ON THE TRANSITION FROM HEAVY TRAFFIC TO HEAVY TAILS FOR THE M/G/1 QUEUE: THE REGULARLY VARYING CASE

• 成果类型: Article
• 署名作者: Olvera-Cravioto, Mariana; Blanchet, Jose; Glynn, Peter
• 署名单位: Columbia University; Stanford University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/10-AAP707
• 发表日期: 2011
• 页码: 645-668
• 关键词: large deviations; Waiting time; approximations; probabilities; networks

摘要

Two of the most popular approximations for the distribution of the steady-state waiting time, W-infinity, of the M/G/1 queue are the so-called heavy-traffic approximation and heavy-tailed asymptotic, respectively. If the traffic intensity, rho, is close to 1 and the processing times have finite variance, the heavy-traffic approximation states that the distribution of W-infinity is roughly exponential at scale O((1 - rho)(-1)), while the heavy tailed asymptotic describes power law decay in the tail of the distribution of W-infinity 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- rho), 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.