Scaling limits for some Mittag-Leffler queues

TL;DR

This paper investigates five heavy-tailed queue models with Mittag-Leffler distributions, analyzing their scaling limits to understand how they behave under different traffic conditions, generalizing classical queue models.

Contribution

It introduces five Mittag-Leffler queue models, derives their inter-arrival and service time distributions, and studies their scaling limits to characterize traffic regimes.

Findings

Derived explicit distributions for inter-arrival and service times.

Established scaling limits for the proposed queue models.

Characterized traffic regimes through limiting process behavior.

Abstract

In this paper, we consider five models of heavy-tailed queues involving Mittag-Leffler distributions that generalize the classical $M/M/1$ queues. These models are suitable modifications of previously defined models in such a way that the classical $M/M/1$ queue can be recovered by a suitable selection of parameters. We provide the distribution of inter-arrival and service times of both the original and modified queueing models. We then study the scaling limits of all the proposed models and we argue that the behaviour of the limiting processes can be used to characterise the traffic regime of the queues.