Consider a system that is subject to a sequence of randomly occurring shocks; each shock causes some damage of random magnitude to the system. Any of the shocks might cause the system to fail, and the probability of such a failure is a function of the sum of the magnitudes of damage caused by all previous shocks. When the system fails, it must be immediately replaced and a failure cost is incurred. If the system is replaced before failure, a smaller replacement cost is incurred, and that cost may depend upon the state of the system at replacement time.
The purpose of this paper is to determine the optimal replacement policy among the class of policies that replace at shock times. The main assumption is that the cumulative damage process is a semi-Markov process. The cost criterion is to minimize the discounted cost of replacement. (See Feldman [Feldman, R. M. 1976. Optimal replacement with semi-Markov shock models. J. Appl. Probab. 13 108117; Feldman, R. M. 1977. The maintenance of systems governed by semi-Markov shock models. Proceedings of the Conference on the Theory and Applications of Reliability with Emphasis on Bayesian and Nonparametric Methods. Edited by C. P. Tsokos. Academic Press.] for the average cost criterion case.) The general approach will be to apply optimal stopping theory to the replacement problem.