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Sojourn Times in a Queueing System with Breakdowns and General Retrial Times

Author

Listed:
  • Ivan Atencia

    (Department of Applied Mathematics, University of Málaga, 29071 Málaga, Spain
    These authors contributed equally to this work.)

  • José Luis Galán-García

    (Department of Applied Mathematics, University of Málaga, 29071 Málaga, Spain
    These authors contributed equally to this work.)

Abstract

This paper centers on a discrete-time retrial queue where the server experiences breakdowns and repairs when arriving customers may opt to follow a discipline of a last-come, first-served (LCFS)-type or to join the orbit. We focused on the extensive analysis of the system, and we obtained the stationary distributions of the number of customers in the orbit and in the system by applying the generation function (GF). We provide the stochastic decomposition law and the application bounds for the proximity between the steady-state distributions for the queueing system under consideration and its corresponding standard system. We developed recursive formulae aimed at the calculation of the steady-state of the orbit and the system. We proved that our discrete-time system approximates M / G /1 with breakdowns and repairs. We analyzed the busy period of an auxiliary system, the objective of which was to study the customer’s delay. The stationary distribution of a customer’s sojourn in the orbit and in the system was the object of a thorough and complete study. Finally, we provide numerical examples that outline the effect of the parameters on several performance characteristics and a conclusions section resuming the main research contributions of the paper.

Suggested Citation

  • Ivan Atencia & José Luis Galán-García, 2021. "Sojourn Times in a Queueing System with Breakdowns and General Retrial Times," Mathematics, MDPI, vol. 9(22), pages 1-25, November.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:22:p:2882-:d:677879
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    References listed on IDEAS

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    1. J. Artalejo & J. Falin, 1994. "Stochastic decomposition for retrial queues," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 2(2), pages 329-342, December.
    2. S. W. Fuhrmann & Robert B. Cooper, 1985. "Stochastic Decompositions in the M / G /1 Queue with Generalized Vacations," Operations Research, INFORMS, vol. 33(5), pages 1117-1129, October.
    3. Harrison, Peter G. & Patel, Naresh M. & Pitel, Edwige, 2000. "Reliability modelling using G-queues," European Journal of Operational Research, Elsevier, vol. 126(2), pages 273-287, October.
    4. Artalejo, J. R., 2000. "G-networks: A versatile approach for work removal in queueing networks," European Journal of Operational Research, Elsevier, vol. 126(2), pages 233-249, October.
    5. Gelenbe, Erol & Labed, Ali, 1998. "G-networks with multiple classes of signals and positive customers," European Journal of Operational Research, Elsevier, vol. 108(2), pages 293-305, July.
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