On derivative estimation of the mean time to failure in simulations of highly reliable Markovian systems

The mean time to failure (MTTF) of a Markovian system can be expressed as a ratio of two expectations. For highly reliable Markovian systems, the resulting ratio formula consists of one expectation that cannot be estimated with bounded relative error when using standard simulation, while the other, which we call a nonrare expectation, can be estimated with bounded relative error. We show that some derivatives of the nonrare expectation cannot be estimated with bounded relative error when using standard simulation, which in turn may lead to an estimator of the derivative of the MTTF that has unbounded relative error. However, if particular importance-sampling methods (e.g., balanced failure biasing) are used, then the estimator of the derivative of the nonrare expectation will have bounded relative error, which (under certain conditions) will yield an estimator of the derivative of the MTTF with bounded relative error.