Reliability Analysis of an n-Spare System with a Single Repair Facility

Abstract

The reliability of standby systems with repair is investigated using semi-Markov processes. The systems are composed of n + 1 identical units having an exponential failure time distribution and of a single repair facility having a general repair time distribution. For these systems, the Laplace-Stieltjes transform of the system failure time distribution is derived in the form of generating functions. For indefinitely large n, asymptotic formulae of the mean time to system failure (MTSF) and limiting distributions of the system failure time are given. Under certain conditions, the system failure time distribution is shown to tend to be exponential as n increases. Numerical examples show that the asymptotic formulae of MTSF are good approximations, even for small n. The results can also be applied to the first passage time problem of the maximal queue size in the system M/G/1.