## On the Transition from Heavy Traffic to Heavy Tails for the M/G/1 Queue: The Regularly Varying Case
*M. Olvera-Cravioto, J. H. Blanchet, and P. W. Glynn*
*Annals of Applied Probability*, Vol. 21(2), 654-668 (2011)
Two of the most popular approximations for the distribution of
the steady-state waiting time, W_{∞}, of the M/G/1 queue are the socalled
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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