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The asymptotic variance of departures in critically loaded queues

Part of: Stochastic processes Operations research and management science

Published online by Cambridge University Press:  01 July 2016

A. Al Hanbali ,
M. Mandjes ,
Y. Nazarathy  and
W. Whitt
A. Al Hanbali*
Affiliation:
University of Twente
M. Mandjes*
Affiliation:
University of Amsterdam and CWI
Y. Nazarathy*
Affiliation:
EURANDOM and Eindoven University of Technology
W. Whitt*
Affiliation:
Columbia University
*
∗ Postal address: School of Management and Governance, University of Twente, Enschede, The Netherlands. Email address: a.m.alhanbali@utwente.nl
∗∗ Postal address: Korteweg-de Vries Institute for Mathematics, University of Amsterdam, The Netherlands. Email address: m.r.h.mandjes@uva.nl
∗∗∗ Postal address: Faculty of Engineering and Industrial Sciences, Swinburne University of Technology, John Street, Hawthorn, Victoria, 3122 Australia. Email address: ynazarathy@swin.edu.au
∗∗∗∗ Postal address: Department of Industrial Engineering and Operations Research, Columbia University, S. W. Mudd Building, 500 West 120th Street, New York, NY 10027, USA. Email address: ww2040@columbia.edu
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Abstract

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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.

Keywords

GI/G/1 queuecritically loaded systemuniform integrabilitydeparture processrenewal theoryBrownian bridgemultiserver queue

MSC classification

Primary: 90B22: Queues and service
Secondary: 60G55: Point processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 1 , March 2011 , pp. 243 - 263
Copyright
Copyright © Applied Probability Trust 2011 

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