The asymptotic variance of departures in critically loaded queues

We consider the asymptotic variance of the departure counting processD(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 % equals 1, and prove that the asymptotic variance rate satisfies lim t→∞ VarD(t) t = λ ( 1− 2 π )( ca + c 2 s ) , where λ is the arrival rate and ca, cs 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 % 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.