System unavailability calculations in biased Monte Carlo simulation: a possible pitfall

The present paper addresses the problem of system unavailability calculations via Monte Carlo simulation. The standard analog procedure amounts to collecting unitary weights in the time channels during which the system is in the failed state. In the case of highly reliable systems, the biased simulation is almost mandatory and it might seem obvious that the natural extension of the procedure would simply require that nonunitary weights, properly devised to account for the biasing, be collected instead of the unitary ones. In this work we show that this procedure leads to erroneous results and suggest tackling the problem by adopting another Monte Carlo simulation procedure. The process originated by the stochastic transitions of the system components may be seen as a random walk within the discrete phase space of the possible system hardware configurations. From the simulation of this random walk, ensemble averages can be computed to estimate the quantities of interest. In particular, the system instantaneous unavailability, at a given time, is a functional, whose estimate can be obtained through the ensemble average of the values of a function representing the probability of the system sojourning in a failed state up to the time instant considered. We show that this way of proceeding remains valid also for biased simulations.