Repair systems with exchangeable items and the longest queue mechanism

R Ravid,D Perry,O J Boxma

doi:10.1007/s11134-012-9319-5

Abstract

We consider a repair facility consisting of one repairman and two arrival streams of failed items, from bases 1 and 2. The arrival processes are independent Poisson processes, and the repair times are independent and identically exponentially distributed. The item types are exchangeable, and a failed item from base 1 could just as well be returned 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. We point out a direct relation between our model and the classical longer queue model. We obtain simple expressions for several probabilities of interest, and show how all two-dimensional queue length probabilities may be obtained. Finally, we derive the sojourn time distributions.

Highlights

In this paper, we consider a repair facility consisting of one repairman and two arrival streams of failed items, from bases 1 and 2
The rule according to which backorders are satisfied by repaired items is the longest queue rule: At the completion of a service, the repaired item is delivered to the base that has the largest number of failed items
Contributions Our main contributions are: (i) We point out a direct relation between our model and the classical longer queue model of Cohen [2]; (ii) we obtain simple expressions for several probabilities of interest, and we show how all two-dimensional probabilities may be obtained; (iii) we derive the sojourn time distributions—this performance measure was not studied in the papers mentioned above; and (iv) we present some methodological ideas which might be more broadly applicable; one example is the use of the “difference busy period.”

Summary

Introduction

We consider a repair facility consisting of one repairman and two arrival streams of failed items, from bases 1 and 2. If the server has completed a service, the customer to be served is the one at the head of the longest queue if the queue lengths are not equal; if both queues have equal length, the customer in service is of type i with some probability αi He determines the generating function of the joint steady-state queue length distribution right after service completions, by solving a boundary value problem of Riemann–. Flatto [5] considers the symmetric exponential case He allows preemption, and derives an expression for the probability generating function of the joint queue length distribution. We study its generating function (GF), deriving various special results like the distribution of the difference of the two queue lengths and the probability that there are n1 customers of one type and none of the other type.

Queue lengths—a generating function approach

Queue lengths—a probabilistic approach

Numerical example

Sojourn times

The LST of the sojourn time distribution

Further results for the sojourn time distribution