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Randomized Threshold Strategy for Providing Flexible Priority in Multi-Server Queueing System with a Marked Markov Arrival Process and Phase-Type Distribution of Service Time

Author

Listed:
  • A. N. Dudin

    (Department of Applied Mathematics and Computer Science, Belarusian State University, 4, Nezavisimosti Ave., 220030 Minsk, Belarus)

  • S. A. Dudin

    (Department of Applied Mathematics and Computer Science, Belarusian State University, 4, Nezavisimosti Ave., 220030 Minsk, Belarus)

  • O. S. Dudina

    (Department of Applied Mathematics and Computer Science, Belarusian State University, 4, Nezavisimosti Ave., 220030 Minsk, Belarus)

Abstract

In this paper, we analyze a multi-server queueing system with a marked Markov arrival process of two types of customers and a phase-type distribution of service time depending on the type of customer. Customers of both types are assumed to be impatient and renege from the buffers after an exponentially distributed number of times. The strategy of flexible provisioning of priorities is analyzed. It assumes a randomized choice of the customers from the buffers, with probabilities dependent on the relation between the number of customers in a priority finite buffer and the fixed threshold value. To simplify the construction of the underlying Markov chain and the derivation of the explicit form of its generator, we use the so-called generalized phase-type distribution. It is shown that the created Markov chain fits the category of asymptotically quasi-Toeplitz Markov chains. Using this fact, we show that the considered Markov chain is ergodic for any value of the system parameters and compute its stationary distribution. Expressions for key performance measures are presented. Numerical results that show how the parameters of the control strategy affect the system’s performance measurements are given. It is shown that the results can be used for managerial purposes and that it is crucial to take correlation in the arrival process into account.

Suggested Citation

  • A. N. Dudin & S. A. Dudin & O. S. Dudina, 2023. "Randomized Threshold Strategy for Providing Flexible Priority in Multi-Server Queueing System with a Marked Markov Arrival Process and Phase-Type Distribution of Service Time," Mathematics, MDPI, vol. 11(12), pages 1-23, June.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:12:p:2669-:d:1169325
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    References listed on IDEAS

    as
    1. Yacov Satin & Rostislav Razumchik & Ivan Kovalev & Alexander Zeifman, 2023. "Ergodicity and Related Bounds for One Particular Class of Markovian Time—Varying Queues with Heterogeneous Servers and Customer’s Impatience," Mathematics, MDPI, vol. 11(9), pages 1-15, April.
    2. Joris Walraevens & Thomas Giel & Stijn Vuyst & Sabine Wittevrongel, 2022. "Asymptotics of waiting time distributions in the accumulating priority queue," Queueing Systems: Theory and Applications, Springer, vol. 101(3), pages 221-244, August.
    3. Che Kim & Vilena Mushko & Alexander Dudin, 2012. "Computation of the steady state distribution for multi-server retrial queues with phase type service process," Annals of Operations Research, Springer, vol. 201(1), pages 307-323, December.
    4. Sergei Dudin & Olga Dudina & Konstantin Samouylov & Alexander Dudin, 2020. "Improvement of the Fairness of Non-Preemptive Priorities in the Transmission of Heterogeneous Traffic," Mathematics, MDPI, vol. 8(6), pages 1-17, June.
    5. Alexander Dudin & Sergei Dudin, 2016. "Analysis of a Priority Queue with Phase-Type Service and Failures," International Journal of Stochastic Analysis, Hindawi, vol. 2016, pages 1-11, July.
    6. Alexander Dudin & Chesoong Kim & Olga Dudina & Sergey Dudin, 2016. "Multi-server queueing system with a generalized phase-type service time distribution as a model of call center with a call-back option," Annals of Operations Research, Springer, vol. 239(2), pages 401-428, April.
    7. 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.
    8. 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.
    9. Kim, Chesoong & Dudin, Alexander & Dudina, Olga & Dudin, Sergey, 2014. "Tandem queueing system with infinite and finite intermediate buffers and generalized phase-type service time distribution," European Journal of Operational Research, Elsevier, vol. 235(1), pages 170-179.
    10. Valentina Klimenok & Alexander Dudin & Vladimir Vishnevsky, 2020. "Priority Multi-Server Queueing System with Heterogeneous Customers," Mathematics, MDPI, vol. 8(9), pages 1-16, September.
    11. Seokjun Lee & Sergei Dudin & Olga Dudina & Chesoong Kim & Valentina Klimenok, 2020. "A Priority Queue with Many Customer Types, Correlated Arrivals and Changing Priorities," Mathematics, MDPI, vol. 8(8), pages 1-20, August.
    12. Joris Walraevens, 2022. "Asymptotics in priority queues: from finite to infinite capacities," Queueing Systems: Theory and Applications, Springer, vol. 100(3), pages 361-363, April.
    13. Kim, Chesoong & Klimenok, V.I. & Dudin, A.N., 2017. "Analysis of unreliable BMAP/PH/N type queue with Markovian flow of breakdowns," Applied Mathematics and Computation, Elsevier, vol. 314(C), pages 154-172.
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