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April 2011 On the transition from heavy traffic to heavy tails for the M/G/1 queue: The regularly varying case
Mariana Olvera-Cravioto, Jose Blanchet, Peter Glynn
Ann. Appl. Probab. 21(2): 645-668 (April 2011). DOI: 10.1214/10-AAP707


Two of the most popular approximations for the distribution of the steady-state waiting time, W, of the M/G/1 queue are the so-called heavy-traffic approximation and heavy-tailed asymptotic, respectively. If the traffic intensity, ρ, is close to 1 and the processing times have finite variance, the heavy-traffic approximation states that the distribution of W is roughly exponential at scale O((1 − ρ)−1), while the heavy tailed asymptotic describes power law decay in the tail of the distribution of W for a fixed traffic intensity. In this paper, we assume a regularly varying processing time distribution and obtain a sharp threshold in terms of the tail value, or equivalently in terms of (1 − ρ), that describes the point at which the tail behavior transitions from the heavy-traffic regime to the heavy-tailed asymptotic. We also provide new approximations that are either uniform in the traffic intensity, or uniform on the positive axis, that avoid the need to use different expressions on the two regions defined by the threshold.


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Mariana Olvera-Cravioto. Jose Blanchet. Peter Glynn. "On the transition from heavy traffic to heavy tails for the M/G/1 queue: The regularly varying case." Ann. Appl. Probab. 21 (2) 645 - 668, April 2011.


Published: April 2011
First available in Project Euclid: 22 March 2011

zbMATH: 1216.41033
MathSciNet: MR2807969
Digital Object Identifier: 10.1214/10-AAP707

Primary: 41A60
Secondary: 60F05 , 60F10 , 60G10 , 60G50

Keywords: heavy tails , heavy traffic , large deviations , M/G/1 queue , uniform approximations

Rights: Copyright © 2011 Institute of Mathematical Statistics

Vol.21 • No. 2 • April 2011
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