Queuing models with Mittag-Leffler inter-event times

TL;DR

This paper investigates three queueing models with Mittag-Leffler distributed inter-event times, analyzing their stability, equilibrium properties, and asymptotic behaviors, extending classical queue theory to heavy-tailed inter-arrival times.

Contribution

It introduces three generalized queue models with Mittag-Leffler inter-event times and provides a comprehensive analysis of their stability, equilibrium, and asymptotic properties.

Findings

Determined conditions for recurrence and transience in each model.

Derived existence and form of equilibrium distributions.

Analyzed mixing times and asymptotic behaviors.

Abstract

We study three non-equivalent queueing models in continuous time that each generalise the classical M/M/1 queue in a different way. Inter-event times in all models are Mittag-Leffler distributed, which is a heavy tail distribution with no moments. For each of the models we answer the question of the queue being at zero infinitely often (the `recurrence' or `stable' regime) or not (the transient regime). Aside from this question, the different analytical properties of each models allow us to answer a number of questions such as existence and description of equilibrium distributions, mixing times, asymptotic behaviour of return probabilities and moments and functional limit theorems.