Analysis of an k-out-of-n:G system with repairman's single vacation and shut off rule

This paper considers an k-out-of-n:G repairable system with repairman's single vacation and shut off rule where the working times and repair times of components follow exponential distributions, and the duration of the repairman's vacation is governed by a phase type distribution. Further, continuous operation as shut off rule in this paper means that all the non-failed components are still subject to failure when the system is down. We derive the transient system availability, the rate of occurrence of failures at time t, the stationary availability, the stationary rate of occurrence of failures along with other system performance measures by using the Markov process theory and the matrix analytical method. Moreover, employing the probabilistic properties of the pH distribution, we deduce the system reliability, the mean time to the first failure of the system and the waiting time distribution of an arbitrary failed component. Based on this, we discuss the time-dependent behavior of the system performance measures under different initial system states. Finally, the Monte Carlo simulation and special cases of the system are provided to show the correctness of our results.