Rare Event Simulation using Monte Carlo Methods

Transcription

1 Rare Event Simulation using Monte Carlo Methods Edited by Gerardo Rubino And Bruno Tuffin INRIA, Rennes, France A John Wiley and Sons, Ltd., Publication

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3 Rare Event Simulation using Monte Carlo Methods

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5 Rare Event Simulation using Monte Carlo Methods Edited by Gerardo Rubino And Bruno Tuffin INRIA, Rennes, France A John Wiley and Sons, Ltd., Publication

6 This edition first published John Wiley & Sons Ltd. Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 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 or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Library of Congress Cataloging-in-Publication Data Rare event simulation using Monte Carlo methods/edited by Gerardo Rubino, Bruno Tuffin. p. cm. Includes bibliographical references and index. ISBN (cloth) 1. Limit theorems (Probability theory) 2. Monte Carlo method. 3. System analysis Data processing. 4. Digital computer simulation. I. Rubino, Gerardo, II. Tuffin, Bruno. QA R dc A catalogue record for this book is available from the British Library ISBN: Typeset in 10/12pt Times by Laserwords Private Limited, Chennai, India Printed in the UK by TJ International, Padstow, Cornwall.

7 Contents Contributors Preface vii ix 1 Introduction to Rare Event Simulation 1 Gerardo Rubino and Bruno Tuffin PART I THEORY 15 2 Importance Sampling in Rare Event Simulation 17 Pierre L Ecuyer, Michel Mandjes and Bruno Tuffin 3 Splitting Techniques 39 Pierre L Ecuyer, François Le Gland, Pascal Lezaud and Bruno Tuffin 4 Robustness Properties and Confidence Interval Reliability Issues 63 Peter W. Glynn, Gerardo Rubino and Bruno Tuffin PART II APPLICATIONS 85 5 Rare Event Simulation for Queues 87 José Blanchet and Michel Mandjes 6 Markovian Models for Dependability Analysis 125 Gerardo Rubino and Bruno Tuffin 7 Rare Event Analysis by Monte Carlo Techniques in Static Models 145 Héctor Cancela, Mohamed El Khadiri and Gerardo Rubino 8 Rare Event Simulation and Counting Problems 171 José Blanchet and Daniel Rudoy

8 vi CONTENTS 9 Rare Event Estimation for a Large-Scale Stochastic Hybrid System with Air Traffic Application 193 Henk A. P. Blom, G. J. (Bert) Bakker and Jaroslav Krystul 10 Particle Transport Applications 215 Thomas Booth 11 Rare Event Simulation Methodologies in Systems Biology 243 Werner Sandmann Index 267

9 Contributors We list the contributors to this volume in alphabetical order, with their respective electronic addresses. José Blanchet, University of Columbia, New York, USA Henk Blom, NLR, The Netherlands Thomas Booth, Los Alamos National Laboratory, USA Héctor Cancela, University of the Republic, Montevideo, Uruguay Mohamed El Khadiri, University of Nantes/IUT of Saint-Nazaire, France Peter Glynn, University of Stanford, USA Pierre L Ecuyer, University of Montréal, Canada lecuyer@iro.umontreal.ca François Legland, INRIA, France legland@irisa.fr Pascal Lezaud, DGAC, France lezaud@cena.fr Michel Mandjes, CWI, The Netherlands mmandjes@science.uva.nl Gerardo Rubino, INRIA,France gerardo.rubino@inria.fr

10 viii CONTRIBUTORS Daniel Rudoy, Harvard University, Cambridge, USA Werner Sandmann, Bamberg University Bruno Tuffin, INRIA, France

11 Preface Rare event simulation has attracted a great deal of attention since the first development of Monte Carlo techniques on computers, at Los Alamos during the production of the first nuclear bomb. It has found numerous applications in fields such as physics, biology, telecommunications, transporting systems, and insurance risk analysis. Despite the amount of work on the topic in the last sixty years, there are still domains needing to be explored because of new applications. A typical illustration is the area of telecommunications, where, with the advent of the Internet, light-tailed processes traditionally used in queuing networks now have to be replaced by heavy-tailed ones, and new developments of rare event simulation theory are required. Surprisingly, we found that not much was written on the subject, in fact only one book was devoted to it, with a special focus on large-deviations theory. The idea of writing this book therefore started from a collaborative project managed in France by Institut National de Recherche en Informatique et Automatique (INRIA) in (see with groups of researchers from INRIA, the University of Nice, the CWI in the Netherlands, Bamberg University in Germany, and the University of Montréal in Canada. In order to cover the broad range of applications in greater depth, we decided to request contributions from authors who were not members of the project. This book is the result of that effort. As editors, we would like to thank the contributors for their effort in writing the chapters of this book. We are also grateful to John Wiley & Sons staff members, in particular Susan Barclay and Heather Kay, for their assistance and patience. Gerardo Rubino and Bruno Tuffin

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13 1 Introduction to rare event simulation Gerardo Rubino and Bruno Tuffin This monograph deals with the analysis by simulation of rare situations in systems of quite different types, that is, situations that happen very infrequently, but important enough to justify their study. A rare event is an event occurring with a very small probability, the definition of small depending on the application domain. These events are of interest in many areas. Typical examples come, for instance, from transportation systems, where catastrophic failures must be rare enough. For instance, a representative specification for civil aircraft is that the probability of failure must be less than, say, 10 9 during an average-length flight (a flight of about 8 hours). Transportation systems are called critical in the dependability area because of the existence of these types of failures, that is, failures that can lead to loss of human life if they occur. Aircraft, trains, subways, all these systems belong to this class. The case of cars is less clear, mainly because the probability of a catastrophic failure is, in many contexts, much higher. Security systems in nuclear plants are also examples of critical systems. Nowadays we also call critical other systems where catastrophic failures may lead to significant loss of money rather than human lives (banking information systems, for example). In telecommunications, modern networks often offer very high speed links. Since information travels in small units or messages (packets in the Internet world, cells in asynchronous transfer mode infrastructures, etc.), the saturation of the memory of a node in the network, even during a small amount of time, may induce a huge amount of losses (in most cases, any unit arriving at Rare Event Simulation using Monte Carlo Methods 2009 John Wiley & Sons, Ltd Edited by G. Rubino and B. Tuffin

14 2 INTRODUCTION a saturated node is lost). For this reason, the designer wants the overflow of such a buffer to be a rare event, with probabilities of the order of Equivalently, the probability of ruin is a central issue for the overall wealth of an insurance company: the (time) evolution of the reserves of the company is represented by a stochastic process, with initial value R 0 ; the reserves may decrease due to incoming claims, but also have a linear positive drift thanks to the premiums paid by customers. A critical issue is to estimate the probability of ruin, that is, the probability of the reserve process reaching zero. In biological systems, molecular reactions may occur on different time scales, and reactions with extremely small occurrence rates are therefore rare events. As a consequence, this stiffness requires the introduction of specific techniques to solve the embedded differential equations. This chapter introduces the general area of rare event simulation, recalls the basic background elements necessary to understand the technical content of the following chapters, and gives an overview of the contents of those chapters. 1.1 Basics in Monte Carlo Solving scientific problems often requires the computation of sums or integrals, or the solution of equations. Direct computations, also called analytic techniques, become quickly useless due to their stringent requirements in terms of complexity and/or assumptions on the model. In that case, approximation techniques can sometimes be used. On the other hand, standard numerical analysis procedures also require assumptions (even if less stringent) on the model, and suffer from inefficiency as soon as the mathematical dimension of the problem increases. A typical illustration is when using quadrature rules for numerical integration. Considering, for instance, the trapezoidal rule with n points in dimension s, the speed of convergence to the exact value is usually O(n 2/s ), therefore slow when s is large. The number of points necessary to reach a given precision increases exponentially with the dimension. To cope with those problems, we can use Monte Carlo simulation techniques, which are statistical approximation techniques, instead of the above mentioned deterministic ones. Let us start with the basic concepts behind Monte Carlo techniques. Suppose that the probability γ of some event A is to be estimated. A model of the system is simulated n times (we say that we build an n-sample of the model) and at each realization we record whether A happens or not. In the simplest (and most usual) case, the n samples are independent (stochastically speaking) of each other. If X i is the (Bernoulli) random variable X i = (A occurs in the nth sample) (i.e., X i = 1ifA occurs in sample i, 0 if not), we estimate γ by γ = (X X n )/n. Observe that E(X i ) = γ and Var(X i ) = γ(1 γ)which we denote by σ 2 (for basic results on probability theory and to verify, for instance, typical results on Bernoulli random variables, the reader can consult textbooks such as [6]). How far will the given estimator γ be from the actual value γ? To answer this question, we can apply the central limit theorem, which says that X 1 + +X n

15 INTRODUCTION 3 is approximately normal (if n is large enough ) [6]. Let us scale things first: E(X 1 + +X n ) = nγ and Var(X 1 + +X n ) = nσ 2, so the random variable Z = [nσ 2 ] 1/2 (X 1 + +X n nγ ) has mean 0 and variance 1. The central limit theorem says that as n, the distribution of Z tends to the standard normal distribution N (0, 1), whose cdf is (x) = (2π) 1/2 x exp( u2 /2)du. We then assume that n is large enough so that Z N (0, 1) in distribution. This means, for instance, that P( z Z z) = 2 (z) 1(using ( z) = 1 (z)), which is equivalent to writing (( P γ zσ, γ + zσ ) ) γ 2 (z) 1. n n The random interval I = ( γ zσn 1/2 ) is called a confidence interval for γ, with level 2 (z) 1. We typically consider, for instance, z = 1.96 because 2 (1.96) 1 = 0.95 (or z = 2.56, for which 2 (2.56) 1 = 0.99). The preceding observations lead to P(γ ( γ 1.96σn 1/2 )) In general, for a confidence interval with level α, 0<α<1, we take ( γ 1 ((1 + α)/2)σ n 1/2 ). From the practical point of view, we build our n-sample (i.e., we perform our n system simulations), we estimate γ by γ and, since σ 2 is unknown, we estimate it using σ 2 = n γ(1 γ )/(n 1). The reason for dividing by n 1 and not by n is to have an unbiased estimator (which means E( σ 2 ) = σ 2,asE( γ)= γ ), although from the practical point of view this is not relevant, since n will be usually large enough. Finally, the result of our estimation work will take the form I = ( γ 1.96 σn 1/2 ), which says that γ is, with high probability (our confidence level), inside this interval and by high we mean The speed of convergence is measured by the size of the confidence interval, that is, 2zσn 1/2. This decreases as the inverse square root of the sample size, independently of the mathematical dimension s of the problem, and therefore faster than standard numerical techniques, even for small values of s. Now, suppose that A is a rare event, that is to say, that γ 1. For very small numbers, the absolute error (given by the size of the confidence interval, or by half this size) is not of sufficient interest: the accuracy of the simulation process is captured by the relative error instead, that is, the absolute error divided by the actual value: RE = zn 1/2 σ/γ. This leads immediately to the main problem with rare events, because if γ 1, then γ(1 γ) RE = z nγ z n γ 1 (unless n is huge enough). To illustrate this, let us assume that we want a relative error less than 10%, and that γ = The constraint RE 0.1 translates into n In words, this means that we need a few hundred billion experiments to get a modest 10% relative error in the answer. If the system being simulated is complex enough, this will be impossible, and something different must be done in order to provide the required estimation. More formally, if we