Importance Sampling for the Simulation of Highly Reliable Markovian Systems

In this paper we investigate importance sampling techniques for the simulation of Markovian systems with highly reliable components. The need for simulation arises because the state space of such systems is typically huge, making numerical computation inefficient. Naive simulation is inefficient due to the rarity of the system failure events. Failure biasing is a useful importance sampling technique for the simulation of such systems. However, until now, this technique has been largely heuristic. We present a mathematical framework for the study of failure biasing. Using this framework we derive variance reduction results which explain the orders of magnitude of variance reduction obtained in practice. We show that in many cases the magnitude of the variance reduction is such that the relative errors of the estimates remain bounded as the failure rates of components tend to zero. We also prove that the failure biasing heuristic in its original form may not give bounded relative error for a large class of systems and that a modification of the heuristic works for the general case. The theoretical results in this paper agree with experiments on the subject which have been reported in a previous paper.