On the importance function in splitting simulation

Abstract

Abstract.The splitting method is a simulation technique for the estimation of very small probabilities. In this technique, the sample paths are split into multiple copies, at various stages in the simulation. Of vital importance to the efficiency of the method is the Importance Function (IF). This function governs the placement of the thresholds or surfaces at which the paths are split. We derive a characterisation of the optimal IF and show that for multi-dimensional models the "natural" choice for the IF is usually not optimal. We also show how nearly optimal splitting surfaces can be derived or simulated using reverse time analysis. Our numerical experiments illustrate that by using the optimal IF, one can obtain a significant improvement in simulation efficiency.

Importance functions for restart simulation of general Jackson networks

“…However, worse results were obtained for valúes of aand b greater than those given by the formulas. Garvels et al (2002) estimated the importance function using the time-reversal method with the restriction of assuming finite capacity of the first two queues, but they could not estimate accurately overflow probabilities lower than 10~1 5 . The method cannot be generalized to bigger networks because, as they mentioned, the bigger state space results in a greater variance of the estimates of the importance function.…”

“…The problem of finding the optimal importance function has been compared by Garvels et al (2002) with the problem of finding a good change of measure in importance sampling, because in both cases "knowledge about the behaviour of the system leading to the rare event is necessary". Glasserman et al (1999) stated that "splitting ultimately relies on a detailed understanding of a process's rare event asymptotic, much as importance sampling does".…”

“…This factor was analysed in Villén-Altamirano and Villén-Altamirano (2006) and guidelines for selecting heuristically such a function were provided. In Garvels et al (2002), an algorithm based on reverse-time simulation was used to obtain an importance function for the two-queue Jackson tándem network, but such algorithm is difficult to implement in models with bigger state space. In Villén-Altamirano (2007) formulas for obtaining effective importance functions were provided for two-stage Markovian networks with any number of nodes in each stage.…”

“…It will be tested by simulation that the formulas are also valid for general Jackson networks without any type of restriction. Even for a network as simple as the three-queue tándem network, it is not possible to calcúlate the optimal importance function, see Garvels et al (2002). Therefore, some approximations, one of them a rough approximation, are needed to derive these formulas.…”

A Randomized Quasi-Monte Carlo Simulation Method for Markov Chains

“…When sorting the states, h-equivalent states can be placed in arbitrary order. A similar type of function h is used in the splitting methodology for rare-event simulation, where it is called the importance function (Garvels et al, 2002;Glasserman et al, 1999). We use a different name, to avoid possible confusion in case the two methods are combined.…”