In this thesis, we present an expanded account of A Discussion in the Analysis R-out-of-N Repairable Systems. an article by Y. Barron and E. Frostig and B. Levikson (2004). These days, one of the main discussion in studing systems, that has been considered much by statisticians and engineers, is calculating availability of R-out-of-N Repairable Systems. An R-out-of-N Repairable System consisting of N components, is operating if at least R components are functioning, otherwise the system freezes. The down-time period terminates at the first time that the number of working components increases to R. Repairable means that faild components are repaired, and upon repair completion they are as good as new. We consider phase-type distribution and its properties in this paper. We studied two R-out-of-N repairable systems with one repairman and 1-out-of-N repairable systems with c repairmen. We use regenerative, semi-regenerative and markov renewal processes are used for analyzing these repairable systems. Numerical example are given for everal life times and repair times distributions.
