On the Nonsymmetric Longer Queue Model: Joint Distribution, Asymptotic Properties, and Heavy Traffic Limits

We consider two parallel queues, each with independent Poisson arrival rates, that are tended by a single server. The exponential server devotes all of its capacity to the longer of the queues. If both queues are of equal length, the server devotes of its capacity to the first queue and the remaining to the second. We obtain exact integral representations for the joint probability distribution of the number of customers in this two-node network. Then we evaluate this distribution in various asymptotic limits, such as large numbers of customers in either/both of the queues, light traffic where arrivals are infrequent, and heavy traffic where the system is nearly unstable. 1. Introduction We consider a nonsymmetric version of the longer queue model. Here there are two parallel queues, each fed by a Poisson arrival stream. There is but a single server who tends to the longer of the two queues. If the number of customers in each queue is the same, then the server devotes of its capacity to the first queue and to the second queue, with . We let ( ) denote the number of customers in the first (second) queue, the two arrival rates are and , and the server works at rate . Note that the total number of customers, , in the two-node network behaves as the standard model, so in the steady state we have the geometric distribution , where , assuming the stability condition . The “symmetric case” corresponds to (thus ) and , and this was analyzed in detail by Flatto [1]. Such models were proposed by Zheng and Zipkin [2] to study problems in inventory control. In [2] finite capacities were assumed in the two queues, and the authors studied numerically the steady state probabilities , in terms of the capacity size and also for different service disciplines, such as the longer queue (LQ) discipline here, and also the first-come-first-served discipline. In [1] the author used two-dimensional generating functions and analyticity arguments and obtained explicit expressions for , in the symmetric case, as contour integrals. Then asymptotic results were derived for the joint distribution , as and/or becomes large, and also the marginal tails and various conditional limit laws were obtained. In this paper we generalize some of the results of [1] to the nonsymmetric model, and we will show that now many of the asymptotic results become quite different. As in [1] we assume that the model is preemptive, so if (and thus the server works on both queues) and a new arrival occurs to the first queue, then the server switches immediately all its capacity to the first queue. The more