IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i12p2669-d1169325.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/12/2669/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/11/12/2669/
    Download Restriction: no
    ---><---

    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. 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.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. Valentina Klimenok & Alexander Dudin & Vladimir Vishnevsky, 2020. "Priority Multi-Server Queueing System with Heterogeneous Customers," Mathematics, MDPI, vol. 8(9), pages 1-16, September.
    9. 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.
    10. 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.
    11. 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.
    12. 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.
    13. 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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. Alexander Dudin & Olga Dudina & Sergei Dudin & Konstantin Samouylov, 2021. "Analysis of Multi-Server Queue with Self-Sustained Servers," Mathematics, MDPI, vol. 9(17), pages 1-18, September.
    3. Sally McClean, 2021. "Using Markov Models to Characterize and Predict Process Target Compliance," Mathematics, MDPI, vol. 9(11), pages 1-12, May.
    4. Vladimir Vishnevsky & Valentina Klimenok & Alexander Sokolov & Andrey Larionov, 2021. "Performance Evaluation of the Priority Multi-Server System MMAP/PH/M/N Using Machine Learning Methods," Mathematics, MDPI, vol. 9(24), pages 1-27, December.
    5. Samira Taleb & Amar Aissani, 2016. "Preventive maintenance in an unreliable M/G/1 retrial queue with persistent and impatient customers," Annals of Operations Research, Springer, vol. 247(1), pages 291-317, December.
    6. K. R. Rejitha & K. P. Jose, 2018. "A stochastic inventory system with two modes of service and retrial of customers," OPSEARCH, Springer;Operational Research Society of India, vol. 55(1), pages 134-149, March.
    7. Janhavi Prabhu & Myron Hlynka, 2025. "A Queueing Model With Servers Disguised as Customers," International Journal of Statistics and Probability, Canadian Center of Science and Education, vol. 13(2), pages 1-55, January.
    8. Lidiya P & K Julia Rose Mary, 2024. "A study on the performance of a queuing system with heterogeneous arrivals and various types of breakdowns under multiple working vacations," Operations Research and Decisions, Wroclaw University of Science and Technology, Faculty of Management, vol. 34(4), pages 125-140.
    9. Pala, Ali & Zhuang, Jun, 2018. "Security screening queues with impatient applicants: A new model with a case study," European Journal of Operational Research, Elsevier, vol. 265(3), pages 919-930.
    10. Sergei A. Dudin & Olga S. Dudina & Olga I. Kostyukova, 2023. "Analysis of a Queuing System with Possibility of Waiting Customers Jockeying between Two Groups of Servers," Mathematics, MDPI, vol. 11(6), pages 1-21, March.
    11. Herwig Bruneel & Arnaud Devos, 2024. "Explicit Solutions for Coupled Parallel Queues," Mathematics, MDPI, vol. 12(15), pages 1-31, July.
    12. 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.
    13. Alexander Dudin & Olga Dudina & Sergei Dudin & Konstantin Samouylov, 2021. "Analysis of Single-Server Multi-Class Queue with Unreliable Service, Batch Correlated Arrivals, Customers Impatience, and Dynamical Change of Priorities," Mathematics, MDPI, vol. 9(11), pages 1-17, May.
    14. Dmitry Efrosinin & Vladimir Vishnevsky & Natalia Stepanova & Janos Sztrik, 2025. "Use Cases of Machine Learning in Queueing Theory Based on a GI / G / K System," Mathematics, MDPI, vol. 13(5), pages 1-36, February.
    15. Dudin, A.N. & Dudin, S.A. & Dudina, O.S. & Samouylov, K.E., 2018. "Analysis of queueing model with processor sharing discipline and customers impatience," Operations Research Perspectives, Elsevier, vol. 5(C), pages 245-255.
    16. Chakravarthy, Srinivas R. & Shruti, & Kulshrestha, Rakhee, 2020. "A queueing model with server breakdowns, repairs, vacations, and backup server," Operations Research Perspectives, Elsevier, vol. 7(C).
    17. V. Vinitha & N. Anbazhagan & S. Amutha & K. Jeganathan & Bhanu Shrestha & Hyoung-Kyu Song & Gyanendra Prasad Joshi & Hyeonjoon Moon, 2022. "Analysis of a Stochastic Inventory Model on Random Environment with Two Classes of Suppliers and Impulse Customers," Mathematics, MDPI, vol. 10(13), pages 1-18, June.
    18. Gabi Hanukov, 2023. "A queueing-inventory model with skeptical and trusting customers," Annals of Operations Research, Springer, vol. 331(2), pages 763-786, December.
    19. Alka Choudhary & Srinivas R. Chakravarthy & Dinesh C. Sharma, 2023. "Impact of the degradation in service rate in MAP/PH/1 queueing system with phase type vacations, breakdowns, and repairs," Annals of Operations Research, Springer, vol. 331(2), pages 1207-1248, December.
    20. Mario Lefebvre, 2024. "A Controlled Discrete-Time Queueing System as a Model for the Orders of Two Competing Companies," Games, MDPI, vol. 15(3), pages 1-8, May.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:11:y:2023:i:12:p:2669-:d:1169325. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.