Multi-point and multi-interval availabilities for Markov repairable systems with history-dependent up and down states

A Markov repairable system with history-dependent up and down states is presented. The state space of the repairable system is divided into three sets, which are up states, down states and changeable states. Using aggregated stochastic processes, we can get the multi-interval availability and multi-point availability for the Markov repairable system. Finally, numerical examples are given to illustrate the obtained results.