Please use this identifier to cite or link to this item: http://hdl.handle.net/1959.3/189594

Title

The asymptotic variance of departures in critically loaded queues

Author(s)

Al Hanbali, A.;  Mandjes, M.;  Nazarathy, Y.;  Whitt, W.

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.

Publication type

Journal article

Source

Advances in Applied Probability, Vol. 43, no. 1 (Mar 2011), pp. 243-263

Publication year

2011

FOR Code(s)

0102 Applied Mathematics;  0104 Statistics

Keyword(s)

Brownian bridge;  Critically loaded system;  Departure process;  GT/G/1 queue;  Multiserver queue;  Renewal theory;  Uniform integrability

Publisher

Applied Probability Trust

ISSN

0001-8678

Publisher URL

http://dx.doi.org/10.1239/aap/1300198521

Copyright

Copyright © 2011.

Peer reviewed