Generalized control-limit preventive repair policies for deteriorating cold and warm standby Markovian systems

Consider a deteriorating repairable Markovian system with N stochastically independent identical units. The lifetime of each unit follows a discrete phase-type distribution. There is one online unit and the others are in standby status. In addition, there is a single repair facility and the repair time of a failed unit has a geometric distribution. The system is inspected at equally spaced points in time. After each inspection, either repair or a full replacement is possible. We consider state-dependent operating costs, repair costs that are dependent on the extent of the repair, and failure penalty costs. Applying dynamic programming, we show that under reasonable conditions on the system’s law of evolution and on the state-dependent costs, a generalized control-limit policy is optimal for the expected total discounted criterion for both cold standby and warm standby systems. Illustrative numerical examples are presented and insights are provided.