March 2011 The asymptotic variance of departures in critically loaded queues
A. Al Hanbali, M. Mandjes, Y. Nazarathy, W. Whitt
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Adv. in Appl. Probab. 43(1): 243-263 (March 2011). DOI: 10.1239/aap/1300198521


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.


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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.


Published: March 2011
First available in Project Euclid: 15 March 2011

zbMATH: 1279.90045
MathSciNet: MR2761156
Digital Object Identifier: 10.1239/aap/1300198521

Primary: 90B22
Secondary: 60G55

Keywords: Brownian bridge , critically loaded system , departure process , GI/G/1 queue , multiserver queue , renewal theory , uniform integrability

Rights: Copyright © 2011 Applied Probability Trust


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Vol.43 • No. 1 • March 2011
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