The asymptotic variance of departures in critically loaded queues

Abstract

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 /. We focus on the case where the system load q equals 1, and prove that the asymptotic variance rate satisfies limt→∞ var D(t)/t = Λ.(l © 2/π)(c2a + c2a), where Λ is the arrival rate, and c2a and c2aare squared coefficients of variation of the interarrival and service times, respectively. As a consequence, the departures variability has a remarkable singularity in the case in which q equals 1, in line with the BRAVO (balancing reduces asymptotic variance of outputs) effect which was previously encountered in finite-capacity birth-death queues. Under certain technical conditions, our result generalizes to multiserver queues, as well as to queues with more general arrival and service patterns. For the M/M/l queue, we present an explicit expression of the variance of D(t) for any t. © Applied Probability Trust 2011.