The asymptotic variance of departures in critically loaded queues

Al Hanbali, A., Mandjes, M., Nazarathy, Y. and Whitt, W. (2011) The asymptotic variance of departures in critically loaded queues. Advances In Applied Probability, 43 1: 243-263. doi:10.1239/aap/1300198521

Author Al Hanbali, A.
Mandjes, M.
Nazarathy, Y.
Whitt, W.
Title The asymptotic variance of departures in critically loaded queues
Journal name Advances In Applied Probability   Check publisher's open access policy
ISSN 0001-8678
Publication date 2011-03
Sub-type Article (original research)
DOI 10.1239/aap/1300198521
Volume 43
Issue 1
Start page 243
End page 263
Total pages 21
Place of publication Sheffield, U.K.
Publisher Applied Probability Trust
Collection year 2012
Language eng
Formatted 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.
Keyword GIG/1 queue
Critically loaded system
Uniform integrability
Departure process
Renewal theory
Brownian bridge
Multiserver queue
Q-Index Code C1
Q-Index Status Confirmed Code
Institutional Status Non-UQ

Document type: Journal Article
Sub-type: Article (original research)
Collections: School of Mathematics and Physics
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Created: Fri, 27 Apr 2012, 12:47:22 EST by Kay Mackie on behalf of School of Mathematics & Physics