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Two of the most popular approximations for the distribution of thesteady-state waiting time, $W_{\infty}$, of the M/G/1 queue are the so-calledheavy-traffic approximation and heavy-tailed asymptotic, respectively. If thetraffic intensity, $\rho$, is close to 1 and the processing times have finitevariance, the heavy-traffic approximation states that the distribution of$W_{\infty}$ is roughly exponential at scale $O((1-\rho)^{-1})$, while theheavy tailed asymptotic describes power law decay in the tail of thedistribution of $W_{\infty}$ for a fixed traffic intensity. In this paper, weassume a regularly varying processing time distribution and obtain a sharpthreshold in terms of the tail value, or equivalently in terms of $(1-\rho)$,that describes the point at which the tail behavior transitions from theheavy-traffic regime to the heavy-tailed asymptotic. We also provide newapproximations that are either uniform in the traffic intensity, or uniform onthe positive axis, that avoid the need to use different expressions on the tworegions defined by the threshold.

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