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

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.