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**On the transition from heavy traffic to heavy tails for the \(M/G/1\) queue: the regularly varying case.**
*(English)*
Zbl 1216.41033

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 \(\rho\) 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 \(O((1-\rho)^{-1})\), 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 \((1-\rho)\), 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.

### MSC:

41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |

60F10 | Large deviations |

60F05 | Central limit and other weak theorems |

60G10 | Stationary stochastic processes |

60G50 | Sums of independent random variables; random walks |

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\textit{M. Olvera-Cravioto} et al., Ann. Appl. Probab. 21, No. 2, 645--668 (2011; Zbl 1216.41033)

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