Technical Note—Probabilistic Analysis of a Single-Servern-unit System with (n− 1) Warm Standbys

Abstract

This paper deals with the availability and reliability of an n -unit system with ( n − 1) warm standbys and a single repair facility. The failure times of the operating unit and of a standby are assumed 1o be exponentially distributed with parameters λ 1 and λ 2 respectively. Initially, a unit is switched on and the other units are kept as warm standbys. The system breaks down when the last operating unit foils. This paper characterizes the system by the probabilities of its being in the up or the down state, sets up integral equations for these probabilities by identifying suitable regenerative stochastic processes, and employs the Laplace-transform technique to solve these equations. It obtains the mean down-time of the system during (0, t ) and the mean time to system failure explicitly. Finally it discusses the cases n = 2 and 3.