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A Double-ended Queue with Catastrophes and Repairs, and a Jump-diffusion Approximation

Author

Listed:
  • Antonio Crescenzo

    (Università di Salerno)

  • Virginia Giorno

    (Università di Salerno)

  • Balasubramanian Krishna Kumar

    (Anna University)

  • Amelia G. Nobile

    (Università di Salerno)

Abstract

Consider a system performing a continuous-time random walk on the integers, subject to catastrophes occurring at constant rate, and followed by exponentially-distributed repair times. After any repair the system starts anew from state zero. We study both the transient and steady-state probability laws of the stochastic process that describes the state of the system. We then derive a heavy-traffic approximation to the model that yields a jump-diffusion process. The latter is equivalent to a Wiener process subject to randomly occurring jumps, whose probability law is obtained. The goodness of the approximation is finally discussed.

Suggested Citation

  • Antonio Crescenzo & Virginia Giorno & Balasubramanian Krishna Kumar & Amelia G. Nobile, 2012. "A Double-ended Queue with Catastrophes and Repairs, and a Jump-diffusion Approximation," Methodology and Computing in Applied Probability, Springer, vol. 14(4), pages 937-954, December.
    • Handle: RePEc:spr:metcap:v:14:y:2012:i:4:d:10.1007_s11009-011-9214-2
    DOI: 10.1007/s11009-011-9214-2
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    References listed on IDEAS

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    1. Randall J. Swift, 2001. "Transient probabilities for a simple birth-death-immigration process under the influence of total catastrophes," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 25, pages 1-4, January.
    2. Economou, Antonis & Fakinos, Demetrios, 2003. "A continuous-time Markov chain under the influence of a regulating point process and applications in stochastic models with catastrophes," European Journal of Operational Research, Elsevier, vol. 149(3), pages 625-640, September.
    3. Di Crescenzo, A. & Giorno, V. & Nobile, A.G. & Ricciardi, L.M., 2008. "A note on birth-death processes with catastrophes," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2248-2257, October.
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    Cited by:

    1. Virginia Giorno & Amelia G. Nobile, 2020. "Bell Polynomial Approach for Time-Inhomogeneous Linear Birth–Death Process with Immigration," Mathematics, MDPI, vol. 8(7), pages 1-29, July.
    2. Heng-Li Liu & Quan-Lin Li, 2023. "Matched Queues with Flexible and Impatient Customers," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-26, March.
    3. Di Crescenzo, Antonio & Giorno, Virginia & Nobile, Amelia G., 2016. "Constructing transient birth–death processes by means of suitable transformations," Applied Mathematics and Computation, Elsevier, vol. 281(C), pages 152-171.
    4. Antonio Di Crescenzo & Virginia Giorno & Balasubramanian Krishna Kumar & Amelia G. Nobile, 2018. "A Time-Non-Homogeneous Double-Ended Queue with Failures and Repairs and Its Continuous Approximation," Mathematics, MDPI, vol. 6(5), pages 1-23, May.
    5. Shi, Ying & Lian, Zhaotong, 2016. "Optimization and strategic behavior in a passenger–taxi service system," European Journal of Operational Research, Elsevier, vol. 249(3), pages 1024-1032.
    6. Giorno, Virginia & Nobile, Amelia G., 2022. "On some integral equations for the evaluation of first-passage-time densities of time-inhomogeneous birth-death processes," Applied Mathematics and Computation, Elsevier, vol. 422(C).
    7. Giorno, Virginia & Nobile, Amelia G., 2020. "On a class of birth-death processes with time-varying intensity functions," Applied Mathematics and Computation, Elsevier, vol. 379(C).
    8. Chai, Xudong & Liu, Liwei & Chang, Baoxian & Jiang, Tao & Wang, Zhen, 2019. "On a batch matching system with impatient servers and boundedly rational customers," Applied Mathematics and Computation, Elsevier, vol. 354(C), pages 308-328.
    9. Ying Shi & Zhaotong Lian, 2016. "Equilibrium Strategies and Optimal Control for a Double-Ended Queue," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 33(03), pages 1-18, June.
    10. Hung Q. Nguyen & Tuan Phung-Duc, 2022. "Strategic customer behavior and optimal policies in a passenger–taxi double-ended queueing system with multiple access points and nonzero matching times," Queueing Systems: Theory and Applications, Springer, vol. 102(3), pages 481-508, December.
    11. Virginia Giorno & Amelia G. Nobile, 2021. "Time-Inhomogeneous Feller-Type Diffusion Process in Population Dynamics," Mathematics, MDPI, vol. 9(16), pages 1-29, August.
    12. Anna Sinitcina & Yacov Satin & Alexander Zeifman & Galina Shilova & Alexander Sipin & Ksenia Kiseleva & Tatyana Panfilova & Anastasia Kryukova & Irina Gudkova & Elena Fokicheva, 2018. "On the Bounds for a Two-Dimensional Birth-Death Process with Catastrophes," Mathematics, MDPI, vol. 6(5), pages 1-17, May.

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