A general replacement of a system subject to shocks

A system is subject to shocks that arrive according to a nonhomogeneous Poisson process. The system is replaced at age T at a fixed cost c₀. If the k-th shock arrives at time S_k<T, it is either a fatal shock with probability p(S_k) or a nonfatal shock with probability 1−p(S_k). The fatal shock causes the system total breakdown, and the system is replacd at a cost c_∞. The nonfatal shock weakens the system and makes it more expensive to run. The aim of the paper is to find the optimal T which minimizes the long run expected cost per unit time of the policy. Various special cases are considered.