The asymptotic variance of departures in critically loaded queues

We consider the asymptotic variance of the departure counting process D(t) of the GI/G/1 queue; D(t) denotes the number of departures up to time t. We focus on the case that the system load $\rho$ equals 1, and prove that the asymptotic variance rate satisfies \[ \lim_{t \rightarrow \infty} \frac{Var D(t)}{t} = \lambda (1-\frac{2}{\pi})(c^2_a+c^2_s) \] , where $\lambda$ is the arrival rate and $c^2_a$, $c^2_s$ are squared coefficients of variation of the inter-arrival and service times respectively. As a consequence, the departures variability has a remarkable singularity in case $\rho$ equals 1, in line with the BRAVO effect (Balancing Reduces Asymptotic Variance of Outputs) which was previously encountered in the finite-capacity birth-death queues. Under certain technical conditions, our result generalizes to multi-server queues, as well as to queues with more general arrival and service patterns. For the M/M/1 queue we present an explicit expression of the variance of D(t) for any t. Keywords: GI/G/1 queues, critically loaded systems, uniform integrability, departure processes, renewal theory, Brownian bridge, multi-server queues