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Transient Behavior of Fractional Queues and Related Processes

Author

Listed:
  • Dexter O. Cahoy

    (Louisiana Tech University)

  • Federico Polito

    (University of Torino)

  • Vir Phoha

    (Louisiana Tech University)

Abstract

We propose a generalization of the classical M/M/1 queue process. The resulting model is derived by applying fractional derivative operators to a system of difference-differential equations. This generalization includes both non-Markovian and Markovian properties which naturally provide greater flexibility in modeling real queue systems than its classical counterpart. Algorithms to simulate M/M/1 queue process and the related linear birth-death process are provided. Closed-form expressions of the point and interval estimators of the parameters of the proposed fractional stochastic models are also presented. These methods are necessary to make these models usable in practice. The proposed fractional M/M/1 queue model and the statistical methods are illustrated using financial data.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metcap:v:17:y:2015:i:3:d:10.1007_s11009-013-9391-2
    DOI: 10.1007/s11009-013-9391-2
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    References listed on IDEAS

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    1. Mark Veillette & Murad S. Taqqu, 2010. "Numerical Computation of First-Passage Times of Increasing Lévy Processes," Methodology and Computing in Applied Probability, Springer, vol. 12(4), pages 695-729, December.
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    Cited by:

    1. Giacomo Ascione & Nikolai Leonenko & Enrica Pirozzi, 2018. "Fractional Queues with Catastrophes and Their Transient Behaviour," Mathematics, MDPI, vol. 6(9), pages 1-26, September.
    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.
    3. Ascione, Giacomo & Leonenko, Nikolai & Pirozzi, Enrica, 2020. "Fractional Erlang queues," Stochastic Processes and their Applications, Elsevier, vol. 130(6), pages 3249-3276.
    4. Souza, Matheus de Oliveira & Rodriguez, Pablo M., 2021. "On a fractional queueing model with catastrophes," Applied Mathematics and Computation, Elsevier, vol. 410(C).
    5. Kulmus, Kathrin & Essex, Christopher & Prehl, Janett & Hoffmann, Karl Heinz, 2019. "The entropy production paradox for fractional master equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 525(C), pages 1370-1378.
    6. Giacomo Ascione & Bruno Toaldo, 2019. "A Semi-Markov Leaky Integrate-and-Fire Model," Mathematics, MDPI, vol. 7(11), pages 1-24, October.

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