Multiple channel queues in heavy traffic

Abstract

The queueing systems considered in this paper consist of r independent arrival channels and s independent service channels, where as usual the arrival and service channels are independent. Arriving customers form a single queue and are served in the order of their arrival without defections. We shall treat two distinct modes of operation for the service channels. In the standard system a waiting customer is assigned to the first available service channel and the servers (servers - service channels) are shut off when they are idle. Thus the classical GI/G/s system is a special case of our standard system. In the modified system a waiting customer is assigned to the service channel that can complete his service first and the servers are not shut off when they are idle. While the modified system is of some interest in its own right, we introduce it primarily as an analytical tool. Let Ai denote the arrival rate (reciprocal of the mean interarrival time) in the ith arrival channel and /j the service rate (reciprocal of the mean service time) in the jth service channel. Then A = E r= Ai is the total arrival rate to the system and yu = E = #ij is the maximum service rate of the system. As a measure of congestion we define the traffic intensity p = A/lt. We shall restrict our attention to systems in which p > 1. Under this condition the systems are of course unstable (a proof of this fact is an easy byproduct of our results). Our principal objective will be to obtain functional central limit theorems (invariance principles) for the stochastic processes characterizing these systems after appropriately scaling and translating the processes. There has been a growing literature on queues in heavy traffic beginning with Kingman ((1961), (1962)); see Kingman (1965) for a summary of his work. We use the term heavy trafficin a broader sense than Kingman. While he considered queueing systems with traffic intensity less than but close to