In this research we consider a repair facility consisting of one repairman and two arrival streams of failed items, from base 1 and base 2. The arrival processes are independet Poisson processes with unequal rates .The repair times are independent from an exponential distribution with the same rate. The item types are exchangeable, and a failed item from base 1could just as well be returened to base 2 and vice versa. 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. In a case of a tie, the item will be delivered to base 1 or base 2 with equal probabilities. We assume that the system is in steady state.
Exchangeable-items repair systems have received considerable attention in the literature because repairable-type items are often essential and expensive. Many organizations extensively use multi-echelon repairable-item systems to support advanced computer systems and sophisticated medical equipment. The items are admitted to one line but the customers wait and marked according to their sources. The main focus of the studies to date has been on the number of backorders (mainly its expectation), which is the customer queue size in the terminology used here. Another important performance measure is the customer's sojourn time distribution.
The main purpose of this research is to express the Laplace Stieltjes transform of the sojourn time distribution of a customer who joins line 1 with a failed item. By interchanging indices 1 and 2 (in particular, the arrival rates) we then obtain the sojourn time Laplace Stieltjes transform for customers who join line 2. We also pointed out a direct relation between the classical longest queue model and our model.
Keywords: Sojourn time, Longest queue system, Exchangeable items
