Fractional Queues with Catastrophes and Their Transient Behaviour
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...
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| Format: | Article |
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MDPI AG
2018-09-01
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| Series: | Mathematics |
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| Online Access: | http://www.mdpi.com/2227-7390/6/9/159 |
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| author | Giacomo Ascione Nikolai Leonenko Enrica Pirozzi |
| author_facet | Giacomo Ascione Nikolai Leonenko Enrica Pirozzi |
| author_sort | Giacomo Ascione |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-12T18:40:43Z |
| format | Article |
| id | doaj.art-ce646bd15c0749e9b54667b99cf9472a |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-12-12T18:40:43Z |
| publishDate | 2018-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-ce646bd15c0749e9b54667b99cf9472a2022-12-22T00:15:39ZengMDPI AGMathematics2227-73902018-09-016915910.3390/math6090159math6090159Fractional Queues with Catastrophes and Their Transient BehaviourGiacomo Ascione0Nikolai Leonenko1Enrica Pirozzi2Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Università degli Studi di Napoli Federico II, 80126 Napoli, ItalySchool of Mathematics, Cardiff University, Cardiff CF24 4AG, UKDipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Università degli Studi di Napoli Federico II, 80126 Napoli, ItalyStarting 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.http://www.mdpi.com/2227-7390/6/9/159fractional differential-difference equationsfractional queuesfractional birth-death processesbusy period |
| spellingShingle | Giacomo Ascione Nikolai Leonenko Enrica Pirozzi Fractional Queues with Catastrophes and Their Transient Behaviour Mathematics fractional differential-difference equations fractional queues fractional birth-death processes busy period |
| title | Fractional Queues with Catastrophes and Their Transient Behaviour |
| title_full | Fractional Queues with Catastrophes and Their Transient Behaviour |
| title_fullStr | Fractional Queues with Catastrophes and Their Transient Behaviour |
| title_full_unstemmed | Fractional Queues with Catastrophes and Their Transient Behaviour |
| title_short | Fractional Queues with Catastrophes and Their Transient Behaviour |
| title_sort | fractional queues with catastrophes and their transient behaviour |
| topic | fractional differential-difference equations fractional queues fractional birth-death processes busy period |
| url | http://www.mdpi.com/2227-7390/6/9/159 |
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