The average optimality of a repair-limit replacement policy

We consider a minimal-repair and replacement problem of a reliability system whose state at a failure is described by a pair of two attributes, i.e., the total number of its past failures and the current failure level. It is assumed that the system is bothered by more frequent and more costly failures as time passes. Our problem is to find and/or characterize a minimal-repair and replacement policy of minimizing the long-run average expected maintenance cost per unit time over the infinite time horizon. Formulating the problem as a semi-Markov decision process, we show that a repairlimit replacement policy is average optimal. That is, for each total number of past system failures, there exists a threshold, called a repair limit, such that it is optimal to repair minimally if the current failure level is lower than the repair limit, and to replace otherwise. Furthermore, the repair limit is decreasing in the total number of past system failures.