Simulation of steady-state availability models of fault-tolerant systems with deferred repair

This paper targets the simulation of continuous-time Markov chain models of fault-tolerant systems with deferred repair. We start by stating sufficient conditions for a given importance sampling scheme to satisfy the bounded relative error property. Using those sufficient conditions, it is noted that many previously proposed importance sampling schemes such as failure biasing and balanced failure biasing satisfy that property. Then, we adapt the importance sampling schemes failure transition distance biasing and balanced failure transition distance biasing so as to develop new importance sampling schemes which can be implemented with moderate effort and at the same time can be proved to be more efficient for balanced systems than the simpler failure biasing and balanced failure biasing schemes. The increased efficiency for balanced and unbalanced systems of the new adapted importance sampling schemes is illustrated using examples.Preprin

A Characterization of the Simple Failure-Biasing Method for Simulations of Highly Reliable Markovian Systems

Simple fadure biasing is an importance-sampling technique used to reduce the variance of estimates of performance measures and them gradients m simulations of highly reliable Marko-vian systems. Although simple fadure biasing yields bounded relative error for the performance measure estimate when the system is balanced, it may not provide bounded relative error when the system is unbalanced. In this article, we provide a characterization of when the simple fadure-biasing method produces estimators of a performance measure and lts derivatives with bounded relative error. We derive a necessary and sufficient condition on the structure of the system for when the performance measure can be estimated with bounded relative error when using simple failure biasing. Furthermore, a similar condition for the derivative estimators is established. One interesting aspect of the conditions is that it shows that to obtain bounded relative error, not only the most hkely paths to system failure must be examined but also some secondary paths leading to failure as well, We also show by example that the necessary and sufficient conditions for a derivative estimator do not imply those for the performance measure estimator; I.e., it is possible to estimate a derivative more efficiently than the performance measure when using simple failure biasing