Next Article in Journal
On Magnifying Elements in E-Preserving Partial Transformation Semigroups
Next Article in Special Issue
Network Reliability Modeling Based on a Geometric Counting Process
Previous Article in Journal
A Game-Theoretic Loss Allocation Approach in Power Distribution Systems with High Penetration of Distributed Generations
Previous Article in Special Issue
Stability Analysis of Cohen–Grossberg Neural Networks with Random Impulses
Article Menu
Issue 9 (September) cover image

Export Article

Open AccessArticle
Mathematics 2018, 6(9), 159;

Fractional Queues with Catastrophes and Their Transient Behaviour

Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Università degli Studi di Napoli Federico II, 80126 Napoli, Italy
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)
PDF [321 KB, uploaded 6 September 2018]


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).

Share & Cite This Article

MDPI and ACS Style

Ascione, G.; Leonenko, N.; Pirozzi, E. Fractional Queues with Catastrophes and Their Transient Behaviour. Mathematics 2018, 6, 159.

Show more citation formats Show less citations formats

Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Related Articles

Article Metrics

Article Access Statistics



[Return to top]
Mathematics EISSN 2227-7390 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top