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Fractional Erlang queues

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  • Ascione, Giacomo
  • Leonenko, Nikolai
  • Pirozzi, Enrica

Abstract

We introduce a fractional generalization of the Erlang Queues M∕Ek∕1. Such process is obtained through a time-change via inverse stable subordinator of the classical queue process. We first exploit the (fractional) Kolmogorov forward equation for such process, then we use such equation to obtain an interpretation of this process in the queuing theory context. Then we also exploit the transient state probabilities and some features of this fractional queue model, such as the mean queue length, the distribution of the busy periods and some conditional distributions of the waiting times. Finally, we provide some algorithms to simulate their sample paths.

Suggested Citation

  • Ascione, Giacomo & Leonenko, Nikolai & Pirozzi, Enrica, 2020. "Fractional Erlang queues," Stochastic Processes and their Applications, Elsevier, vol. 130(6), pages 3249-3276.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:6:p:3249-3276
    DOI: 10.1016/j.spa.2019.09.012
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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. Erhan Cinlar, 1974. "Markov Additive Processes and Semi-Regeneration," Discussion Papers 118, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    3. Donald P. Gaver, 1954. "The Influence of Servicing Times in Queuing Processes," Operations Research, INFORMS, vol. 2(2), pages 139-149, May.
    4. George Luchak, 1956. "The Solution of the Single-Channel Queuing Equations Characterized by a Time-Dependent Poisson-Distributed Arrival Rate and a General Class of Holding Times," Operations Research, INFORMS, vol. 4(6), pages 711-732, December.
    5. 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.
    6. Veillette, Mark & Taqqu, Murad S., 2010. "Using differential equations to obtain joint moments of first-passage times of increasing Lévy processes," Statistics & Probability Letters, Elsevier, vol. 80(7-8), pages 697-705, April.
    7. Changpin Li & Deliang Qian & YangQuan Chen, 2011. "On Riemann-Liouville and Caputo Derivatives," Discrete Dynamics in Nature and Society, Hindawi, vol. 2011, pages 1-15, March.
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    Cited by:

    1. 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.
    2. 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.

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