Some Results on an Optimal Maintenance Policy for a Markovian Deteriorating System Subject to Random Shocks

This paper considers a system whose deterioration is modeled as a discrete-time and discrete-state Markov chain and is subject to randomly occurring shocks. The system is more likely to deteriorate after the occurrence of shocks. At each time epoch, we can select from one of three possible actions: operate, repair, or replace. The difference between repair and replace is whether the influence of shocks is removed or not. While repair does not change the number of shocks that the system has suffered, replacement can return it to zero. We derive an expected discounted cost for an unbounded horizon and show that a generalized control limit policy holds under certain assumptions on the costs and the transition probability. Several structural properties of the optimal maintenance policy are also investigated under different assumptions for the occurrence of shocks. These results are useful for numerically determining the optimal maintenance policy. Finally, we consider some special cases by imposing constraints on the costs and the probabilities.