The Annals of Applied Probability

On the transition from heavy traffic to heavy tails for the M/G/1 queue: The regularly varying case

Mariana Olvera-Cravioto, Jose Blanchet, and Peter Glynn

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Abstract

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.

Article information

Source
Ann. Appl. Probab., Volume 21, Number 2 (2011), 645-668.

Dates
First available in Project Euclid: 22 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1300800984

Digital Object Identifier
doi:10.1214/10-AAP707

Mathematical Reviews number (MathSciNet)
MR2807969

Zentralblatt MATH identifier
1216.41033

Subjects
Primary: 41A60: Asymptotic approximations, asymptotic expansions (steepest descent, etc.) [See also 30E15]
Secondary: 60F10: Large deviations 60F05: Central limit and other weak theorems 60G10: Stationary processes 60G50: Sums of independent random variables; random walks

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

Citation

Olvera-Cravioto, Mariana; Blanchet, Jose; Glynn, Peter. On the transition from heavy traffic to heavy tails for the M/G/1 queue: The regularly varying case. Ann. Appl. Probab. 21 (2011), no. 2, 645--668. doi:10.1214/10-AAP707. https://projecteuclid.org/euclid.aoap/1300800984


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