbox 339 A.1. Probability spaces and processes 339 A.2. Conditional expectation 347 A.3. Vector spaces and orders 352 A.4. Bounded variation processes 356 A.5. Martingales 363 A.6. Laplace transform 378 A.7. Notes and comments 379 Bibliography 381 Index 385 www.it-ebooks.info 1 Preface In mobile telecommunications, ARCEP (the French Regulatory Authority for Electronic Communicationsand Postalservices) publishesan annualanalysis ofquality of different mobile radio networks. For voice, the two criteria are the ability to start up a communication and to hold it for 2 or 5 minutes as well as the audio quality of the communication. For each data service, the transmission time and integrity of the message (SMS, MMS) are tested in different situations: urban, semi-urban, for pedestrians, cars, high-speed train, etc. The results of these tests are often used as commercial arguments. On the contrary, bad results may rapidly alter the image of a telecom operator in the public opinion and thus lead to an economic disaster. Hence, these performance tests are a major challenge for the whole telecom industry. The satisfaction of some of these criteria depends directly on the number of resources allocated to the network, including the capacity of the so-called base stations. The operator must have some quantitative means to anticipate demand and its impact on the design of its network. If we want to move beyond the phase of divination, then modelization is needed. This is about putting into equations, although sometimes with a kabbalistic aspect, the phenomenon which we want to study. To each situation may correspond several models depending on whether one is interested in the microscopic or macroscopic scale, the long or short time behavior, and so on. Ideally, the choice should be made only based on purpose but it is also conditioned by the technical and mathematical knowledge of the people who build the model. Once the problem is raised, it must be solved: in other words, if numbers are given in input, then some numbers should pop up in output. Thanks to advances in computing, the situation has changed dramatically in the last twenty years. It is now possible to calculate quantities that are not only defined by explicit mathematical formulas, but that may result from more or less sophisticated algorithms. www.it-ebooks.info x Networks Modeling and Analysis A model is also often a support for simulation, in this way it creates an artificial simplification of reality. If this method gives very often only approximate results and is costly in computation time, it is also often the only possible. We tried in this book to show for what purpose could stochastic models be used in telecommunications networks, with quantitative as well as qualitive points of view. We wanted to vary the possible approaches (discrete time, continuous time Markov chains or processes, recurrent sequences, spatial modeling) to allow the reader to proceed with his modelization works himself. We have not, far from it, addressed all themes and all the technicalities on which the researchers are currently working. In particular, we did not discuss fluid limits and Palm measures, but we hope that our readers can take the rich literature to extend their thinking. We have tried to be as complete as possible in the mathematical prerequisites. Proofs and results that are missing can be found easily in many books that appear in the references. To emphasize the computational aspects and to help our student readers, we have very often explained the algorithms that to be implemented in order to solve a particular problem. Languages such as Octave, Scilab or Scipy/Numpy (available through the SAGE platform) are particularly well suited to the vector computations that appear here and allow us to instantiate the algorithms described in a few lines only. This book would not exist without the assistance of a considerable number of people. The first draft of this book is a handout from Telecom ParisTech written by L. Decreusefond, D. Kofman, H. Korezlioglu and S. Tohme. The introduction to the martingale theory owes much to a handout from A.S. Üstünel. We have tried as much as possible to present the underlying network protocols. It must be noted that the decryption of standards of thousands of pages and their translation into human language require much work and fine knowledge in a wide variety of disciplines, and as well as infinite patience. We wish to thank C. Rigault and especially P. Martins, without whom we would not know what POTS was and even less OFDMA. We heartily thank N.Limnios, who offered us a beginning on this long-term venture, as well as our colleagues C. Graham, Ph. Robert and F. Baccelli, with whom we have had much interaction on these topics for several years. This book would not have been what it is without the inspiration born from reading their books on these topics. A big thanks to our partners for having supported us at difficult times. Thanks to Adele for her help. Our students or colleagues, E. Ferraz, I. Flint, P. Martins, A. Vergne, T. T. Vu have reviewed and amended all or part of this opus. We thank them for participating in an often thankless task. The residual errors are ours. Paris, February 2012. www.it-ebooks.info Chapter 1 Introduction 1.1. Traffic, load, Erlang, etc. In electricity, we count the amps or volts; in meteorology, we measure the pressure; in telecommunications, we count the Erlangs. The telephone came into existence in 1870. Most of the concepts and notations were derived during this period. Looking at a telephone connection over a time period of length T, we define its observed traffic flow as the percentage of time during which the connection is busy ρ =  i t i T . A priori, traffic is a dimensionless quantity since it is the ratio of the occupation time to the total time. However, it still has a unit, Erlang, in remembrance of Erlang who, along with Palm, was one of the pioneers of the performance assessment of telephone networks. Therefore, a load of 1 Erlang corresponds to an always busy connection. Time 0 T t 1 t 2 t 3 t 4 Figure 1.1. Traffic of a connection: ratio of the occupation time to the total time of observation Stochastic Modeling and Analysis of Telecom Networks Laurent Decreusefond and Pascal Moyal © 2012 ISTE Ltd. Published 2012 by ISTE Ltd. www.it-ebooks.info 2 Networks Modeling and Analysis Looking at several connections, the traffic carried by this trunk is the sum of the traffic of each connection ρ trunk =  connections ρ connection . This is no longer a percentage, but we can give a physical interpretation to this quantity according to the ergodic hypothesis. In fact, assume that the number of junctions is large, then we can calculate the average occupation rate in two different ways: either by calculating the percentage of the occupation time of a particular connection over a large period of time; or by computing the percentage of busy connections at a given time. In statistical physics, the ergodicity of a set of gas molecules implies that the spatial averages (for example, averages calculated on the set of gas molecules) are equal to time averages (i.e. averages calculated over a molecule for a long period of time). By analogy, we now assume that the same holds true for the occupation rate of telephone connections. We, therefore, have p = lim T →∞ 1 T  j t j = lim N→∞ 1 N  n X n (t), [1.1] where X n =1if the junction n is busy at time t, X n =0,otherwise. Note that on the right-hand side, the value of t is arbitrary. This implies that we have implicitly assumed that the system is in steady state, that is statistically, its behavior does not change with time. When the number of junctions is large, it is unrealistic to try to define a structure of correlation between them. It is therefore reasonable to assume that a connection is free or busy, irrespective of the situation of other connections. Therefore, at a given time t, the number of busy connections follows a binomial distribution with parameters N (the total number of connections) and p (calculated by equation [1.1]). The average number of busy connections is Np at each moment. This relation provides a simple and efficient way to estimate p. Telephone switches have among other functions to count the number of ongoing calls at each moment. By averaging this number over 15 seconds, we obtain a fairly accurate estimation of the average number of simultaneous calls, that is an estimation of p. This raises a question: How to choose T and when to carry out the measurements? It is in fact clear that the traffic fluctuates throughout the day based on the human activities. For we want to reduce and ensure a low failure rates, it is necessary to consider the worst case and conduct measurements during heavy traffic periods. For generations, the observation period has been referred to as one hour and we look at the traffic at the busiest hour of the day. Let us imagine for a moment that calls occur every 1/λ seconds and last exactly 1/µ seconds with µ>λ. www.it-ebooks.info [...]... distribution Stochastic Modeling and Analysis of Telecom Networks © 2012 ISTE Ltd Published 2012 by ISTE Ltd Laurent Decreusefond and Pascal Moyal www.it-ebooks.info 24 Networks Modeling and Analysis More generally, the asymptotic study is essentially based on the properties of the recurrence function (monotonicity, continuity, etc.), on criteria of comparison with other sequences and on the resolution, in a stochastic. .. S(t) = Yn Tn ≤t where (Tn , n ≥ 1) is the sequence of arrival times distributed according to a Poisson process and (Yn , n ≥ 1) is a sequence of independent random variables that are identically distributed www.it-ebooks.info 18 Networks Modeling and Analysis 1.4 Notes and comments For more details on the design of telephone networks, their history and future developments, we may refer to [RIG 98] The... arrival of a customer in an empty system and ends on the next arrival of a customer in an empty system This is the concatenation of a busy period and an idle period, that is the time elapsed between the departure of the last customer of the busy period and the arrival of the next customer Note.– In Figure 1.8, a busy period begins at T1 and ends at D4 The corresponding cycle begins at T1 and ends... that L increases only in the set of zeros of W Let T0 be a point of − increase of L, then for any non-negative h, there exists th such that Lt0 −h ≤ Xth ≤ Lt0 www.it-ebooks.info 14 Networks Modeling and Analysis − When h tends toward 0, we have by left-continuity, Lt0 = Xt0 = −Xt0 , therefore Xt0 + Lt0 = Wt0 = 0 The proof is thus complete t Figure 1.9 An example of a reflected process The dark color... and is fed by some continuous data flow We can then obtain qualitative results on models whose study supports no other approaches On the one hand, the method does not require precise knowledge about the rate of the input process, and on the other hand, it is particularly well adapted to the study of extreme cases: low and high loads, superposition of heterogeneous traffic We work in continuous time and. .. times, independent and identically distributed inter-arrivals www.it-ebooks.info 10 Networks Modeling and Analysis 1.3 Lindley and Beneˇ s 1.3.1 Discrete model We often consider the number of customers present in the system but the quantity that contains the most information is the system load, defined at each moment as the time required for the system to empty itself in the absence of new arrivals The... the absence of information on the number of resources or the service discipline, it is understood that the number of resources is infinite and that the service discipline is the FIFO discipline Example.– The M/M/1 queue is the queuing system where the inter-arrivals and the service times are independent of exponential distribution and there is one server The waiting room is of infinite size and the service... queues, and to a lesser extent, the M/GI/1 and GI/M/1 queues For the others (other distributions of interarrivals or service times, other disciplines), we often have only partial or asymptotic results www.it-ebooks.info Part 1 Discrete-time Modeling www.it-ebooks.info Chapter 2 Stochastic Recursive Sequences The modeling of discrete-time deterministic dynamical systems is based on recursive sequences of. .. case of a FIFO discipline, therefore to represent the change in the number of customers in the system They are also used to qualitatively analyze the stability of the system in very general cases This will be dealt with in Chapter 4 www.it-ebooks.info 12 Networks Modeling and Analysis 1.3.2 Fluid model A fluid model consists of replacing a queue which is a discrete-time event system by a reservoir of. .. obvious that the number of calls between 0 and T is about λT and then the occupation rate of such a line is given by 1 (λT × 1/µ) = λ/µ T Of course, in reality, neither the inter-arrivals, nor the holding times are deterministic Let us imagine a situation in which the holding times and the idle times are independent of each other with a distribution that is common to all busy periods and idle periods, respectively . Stochastic Modeling and Analysis of Telecom Networks www.it-ebooks.info Stochastic Modeling and Analysis of Telecom Networks Laurent Decreusefond Pascal Moyal Series. observation Stochastic Modeling and Analysis of Telecom Networks Laurent Decreusefond and Pascal Moyal © 2012 ISTE Ltd. Published 2012 by ISTE Ltd. www.it-ebooks.info 2 Networks Modeling and Analysis Looking. noted that the decryption of standards of thousands of pages and their translation into human language require much work and fine knowledge in a wide variety of disciplines, and as well as infinite