Optimality of randomized strategies in a Markovian replacement model

We study a replacement system with discrete-time Markovian deterioration and finite state space {0,...,N}. State 0 stands for a new system, and the larger the state the worse the condition of the system with N as the failure state. We impose the condition that the long-term fraction of replacements in state N should not be larger than some fixed number. We prove that a generalized control-limit policy maximizes the expected time between two successive replacements and we explain explicitly how to derive this (randomized) optimal policy. Some numerical examples are given.