Importance Sampling for Failure Probabilities in Computing and Data Transmission

Abstract

In this paper we study efficient simulation algorithms for estimating P(X›x), whereXis 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 ofTgivenX›x. IfT≡tis 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. IfTis random, particular attention is given toThaving 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 forTusing these asymptotic descriptions have bounded relative error asx→∞ when combined with the ideas used for a fixedt. 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.