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’ 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.

Queuing models with Mittag-Leffler inter-event times / Butt, Jacob; Georgiou, Nicos; Scalas, Enrico. - In: FRACTIONAL CALCULUS & APPLIED ANALYSIS. - ISSN 1314-2224. - 26:(2023), pp. 1465-1503.

Queuing models with Mittag-Leffler inter-event times

Enrico Scalas
2023

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’ 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.
2023
GI/ GI/1 queue; Mittag-Leffler queues; queue length; recurrence; transience; fractional derivatives; time-changed queue; semi-Markov process; mixing times; scaling limits for the queue length
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Queuing models with Mittag-Leffler inter-event times / Butt, Jacob; Georgiou, Nicos; Scalas, Enrico. - In: FRACTIONAL CALCULUS & APPLIED ANALYSIS. - ISSN 1314-2224. - 26:(2023), pp. 1465-1503.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1685149
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