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Fractional Queues with Catastrophes and Their Transient Behaviour

Author

Listed:
  • 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)

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.

Suggested Citation

  • Giacomo Ascione & Nikolai Leonenko & Enrica Pirozzi, 2018. "Fractional Queues with Catastrophes and Their Transient Behaviour," Mathematics, MDPI, vol. 6(9), pages 1-26, September.
    • Handle: RePEc:gam:jmathe:v:6:y:2018:i:9:p:159-:d:168128
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    References listed on IDEAS

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    1. Dexter O. Cahoy & Federico Polito & Vir Phoha, 2015. "Transient Behavior of Fractional Queues and Related Processes," Methodology and Computing in Applied Probability, Springer, vol. 17(3), pages 739-759, September.
    2. H. J. Haubold & A. M. Mathai & R. K. Saxena, 2011. "Mittag-Leffler Functions and Their Applications," Journal of Applied Mathematics, Hindawi, vol. 2011, pages 1-51, May.
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    Cited by:

    1. Giacomo Ascione & Nikolai Leonenko & Enrica Pirozzi, 2022. "Non-local Solvable Birth–Death Processes," Journal of Theoretical Probability, Springer, vol. 35(2), pages 1284-1323, June.
    2. Enrica Pirozzi, 2022. "On a Fractional Stochastic Risk Model with a Random Initial Surplus and a Multi-Layer Strategy," Mathematics, MDPI, vol. 10(4), pages 1-18, February.
    3. Nicos Georgiou & Enrico Scalas & Vladislav Vysotsky, 2026. "A Restless Time-Fractional Multiclass Queue," Journal of Theoretical Probability, Springer, vol. 39(2), pages 1-31, June.

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