On terminating Poisson processes in some shock models

A system subject to a point process of shocks is considered. Shocks occur in accordance with the homogeneous Poisson process. Different criteria of system failure (termination) are discussed and the corresponding probabilities of failure (accident)-free performance are derived. The described analytical approach is based on deriving integral equations for each setting and solving these equations through the Laplace transform. Some approximations are analyzed and further generalizations and applications are discussed.