Optimal replacement policy for the case where the damage process is a one-sided Lévy process

A production system is subject to random failure. The system continuously accumulates damage through a "wear" process and the failure time depends on the accumulated damage in the system. Upon failure the system is replaced by a new identical one and the replacement cycles are repeated. A cost is associated with each replacement, and an additional cost is incurred at each failure in service. We allow a controller to replace the system at any Markov time T before failure time. The problem is to find an optimal control policy that minimizes the total long run average cost per unit of time. We will treat the case when the damage process is a one-sided Lévy process. The system fails when the accumulated damage first exceeds V, where V is a random variable having a known distribution. We suppose that the accumulated damage is observable. Assuming that the hazard rate associated with the random variable V is monotonically nondecreasing, we show that the optimal policy is a control limit policy. An example is presented to illustrate computational procedures.