On a singular feature of critical G/M/1 queues

In this paper, we consider a critically loaded G/M/1 queue and contrast its transient behaviour with the transient behaviour of stable (or unstable) G/M/1 queues. We show that the departure process from a critical G/M/1 queue converges weakly to a Poisson process. However, as opposed to the stable (or unstable) case, we show that the departure process of a critical G1/M/1 queue does not couple in finite time with a Poisson process (even though it converges weakly to one). Thus, as the traffic intensity (ratio of arrival to service rates), rho, ranges over (0, infinity), the point rho = 1 represents a singularity with regard to the convergence mode of the departure process.