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A Two-Server Queue with Interdependence between Arrival and Service Processes

Author

Listed:
  • Sindhu S

    (Department of Mathematics, Model Engineering College, Ernakulam 682021, India
    Department of Mathematics, Cochin University of Science and Technology, Ernakulam 682022, India)

  • Achyutha Krishnamoorthy

    (Centre for Research in Mathematics, CMS College, Kottayam 686001, India
    Department of Mathematics, Central University of Kerala, Kasargod 671316, India)

  • Dmitry Kozyrev

    (V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, 65 Profsoyuznaya Street, 117997 Moscow, Russia
    Probability Theory and Cyber Security Department, Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya Street, 117198 Moscow, Russia)

Abstract

In this paper, we analyse a queueing system with two servers where the arrival and service processes are interdependent. The evolution of these processes is governed by transitions on the product space of three Markov chains, which are descriptors of the arrival and service processes. The transitions in this Markov chain follow a semi-Markov rule and the exponential distribution governs the sojourn times in the states. The stability condition of the system is derived and the stationary distribution is calculated for the system in equilibrium. Several important performance measures are provided, and numerical illustrations of the model are presented.

Suggested Citation

  • Sindhu S & Achyutha Krishnamoorthy & Dmitry Kozyrev, 2023. "A Two-Server Queue with Interdependence between Arrival and Service Processes," Mathematics, MDPI, vol. 11(22), pages 1-25, November.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:22:p:4692-:d:1283038
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    References listed on IDEAS

    as
    1. S Sindhu & Achyutha Krishnamoorthy & Dmitry Kozyrev, 2023. "On Queues with Working Vacation and Interdependence in Arrival and Service Processes," Mathematics, MDPI, vol. 11(10), pages 1-16, May.
    2. Boxma, O. J. & Perry, D., 2001. "A queueing model with dependence between service and interarrival times," European Journal of Operational Research, Elsevier, vol. 128(3), pages 611-624, February.
    3. Peter Buchholz & Jan Kriege, 2017. "Fitting correlated arrival and service times and related queueing performance," Queueing Systems: Theory and Applications, Springer, vol. 85(3), pages 337-359, April.
    4. Zhao, Chen & Wang, Zhongbin, 2023. "The impact of line-sitting on a two-server queueing system," European Journal of Operational Research, Elsevier, vol. 308(2), pages 782-800.
    5. Tian, Naishuo & Zhang, Zhe George, 2006. "A two threshold vacation policy in multiserver queueing systems," European Journal of Operational Research, Elsevier, vol. 168(1), pages 153-163, January.
    6. Konstantin Samouylov & Olga Dudina & Alexander Dudin, 2023. "Analysis of Multi-Server Queueing System with Flexible Priorities," Mathematics, MDPI, vol. 11(4), pages 1-22, February.
    7. Weimin Dai & Jian-Qiang Hu, 2022. "Correlated queues with service times depending on inter-arrival times," Queueing Systems: Theory and Applications, Springer, vol. 100(1), pages 41-60, February.
    8. Alexander Dudin & Sergei Dudin & Olga Dudina, 2023. "Analysis of a Queueing System with Mixed Service Discipline," Methodology and Computing in Applied Probability, Springer, vol. 25(2), pages 1-19, June.
    9. C. R. Mitchell & A. S. Paulson, 1979. "M/M/1 queues with interdependent arrival and service processes," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 26(1), pages 47-56, March.
    10. Vlasiou, M. & Adan, I.J.B.F. & Boxma, O.J., 2009. "A two-station queue with dependent preparation and service times," European Journal of Operational Research, Elsevier, vol. 195(1), pages 104-116, May.
    11. Alexander Moiseev & Maria Shklennik & Evgeny Polin, 2023. "Infinite-server queueing tandem with Markovian arrival process and service depending on its state," Annals of Operations Research, Springer, vol. 326(1), pages 261-279, July.
    12. Amina Angelika Bouchentouf & Aicha Messabihi, 2018. "Heterogeneous two-server queueing system with reverse balking and reneging," OPSEARCH, Springer;Operational Research Society of India, vol. 55(2), pages 251-267, June.
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