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Priority Multi-Server Queueing System with Heterogeneous Customers

Author

Listed:
  • Valentina Klimenok

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

  • Alexander Dudin

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

  • Vladimir Vishnevsky

    (Sciences and Closed Corporation Information and Networking Technologies, Institute of Control Sciences of Russian Academy, 119991 Moscow, Russia)

Abstract

In this paper, we analyze a multi-server queueing system with heterogeneous customers that arrive according to a marked Markovian arrival process. Customers of two types differ in priorities and parameters of phase type distribution of their service time. The queue under consideration can be used to model the processes of information transmission in telecommunication networks in which often the flow of information is the superposition of several types of flows with correlation of inter-arrival times within each flow and cross-correlation. We define the process of information transmission as the multi-dimensional Markov chain, derive the generator of this chain and compute its stationary distribution. Expressions for computation of various performance measures of the system, including the probabilities of loss of customers of different types, are presented. Output flow from the system is characterized. The presented numerical results confirm the high importance of account of correlation in the arrival process. The values of important performance measures for the systems with the correlated arrival process are essentially different from the corresponding values for the systems with the stationary Poisson arrival process. Measurements in many real world systems show poor approximation of real flows by such an arrival process. However, this process is still popular among the telecommunication engineers due to the evident existing gap between the needs of adequately modeling the real life systems and the current state of the theory of algorithmic methods of queueing theory.

Suggested Citation

  • Valentina Klimenok & Alexander Dudin & Vladimir Vishnevsky, 2020. "Priority Multi-Server Queueing System with Heterogeneous Customers," Mathematics, MDPI, vol. 8(9), pages 1-16, September.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:9:p:1501-:d:408828
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    References listed on IDEAS

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    1. Qi‐Ming He & Jingui Xie & Xiaobo Zhao, 2012. "Priority queue with customer upgrades," Naval Research Logistics (NRL), John Wiley & Sons, vol. 59(5), pages 362-375, August.
    2. Bong Dae Choi & Gang Uk Hwang, 1997. "The MAP, M / G 1 , G 2 / 1 queue with preemptive priority," International Journal of Stochastic Analysis, Hindawi, vol. 10, pages 1-15, January.
    3. Di Lin & Jonathan Patrick & Fabrice Labeau, 2014. "Estimating the waiting time of multi-priority emergency patients with downstream blocking," Health Care Management Science, Springer, vol. 17(1), pages 88-99, March.
    4. Bin Sun & Moon Ho Lee & Sergey A. Dudin & Alexander N. Dudin, 2014. "Analysis of Multiserver Queueing System with Opportunistic Occupation and Reservation of Servers," Mathematical Problems in Engineering, Hindawi, vol. 2014, pages 1-13, May.
    5. de Lange, Robert & Samoilovich, Ilya & van der Rhee, Bo, 2013. "Virtual queuing at airport security lanes," European Journal of Operational Research, Elsevier, vol. 225(1), pages 153-165.
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    Cited by:

    1. 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.
    2. 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.
    3. 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.

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