A multiple warm standby repairable system under -policy with multiple vacations following Markovian arrival process

The paper considers a warm standby (Formula presented.) -unit system under (Formula presented.) -policy with one repairman taking multiple vacations. The system is operational as long as one of the units is normal. The online unit is subject to internal failures and external shocks. The lifetime of the online unit due to an internal failure follows a phase-type (PH) distribution. The inter-arrival time of shocks and the random amount of shock damage are governed by other two different PH distributions, respectively. When the damage of a shock is larger than the given threshold, the online unit is considered to be failed. Standby units are subject to shocks from a different cause, which are governed by a Markovian arrival process (MAP). A repairman who can take multiple vacations repairs the failed units based on the (Formula presented.) -policy. The successive repair times and the successive vacation times of the repairman are governed by other two different MAPs, respectively. In this paper, first, a Markov process is constructed to describe the warm standby (Formula presented.) -unit system. Then the system is studied in a transient and stationary regime using matrix-analytical approaches. Finally, two numerical applications are given to illustrate the results obtained in the paper.