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Mathematics 2018, 6(9), 159; https://doi.org/10.3390/math6090159

Fractional Queues with Catastrophes and Their Transient Behaviour

1
Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Università degli Studi di Napoli Federico II, 80126 Napoli, Italy
2
School of Mathematics, Cardiff University, Cardiff CF24 4AG, UK
*
Author to whom correspondence should be addressed.
Received: 27 July 2018 / Revised: 30 August 2018 / Accepted: 31 August 2018 / Published: 6 September 2018
(This article belongs to the Special Issue Stochastic Processes with Applications)
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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. View Full-Text
Keywords: fractional differential-difference equations; fractional queues; fractional birth-death processes; busy period fractional differential-difference equations; fractional queues; fractional birth-death processes; busy period
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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Ascione, G.; Leonenko, N.; Pirozzi, E. Fractional Queues with Catastrophes and Their Transient Behaviour. Mathematics 2018, 6, 159.

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