id: 05886813
dt: j
an: 1216.41033
au: Olvera-Cravioto, Mariana; Blanchet, Jose; Glynn, Peter
ti: On the transition from heavy traffic to heavy tails for the $M/G/1$ queue:
    the regularly varying case.
so: Ann. Appl. Probab. 21, No. 2, 645-668 (2011).
py: 2011
pu: Institute of Mathematical Statistics, Hayward, CA
la: EN
cc: 41A60 60F10 60F05 60G10 60G50
ut: $M/G/1$ queue; heavy traffic; heavy tails; uniform approximations; large
    deviations
ci: 
li: doi:10.1214/10-AAP707
ab: Summary: Two of the most popular approximations for the distribution of the
    steady-state waiting time $W_\infty$ 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 $W_\infty$ is roughly exponential at
    scale $O((1-ρ)^{-1})$, while the heavy tailed asymptotic describes the
    power law decay in the tail of the distribution of $W_\infty$ 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.
rv: