Reliability analysis is a field of uncertainty quantification that focuses on calculating the failure probability of a system. In this context, the limit-state function is crucial, as it predicts system failure. Traditionally, this function is assumed to yield the same output when repeatedly evaluated on a given set of input parameters. Recently, non-deterministic limit-state functions have attracted increasing interest in the field. These yield random responses for each set of input parameters. This behavior introduces additional complexity to reliability analysis, as both safe and failed states may be observed for the same input parameters. Our research explores how reliability analysis differs when applied to stochastic simulators. More specifically, we aim to expand the definition of failure probability to cope with these models. In this contribution, we propose simulation methods for dealing with these cases. These, however, lead to a great computational burden. Therefore, we also propose surrogate-based methods to efficiently compute the failure probability.