A note on repair systems with exchangeable items and the longest queue mechanism

Abstract

A repair facility consisting of one repairman and two arrival streams of failed items, from bases 1 and 2 is considered. The arrival processes are independent Poisson processes, and the repair times are independent and identically exponentially distributed. The item types are exchangeable in the sense that a customer leaves the system with a fixed item, not necessarily the item he has brought. The rule according to which backorders are satisfied by repaired items is the longest queue rule: at the completion of a service (repair), the repaired item is delivered to the base that has the largest number of failed items. We obtain simple expressions for the marginal queue length distributions and for the probability mass function of the difference between the queue lengths. Finally we derive a recursive expression for the Laplace transform of the customer’s sojourn time in system, given the length of the queue at the arrival time point.