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Asymptotic sojourn time analysis of finite-source M/M/1 retrial queueing system with collisions and server subject to breakdowns and repairs

Author

Listed:
  • Anatoly Nazarov

    (National Research Tomsk State University)

  • János Sztrik

    (University of Debrecen)

  • Anna Kvach

    (National Research Tomsk State University)

  • Ádám Tóth

    (University of Debrecen)

Abstract

The aim of the present paper is to investigate the steady-state distribution of response and waiting time in a finite-source M / M / 1 retrial queuing system with collision of customers where the server is subjects to random breakdowns and repairs depending on whether it is idle or busy. An asymptotic method is applied under the condition that the number of sources tends to infinity, the primary request generation rate, retrial rate tend to zero while service rate, failure rates, repair rate are fixed. As the result of the analysis it is shown that the steady-state probability distribution of the number of transitions/retrials of the customer into the orbit is geometric with a given parameter, and the normalized sojourn time of the customer in the system follows a generalized exponential distribution. It is also proved that the limiting distributions of the normalized sojourn time of the customer in the system and the normalized sojourn/waiting time of the customer in the orbit coincide. The novelty of this investigation is the introduction of failure and repair of the server. Approximations of prelimit distributions obtained with the help of stochastic simulation by asymptotic one are considered and several illustrative examples show the accuracy and range of applicability of the proposed asymptotic method.

Suggested Citation

  • Anatoly Nazarov & János Sztrik & Anna Kvach & Ádám Tóth, 2020. "Asymptotic sojourn time analysis of finite-source M/M/1 retrial queueing system with collisions and server subject to breakdowns and repairs," Annals of Operations Research, Springer, vol. 288(1), pages 417-434, May.
    • Handle: RePEc:spr:annopr:v:288:y:2020:i:1:d:10.1007_s10479-019-03463-0
    DOI: 10.1007/s10479-019-03463-0
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    References listed on IDEAS

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    1. Jeongsim Kim & Bara Kim, 2016. "A survey of retrial queueing systems," Annals of Operations Research, Springer, vol. 247(1), pages 3-36, December.
    2. Antonio Gómez-Corral & Tuan Phung-Duc, 2016. "Retrial queues and related models," Annals of Operations Research, Springer, vol. 247(1), pages 1-2, December.
    3. Velika I. Dragieva, 2014. "Number Of Retrials In A Finite Source Retrial Queue With Unreliable Server," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 31(02), pages 1-23.
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    5. Velika I. Dragieva, 2016. "Steady state analysis of the M/G/1//N queue with orbit of blocked customers," Annals of Operations Research, Springer, vol. 247(1), pages 121-140, December.
    6. Tien V. Do & Patrick Wüchner & Tamás Bérczes & János Sztrik & Hermann De Meer, 2014. "A New Finite-Source Queueing Model For Mobile Cellular Networks Applying Spectrum Renting," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 31(02), pages 1-19.
    7. Falin, G. I. & Artalejo, J. R., 1998. "A finite source retrial queue," European Journal of Operational Research, Elsevier, vol. 108(2), pages 409-424, July.
    8. Lyes Ikhlef & Ouiza Lekadir & Djamil Aïssani, 2016. "MRSPN analysis of Semi-Markovian finite source retrial queues," Annals of Operations Research, Springer, vol. 247(1), pages 141-167, December.
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    Cited by:

    1. Agassi Melikov & Sevinj Aliyeva & Janos Sztrik, 2021. "Retrial Queues with Unreliable Servers and Delayed Feedback," Mathematics, MDPI, vol. 9(19), pages 1-23, September.
    2. V., Saravanan & V., Poongothai & P., Godhandaraman, 2023. "Performance analysis of a multi server retrial queueing system with unreliable server, discouragement and vacation model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 214(C), pages 204-226.
    3. Wang, Kuo-Hsiung & Wu, Chia-Huang & Yen, Tseng-Chang, 2022. "Comparative cost-benefit analysis of four retrial systems with preventive maintenance and unreliable service station," Reliability Engineering and System Safety, Elsevier, vol. 221(C).
    4. Wu, Chia-Huang & Yen, Tseng-Chang & Wang, Kuo-Hsiung, 2021. "Availability and Comparison of Four Retrial Systems with Imperfect Coverage and General Repair Times," Reliability Engineering and System Safety, Elsevier, vol. 212(C).

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