Multiple channel queues in heavy traffic. I

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Abstract
The queueing systems considered in this paper consist ofrindependent arrival channels andsindependent 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 classicalGI/G/ssystem 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. Letλidenote the arrival rate (reciprocal of the mean interarrival time) in theith arrival channel andμjthe service rate (reciprocal of the mean service time) in thejth service channel. Thenis the total arrival rate to the system andis the maximum service rate of the system. As a measure of congestion we define the traffic intensityρ=λ/μ.

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