Optimal replacement policies with continuously varying observable damage

A system is subject to random failure with failure rate k(Xt) dependent upon the level Xt of accumulated damage at time t. Given the replacement cost C and the additional cost K for replacement after failure, an optimum level of damage at which replacement should be made is investigated when the damage process is (a) a Wiener process with drift, (b) the integral of a Markov chain and (c) a linearly increasing deterministic process. The solution of (c) is used to derive approximations for cases (a) and (b) when the damage process is 'nearly deterministic'.