Two Parallel M/G/1 Queues where Arrivals Join the System with the Smaller Buffer Content

Abstract

We consider two parallel, infinite capacity,M/G/1queues characterized by (U_{1}(t), U_{2}(t)) withU_{j}(t)denoting the unfinished work (buffer content) in queuej. A new arrival is assigned to the queue with the smaller buffer content. We construct formal (as opposed to rigorous) asymptotic approximations to the Joint stationary distribution of the Markov process (U_{1}(t), U_{2}(t)), treating separately the asymptotic limits of heavy traffic, light traffic, and large buffer contents. In heavy traffic, the stochastic processesU_{1}(t) + U_{2}(t)andU_{2}(t) - U_{1}(t)become independent, with the distribution ofU_{1}(t) + U_{2}(t)identical to the heavy traffic waiting time distribution in the standardM/G/2queue, and the distribution ofU_{2}(t) - U_{1}(t)closely related to the tail of the service time density. In light traffic, we obtain a formal expansion of the stationary distribution in powers of the arrival rate.

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