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/1 queues characterized by ( U_{1}(t), U_{2}(t) ) with U_{j}(t) denoting the unfinished work (buffer content) in queue j . 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 processes U_{1}(t) + U_{2}(t) and U_{2}(t) - U_{1}(t) become independent, with the distribution of U_{1}(t) + U_{2}(t) identical to the heavy traffic waiting time distribution in the standard M/G/2 queue, and the distribution of U_{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.