Fractional Queues with Catastrophes and Their Transient Behaviour

Giacomo Ascione, Nikolai Leonenko and Enrica Pirozzi
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Giacomo Ascione: Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Università degli Studi di Napoli Federico II, 80126 Napoli, Italy
Nikolai Leonenko: School of Mathematics, Cardiff University, Cardiff CF24 4AG, UK
Enrica Pirozzi: Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Università degli Studi di Napoli Federico II, 80126 Napoli, Italy

Mathematics, 2018, vol. 6, issue 9, 1-26

Abstract: Starting from the definition of fractional M/M/1 queue given in the reference by Cahoy et al. in 2015 and M/M/1 queue with catastrophes given in the reference by Di Crescenzo et al. in 2003, we define and study a fractional M/M/1 queue with catastrophes. In particular, we focus our attention on the transient behaviour, in which the time-change plays a key role. We first specify the conditions for the global uniqueness of solutions of the corresponding linear fractional differential problem. Then, we provide an alternative expression for the transient distribution of the fractional M/M/1 model, the state probabilities for the fractional queue with catastrophes, the distributions of the busy period for fractional queues without and with catastrophes and, finally, the distribution of the time of the first occurrence of a catastrophe.

Keywords: fractional differential-difference equations; fractional queues; fractional birth-death processes; busy period (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2018
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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