Rare Event Simulation usingMonte Carlo Methods

Edited by

Gerardo Rubino

And

Bruno Tuffin

INRIA, Rennes, France

A John Wiley and Sons, Ltd., Publication

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Rare Event Simulation usingMonte Carlo Methods

Rare Event Simulation usingMonte Carlo Methods

Edited by

Gerardo Rubino

And

Bruno Tuffin

INRIA, Rennes, France

A John Wiley and Sons, Ltd., Publication

This edition first published 2009 2009 John Wiley & Sons Ltd.

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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 978-0-470-77269-0 (cloth)1. Limit theorems (Probability theory) 2. Monte Carlo method. 3. System analysis – Data

processing. 4. Digital computer simulation. I. Rubino, Gerardo, 1955- II. Tuffin, Bruno.QA273.67.R37 2009519.2 – dc22

2009004186

A catalogue record for this book is available from the British Library

ISBN: 978-0-470-77269-0

Typeset in 10/12pt Times by Laserwords Private Limited, Chennai, IndiaPrinted in the UK by TJ International, Padstow, Cornwall.

www.wiley.com

Contents

Contributors vii

Preface ix

1 Introduction to Rare Event Simulation 1Gerardo Rubino and Bruno Tuffin

PART I THEORY 15

2 Importance Sampling in Rare Event Simulation 17Pierre L’Ecuyer, Michel Mandjes and Bruno Tuffin

3 Splitting Techniques 39Pierre L’Ecuyer, François Le Gland, Pascal Lezaudand Bruno Tuffin

4 Robustness Properties and Confidence Interval ReliabilityIssues 63Peter W. Glynn, Gerardo Rubino and Bruno Tuffin

PART II APPLICATIONS 85

5 Rare Event Simulation for Queues 87José Blanchet and Michel Mandjes

6 Markovian Models for Dependability Analysis 125Gerardo Rubino and Bruno Tuffin

7 Rare Event Analysis by Monte Carlo Techniques in Static Models 145Héctor Cancela, Mohamed El Khadiri and Gerardo Rubino

8 Rare Event Simulation and Counting Problems 171José Blanchet and Daniel Rudoy

vi CONTENTS

9 Rare Event Estimation for a Large-Scale Stochastic HybridSystem with Air Traffic Application 193Henk A. P. Blom, G. J. (Bert) Bakker and Jaroslav Krystul

10 Particle Transport Applications 215Thomas Booth

11 Rare Event Simulation Methodologies in Systems Biology 243Werner Sandmann

Index 267

Contributors

We list the contributors to this volume in alphabetical order, with their respectiveelectronic addresses.

José Blanchet, University of Columbia, New York, USAjose.blanchet@columbia.edu

Henk Blom, NLR, The Netherlandsblom@nlr.nl

Thomas Booth, Los Alamos National Laboratory, USAteb@lanl.gov

Héctor Cancela, University of the Republic, Montevideo, Uruguaycancela@fing.edu.uy

Mohamed El Khadiri, University of Nantes/IUT of Saint-Nazaire, Francemohamed.elkhadiri@iutsn.univ-nantes.fr

Peter Glynn, University of Stanford, USAglynn@stanford.edu

Pierre L’Ecuyer, University of Montréal, Canadalecuyer@iro.umontreal.ca

François Legland, INRIA, Francelegland@irisa.fr

Pascal Lezaud, DGAC, Francelezaud@cena.fr

Michel Mandjes, CWI, The Netherlandsmmandjes@science.uva.nl

Gerardo Rubino, INRIA, Francegerardo.rubino@inria.fr

viii CONTRIBUTORS

Daniel Rudoy, Harvard University, Cambridge, USArudoy@seas.harvard.edu

Werner Sandmann, Bamberg Universitywerner.sandmann@wiai.uni-bamberg.de

Bruno Tuffin, INRIA, Francebruno.tuffin@inria.fr

Preface

Rare event simulation has attracted a great deal of attention since the first devel-opment of Monte Carlo techniques on computers, at Los Alamos during theproduction of the first nuclear bomb. It has found numerous applications in fieldssuch as physics, biology, telecommunications, transporting systems, and insur-ance 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. Atypical illustration is the area of telecommunications, where, with the advent ofthe Internet, light-tailed processes traditionally used in queuing networks nowhave to be replaced by heavy-tailed ones, and new developments of rare eventsimulation theory are required.

Surprisingly, we found that not much was written on the subject, in factonly one book was devoted to it, with a special focus on large-deviations the-ory. The idea of writing this book therefore started from a collaborative projectmanaged in France by Institut National de Recherche en Informatique et Automa-tique (INRIA) in 2005–2006 (see http://www.irisa.fr/armor/Rare/), with groupsof researchers from INRIA, the University of Nice, the CWI in the Netherlands,Bamberg University in Germany, and the University of Montréal in Canada. Inorder to cover the broad range of applications in greater depth, we decided torequest contributions from authors who were not members of the project. Thisbook is the result of that effort.

As editors, we would like to thank the contributors for their effort in writingthe chapters of this book. We are also grateful to John Wiley & Sons staffmembers, in particular Susan Barclay and Heather Kay, for their assistance andpatience.

Gerardo Rubino and Bruno Tuffin

1

Introduction to rare eventsimulationGerardo Rubino and Bruno Tuffin

This monograph deals with the analysis by simulation of ‘rare situations’ in sys-tems of quite different types, that is, situations that happen very infrequently, butimportant enough to justify their study. A rare event is an event occurring witha very small probability, the definition of ‘small’ depending on the applicationdomain. These events are of interest in many areas. Typical examples come, forinstance, from transportation systems, where catastrophic failures must be rareenough. For instance, a representative specification for civil aircraft is that theprobability 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 inthe dependability area because of the existence of these types of failures, thatis, 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, muchhigher. Security systems in nuclear plants are also examples of critical systems.Nowadays we also call critical other systems where catastrophic failures maylead to significant loss of money rather than human lives (banking informationsystems, for example). In telecommunications, modern networks often offer veryhigh speed links. Since information travels in small units or messages (packets inthe Internet world, cells in asynchronous transfer mode infrastructures, etc.), thesaturation of the memory of a node in the network, even during a small amountof time, may induce a huge amount of losses (in most cases, any unit arriving at

Rare Event Simulation using Monte Carlo Methods Edited by G. Rubino and B. Tuffin 2009 John Wiley & Sons, Ltd

2 INTRODUCTION

a saturated node is lost). For this reason, the designer wants the overflow of sucha buffer to be a rare event, with probabilities of the order of 10−9. Equivalently,the probability of ruin is a central issue for the overall wealth of an insurancecompany: the (time) evolution of the reserves of the company is representedby a stochastic process, with initial value R0; the reserves may decrease due toincoming claims, but also have a linear positive drift thanks to the premiums paidby customers. A critical issue is to estimate the probability of ruin, that is, theprobability of the reserve process reaching zero. In biological systems, molec-ular reactions may occur on different time scales, and reactions with extremelysmall occurrence rates are therefore rare events. As a consequence, this stiffnessrequires the introduction of specific techniques to solve the embedded differentialequations.

This chapter introduces the general area of rare event simulation, recalls thebasic background elements necessary to understand the technical content of thefollowing 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 complexityand/or assumptions on the model. In that case, approximation techniques cansometimes be used. On the other hand, standard numerical analysis proceduresalso require assumptions (even if less stringent) on the model, and suffer frominefficiency 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, thespeed of convergence to the exact value is usually O(n−2/s), therefore slowwhen s is large. The number of points necessary to reach a given precisionincreases exponentially with the dimension. To cope with those problems, wecan use Monte Carlo simulation techniques, which are statistical approximationtechniques, instead of the above mentioned deterministic ones.

Let us start with the basic concepts behind Monte Carlo techniques. Supposethat the probability γ of some event A is to be estimated. A model of the systemis simulated n times (we say that we build an n-sample of the model) and at eachrealization we record whether A happens or not. In the simplest (and most usual)case, the n samples are independent (stochastically speaking) of each other. IfXi is the (Bernoulli) random variable Xi = (A occurs in the nth sample) (i.e.,Xi = 1 if A occurs in sample i, 0 if not), we estimate γ by γ̂ = (X1 + · · · +Xn)/n. Observe that E(Xi) = γ and Var(Xi) = γ (1 − γ ) which we denote by σ 2(for basic results on probability theory and to verify, for instance, typical resultson Bernoulli random variables, the reader can consult textbooks such as [6]).

How far will the given estimator γ̂ be from the actual value γ ? To answer thisquestion, we can apply the central limit theorem, which says that X1 + · · · +Xn

INTRODUCTION 3

is approximately normal (if n is ‘large enough’) [6]. Let us scale things first:E(X1 + · · · +Xn) = nγ and Var(X1 + · · · +Xn) = nσ 2, so the random variableZ = [nσ 2]−1/2(X1 + · · · +Xn − nγ ) has mean 0 and variance 1. The centrallimit theorem says that as n→ ∞, the distribution of Z tends to the standardnormal 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

((γ̂ − zσ√

n, γ̂ + zσ√

n

)� γ)≈ 2�(z)− 1.

The random interval I = (γ̂ ∓ zσn−1/2) is called a confidence interval for γ ,with level 2�(z)− 1. We typically consider, for instance, z = 1.96 because2�(1.96)− 1 = 0.95 (or z = 2.56, for which 2�(2.56)− 1 = 0.99). The preced-ing observations lead to P(γ ∈ (γ̂ ∓ 1.96σn−1/2)) ≈ 0.95. In general, for a con-fidence interval with level α, 0 < α < 1, we take (γ̂ ∓�−1((1 + α)/2)σn−1/2).

From the practical point of view, we build our n-sample (i.e., we perform ourn system simulations), we estimate γ by γ̂ and, since σ 2 is unknown, we estimateit using σ̂ 2 = nγ̂ (1 − γ̂ )/(n− 1). The reason for dividing by n− 1 and not byn is to have an unbiased estimator (which means E(̂σ 2) = σ 2, as E(γ̂ ) = γ ),although from the practical point of view this is not relevant, since n will beusually large enough. Finally, the result of our estimation work will take theform I = (γ̂ ∓ 1.96σ̂ n−1/2), which says that ‘γ is, with high probability (ourconfidence level ), inside this interval’–and by ‘high’ we mean 0.95.

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 thereforefaster 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 smallnumbers, the absolute error (given by the size of the confidence interval, or byhalf this size) is not of sufficient interest: the accuracy of the simulation processis captured by the relative error instead, that is, the absolute error divided bythe actual value: RE = zn−1/2σ/γ . This leads immediately to the main problemwith rare events, because if γ 1, then

RE = z√γ (1 − γ )√nγ

≈ z√n√γ

1

(unless n is ‘huge’ enough). To illustrate this, let us assume that we want a rela-tive error less than 10%, and that γ = 10−9. The constraint RE ≤ 0.1 translatesinto n ≥ 3.84 × 1011. In words, this means that we need a few hundred billionexperiments to get a modest 10% relative error in the answer. If the system beingsimulated is complex enough, this will be impossible, and something differentmust be done in order to provide the required estimation. More formally, if we