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Importance Sampling for Failure Probabilities in Computing and Data Transmission

Part of: Limit theorems Probabilistic methods, simulation and stochastic differential equations Computer system organization

Published online by Cambridge University Press:  14 July 2016

Søren Asmussen
Søren Asmussen*
Affiliation:
University of Aarhus
*
∗Postal address: Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark. Email address: asmus@imf.au.dk
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Abstract

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In this paper we study efficient simulation algorithms for estimating P(X›x), where X is the total time of a job with ideal time T that needs to be restarted after a failure. The main tool is importance sampling, where a good importance distribution is identified via an asymptotic description of the conditional distribution of T given X›x. If T≡t is constant, the problem reduces to the efficient simulation of geometric sums, and a standard algorithm involving a Cramér-type root, γ(t), is available. However, we also discuss an algorithm that avoids finding the root. If T is random, particular attention is given to T having either a gamma-like tail or a regularly varying tail, and to failures at Poisson times. Different types of conditional limit occur, in particular exponentially tilted Gumbel distributions and Pareto distributions. The algorithms based upon importance distributions for T using these asymptotic descriptions have bounded relative error as x→∞ when combined with the ideas used for a fixed t. Nevertheless, we give examples of algorithms carefully designed to enjoy bounded relative error that may provide little or no asymptotic improvement over crude Monte Carlo simulation when the computational effort is taken into account. To resolve this problem, an alternative algorithm using two-sided Lundberg bounds is suggested.

Keywords

Communications engineeringcompound sumcomputer reliabilityconditioned limit theoremCramér rootexponential tiltinggeometric sumGumbel distributionintegral asymptoticsLundberg's inequalityrare event simulationregular variationRESTART

MSC classification

Secondary: 65C05: Monte Carlo methods 68M15: Reliability, testing and fault tolerance 60F05: Central limit and other weak theorems
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 3 , September 2009 , pp. 768 - 790
Copyright
Copyright © Applied Probability Trust 2009 

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