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 t. We focus on the case where the system load ϱ equals 1, and prove that the asymptotic variance rate satisfies limt→∞varD(t) / t = λ(1 - 2 / π)(ca2 + cs2), where λ is the arrival rate, and ca2 and cs2 are 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 ϱ 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/1 queue, we present an explicit expression of the variance of D(t) for any t.
Citation
A. Al Hanbali. M. Mandjes. Y. Nazarathy. W. Whitt. "The asymptotic variance of departures in critically loaded queues." Adv. in Appl. Probab. 43 (1) 243 - 263, March 2011. https://doi.org/10.1239/aap/1300198521
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