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Retrial queues with constant retrial times

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  • Dieter Fiems

    (Ghent University)

Abstract

We consider the M/D/1 retrial queueing system with constant retrial times, which makes up a natural abstraction for optical fibre delay line buffers. Drawing on a time-discretisation approach and on an equivalence with polling systems, we find explicit expressions for the distribution of the number of retrials, and the probability generating function of the number of customers in orbit. While the state space of the queueing system at hand is complicated, the results are strikingly simple. The number of retrials follows a geometric distribution, while the orbit size decomposes into two independent random variables: the system content of the M/D/1 queue at departure times and the orbit size of the M/D/1 retrial queue when the server is idle. We finally obtain explicit expressions for the retrial rate after a departure and for the distribution of the time until the nth retrial after a departure.

Suggested Citation

  • Dieter Fiems, 2023. "Retrial queues with constant retrial times," Queueing Systems: Theory and Applications, Springer, vol. 103(3), pages 347-365, April.
    • Handle: RePEc:spr:queues:v:103:y:2023:i:3:d:10.1007_s11134-022-09866-4
    DOI: 10.1007/s11134-022-09866-4
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    References listed on IDEAS

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    1. Veeraruna Kavitha & Eitan Altman, 2012. "Continuous polling models and application to ferry assisted WLAN," Annals of Operations Research, Springer, vol. 198(1), pages 185-218, September.
    2. Atencia, I., 2017. "A Geo/G/1 retrial queueing system with priority services," European Journal of Operational Research, Elsevier, vol. 256(1), pages 178-186.
    3. Dieter Fiems, 2022. "Retrial queues with generally distributed retrial times," Queueing Systems: Theory and Applications, Springer, vol. 100(3), pages 189-191, April.
    4. Yang, T. & Posner, M. J. M. & Templeton, J. G. C. & Li, H., 1994. "An approximation method for the M/G/1 retrial queue with general retrial times," European Journal of Operational Research, Elsevier, vol. 76(3), pages 552-562, August.
    5. Bin Liu & Yiqiang Q. Zhao, 2020. "Tail asymptotics for the $$M_1,M_2/G_1,G_2/1$$ M 1 , M 2 / G 1 , G 2 / 1 retrial queue with non-preemptive priority," Queueing Systems: Theory and Applications, Springer, vol. 96(1), pages 169-199, October.
    6. Shin, Yang Woo & Moon, Dug Hee, 2011. "Approximation of M/M/c retrial queue with PH-retrial times," European Journal of Operational Research, Elsevier, vol. 213(1), pages 205-209, August.
    7. Pourbabai, Behnam, 1993. "Tandem behavior of a telecommunication system with repeated calls: II, A general case without buffers," European Journal of Operational Research, Elsevier, vol. 65(2), pages 247-258, March.
    8. Jeongsim Kim & Bara Kim, 2016. "A survey of retrial queueing systems," Annals of Operations Research, Springer, vol. 247(1), pages 3-36, December.
    9. Rein Nobel, 2016. "Retrial queueing models in discrete time: a short survey of some late arrival models," Annals of Operations Research, Springer, vol. 247(1), pages 37-63, December.
    10. Kernane, Tewfik, 2008. "Conditions for stability and instability of retrial queueing systems with general retrial times," Statistics & Probability Letters, Elsevier, vol. 78(18), pages 3244-3248, December.
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    Cited by:

    1. Wei Xu & Liwei Liu & Linhong Li & Zhen Wang & Sabine Wittevrongel, 2023. "Analysis of a Collision-Affected M/GI/1/ /N Retrial Queuing System Considering Negative Customers and Transmission Errors," Mathematics, MDPI, vol. 11(16), pages 1-23, August.

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