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Approximations in Performance Analysis of a Controllable Queueing System with Heterogeneous Servers

Author

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  • Dmitry Efrosinin

    (Insitute for Stochastics, Johannes Kepler University, 4040 Linz, Austria
    Department of Information Technologies, Faculty of Mathematics and Natural Sciences, Peoples’ Friendship University of Russia (RUDN University), 117198 Moscow, Russia)

  • Natalia Stepanova

    (Laboratory N17, Trapeznikov Institute of Control Sciences of RAS, 117997 Moscow, Russia)

  • Janos Sztrik

    (Department of Informatics and Networks, Faculty of Informatics, University of Debrecen, 4032 Debrecen, Hungary)

  • Andreas Plank

    (Insitute for Stochastics, Johannes Kepler University, 4040 Linz, Austria)

Abstract

The paper studies a controllable multi-server heterogeneous queueing system where servers operate at different service rates without preemption, i.e., the service times are uninterrupted. The optimal control policy allocates the customers between the servers in such a way that the mean number of customers in the system reaches its minimal value. The Markov decision model and the policy-iteration algorithm are used to calculate the optimal allocation policy and corresponding mean performance characteristics. The optimal policy, when neglecting the weak influence of slow servers, is of threshold type defined as a sequence of threshold levels which specifies the queue lengths for the usage of any slower server. To avoid time-consuming calculations for systems with a large number of servers, we focus here on a heuristic evaluation of the optimal thresholds and compare this solution with the real values. We develop also the simple lower and upper bound methods based on approximation by an equivalent heterogeneous queueing system with a preemption to measure the mean number of customers in the system operating under the optimal policy. Finally, the simulation technique is used to provide sensitivity analysis of the heuristic solution to changes in the form of inter-arrival and service time distributions.

Suggested Citation

  • Dmitry Efrosinin & Natalia Stepanova & Janos Sztrik & Andreas Plank, 2020. "Approximations in Performance Analysis of a Controllable Queueing System with Heterogeneous Servers," Mathematics, MDPI, vol. 8(10), pages 1-18, October.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:10:p:1803-:d:429103
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    References listed on IDEAS

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    1. Editorial Article, 0. "The Information for Authors," Economics of Contemporary Russia, Regional Public Organization for Assistance to the Development of Institutions of the Department of Economics of the Russian Academy of Sciences, issue 2.
    2. Editorial Article, 0. "The Information for Authors," Economics of Contemporary Russia, Regional Public Organization for Assistance to the Development of Institutions of the Department of Economics of the Russian Academy of Sciences, issue 4.
    3. Editorial Article, 0. "The Information for Authors," Economics of Contemporary Russia, Regional Public Organization for Assistance to the Development of Institutions of the Department of Economics of the Russian Academy of Sciences, issue 3.
    4. Editorial Article, 0. "The Information for Authors," Economics of Contemporary Russia, Regional Public Organization for Assistance to the Development of Institutions of the Department of Economics of the Russian Academy of Sciences, issue 2.
    5. Dmitry Efrosinin, 2013. "Queueing model of a hybrid channel with faster link subject to partial and complete failures," Annals of Operations Research, Springer, vol. 202(1), pages 75-102, January.
    6. Shalit, Haim, 1985. "Calculating the Gini Index of Inequality for Individual Data," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 47(2), pages 185-189, May.
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    9. Mor Armony & Amy R. Ward, 2010. "Fair Dynamic Routing in Large-Scale Heterogeneous-Server Systems," Operations Research, INFORMS, vol. 58(3), pages 624-637, June.
    10. Editorial Article, 0. "The Information for Authors," Economics of Contemporary Russia, Regional Public Organization for Assistance to the Development of Institutions of the Department of Economics of the Russian Academy of Sciences, issue 1.
    11. Laoucine Kerbache & F. S. Q. Alves & H. C. Yehia & L.A.C Pedrosa & F.R.B. Cruz, 2011. "Upper Bounds on Performance Measures of Heterogeneous M/M/c Queues," Post-Print hal-00609498, HAL.
    12. Editorial Article, 0. "The Information for Authors," Economics of Contemporary Russia, Regional Public Organization for Assistance to the Development of Institutions of the Department of Economics of the Russian Academy of Sciences, issue 4.
    13. Efrosinin, Dmitry & Sztrik, Janos, 2018. "An algorithmic approach to analysing the reliability of a controllable unreliable queue with two heterogeneous servers," European Journal of Operational Research, Elsevier, vol. 271(3), pages 934-952.
    14. Editorial Article, 0. "The Information for Authors," Economics of Contemporary Russia, Regional Public Organization for Assistance to the Development of Institutions of the Department of Economics of the Russian Academy of Sciences, issue 3.
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    Cited by:

    1. Isaac Grosof & Mor Harchol-Balter & Alan Scheller-Wolf, 2022. "WCFS: a new framework for analyzing multiserver systems," Queueing Systems: Theory and Applications, Springer, vol. 102(1), pages 143-174, October.
    2. Dmitry Efrosinin & Natalia Stepanova & Janos Sztrik, 2021. "Algorithmic Analysis of Finite-Source Multi-Server Heterogeneous Queueing Systems," Mathematics, MDPI, vol. 9(20), pages 1-24, October.
    3. Asmita Tamuli & Dhruba Das & Amit Choudhury, 2024. "Optimizing the Performance of Multi-server Heterogeneous Queueing Systems with Dynamic Customer Behaviour," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 86(2), pages 366-414, November.
    4. Nika Ivanova, 2023. "On Importance of Sensitivity Analysis on an Example of a k -out-of- n System," Mathematics, MDPI, vol. 11(5), pages 1-18, February.

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