On the estimation of the mean time to failure by simulation

The mean time to failure (MTTF) of a stochastic system is often estimated by simulation. One natural estimator, which we call the direct estimator, simply averages independent and identically distributed copies of simulated times to failure. When the system is regenerative, an alternative approach is based on a ratio representation of the MTTF. The purpose of this paper is to compare the two estimators. We first analyze them in the setting of crude simulation (i.e., no importance sampling), showing that they are actually asymptotically identical in a rare-event context. The two crude estimators are inefficient in different but closely related ways: the direct estimator requires a large computational time because times to failure often include many transitions, whereas the ratio estimator entails estimating a rare-event probability. We then discuss the two approaches when employing importance sampling; for highly reliable Markovian systems, we show that using a ratio estimator is advised.