Approximations for the repairman problem with two repair facilities, II: Spares

Abstract

The model considered here consists of n operating units which are subject to stochastic failure according to an exponential failure time distribution. These operating units are backed up by m_n spare units. Failures can be of two types. With probability p (q) a failure is of type 1(2) and is sent to repair facility 1(2) for repair. Repair facility 1(2) operates as a -server queue with exponential repair times having parameter μ ₁ (μ ₂). The number of units waiting for or undergoing repair at each of the two facilities is a continuous parameter Markov chain with finite state space. This paper derives limit theorems for the stationary distribution of this Markov chain as n becomes large under the assumption that and m_n grow linearly with n. These limit theorems give very useful approximations, in terms of the seven parameters characterizing the model, to a distribution that would be difficult to calculate in practice.