Optimum replacement of a system subject to shocks

A system is subject to shocks. Each shock weakens the system and makes it more expensive to run. It is desirable to determine a replacement time for the system. Boland and Proschan 4 consider periodic replacement of the system and give sufficient conditions for the existence of an optimal finite period, assuming that the shock process is a non-homogeneous Poisson process and the cost structure does not depend on time. Block et al. 3 establish similar results assuming that cost structure is time dependent, still requiring that the shock process is a non-homogeneous Poisson process. We show via a sample path argument that the results of 3 and 4 hold for any counting process whose jump size is of one unit magnitude.