Consider a system subject to periods of deterioration. The system might fail at any time within the set of deterioration time, and the probability of failure is a function of the accumulated damage caused from past deterioration. When the system fails, it is immediately replaced and a failure cost is incurred; if replacement is made before failure, a lesser cost is incurred and that cost may depend upon the amount of accumulated damage at the replacement time. The purpose of this paper is to derive the optimal replacement policy for such a system whose set of deterioration times contains no isolated points and whose cumulative damage process is a semi-Markov process. Only those policies which make a replacement within the set of deterioration times are considered. Optimality is based on a discounted cost criterion.
"Optimal Replacement for Systems Governed by Markov Additive Shock Processes." Ann. Probab. 5 (3) 413 - 429, June, 1977. https://doi.org/10.1214/aop/1176995802