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.
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机译：M / G / 1队列的稳态等待时间分布中最流行的两种近似值$ W _ {\ infty} $分别是所谓的重交通近似和重尾渐近。如果流量强度$ \ rho $接近1，并且处理时间具有有限方差，则重流量近似表示$ W _ {\ infty} $的分布在$ O（（1- \ rho ）^ {-1}）$，而重尾渐近线则描述了固定流量强度下$ W _ {\ infty} $分布尾部的幂律衰减。在本文中，我们假设处理时间分布有规律地变化，并且以尾部值（或等价地以$（1- \ rho）$）获得尖锐的阈值，该阈值描述了尾部行为从重交通转变到的点重尾渐近政权。我们还提供了交通强度一致或正轴一致的新近似值，从而避免了在阈值定义的两个区域上使用不同表达式的需要。
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