Optimal Replacement for Shock Models with General Failure Rate

A device is subject to a series of shocks which arrive as a semi-Markov process. Shocks cause damage and eventually failure will occur at the time of arrival of one of the shocks. The device must be replaced upon failure at some cost but it also can be replaced before failure at a lower cost. We consider the general case where the failure rate need not be increasing and replacement can be made at any time. The form of the optimal replacement policy is found and conditions are determined for which a control limit policy is optimal.