Stochastic modeling for systems with delayed failures

We present stochastic description for repairable systems with delayed failures. Initial defects in a system result in its failure only after some delay. Specifically, it happens in repairable systems when the defects are not repaired within some period of time that can be either deterministic or random. Therefore, operation of a system is described by the corresponding terminating renewal process. We derive and analyze relationships for the survival probability and the mean time to failure for these systems. The results are derived in the form of the corresponding Laplace transforms that are numerically inverted for illustrative examples. Simple fast repair approximations are considered and their accuracy is also discussed.