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A Markovian Queue with N Servers Subject to Breakdowns and Repairs

Author

Listed:
  • Marcel F. Neuts

    (University of Delaware)

  • David M. Lucantoni

    (University of Delaware)

Abstract

We study a queue with N servers, who may break down and require repair at a facility, which has c repair crews. Under exponential assumptions, this model has an algorithmically tractable solution. It is then in fact a particular case of the M/M/N queue in a Markovian environment. The purpose of this paper is twofold. As a novel methodological contribution, the stationary distributions of various waiting times are discussed. Although fairly involved, these distributions may be computed by classical numerical methods. The second and primary purpose is to demonstrate the utility of interactive computation in answering questions on the behavior, design and control of certain service systems. By numerical examples, we shall show that during periods when most servers are down, large build-ups may occur which affect the queue adversely for a long time afterwards. We also find that such build-ups are aggravated by reducing the number of repair crews, but may be attenuated by reducing the arrival rates during periods when the service has deteriorated. Potential applications are in manpower planning, as in a typing pool where persons may be absent and in determining the size of a battery of machines, where machines may be inoperative due to maintenance and repair. More generally, this paper draws attention to qualitative behavior of queues with randomly varying service rates. Such queues may exhibit substantial and long-lived fluctuations, which are not always apparent from simulation runs and which are also typically not present in the more classical queueing models.

Suggested Citation

  • Marcel F. Neuts & David M. Lucantoni, 1979. "A Markovian Queue with N Servers Subject to Breakdowns and Repairs," Management Science, INFORMS, vol. 25(9), pages 849-861, September.
    • Handle: RePEc:inm:ormnsc:v:25:y:1979:i:9:p:849-861
    DOI: 10.1287/mnsc.25.9.849
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    Citations

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    Cited by:

    1. A. Krishnamoorthy & P. Pramod & S. Chakravarthy, 2014. "Queues with interruptions: a survey," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(1), pages 290-320, April.
    2. 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.
    3. Dudin, Alexander & Kim, Chesoong & Dudin, Sergey & Dudina, Olga, 2015. "Priority retrial queueing model operating in random environment with varying number and reservation of servers," Applied Mathematics and Computation, Elsevier, vol. 269(C), pages 674-690.
    4. Liu, Gia-Shie, 2011. "Dynamic group instantaneous replacement policies for unreliable Markovian service systems," International Journal of Production Economics, Elsevier, vol. 130(2), pages 203-217, April.
    5. Nadav Lavi & Hanoch Levy, 2020. "Admit or preserve? Addressing server failures in cloud computing task management," Queueing Systems: Theory and Applications, Springer, vol. 94(3), pages 279-325, April.
    6. Gia-Shie Liu, 2019. "A Group Replacement Decision Support System Based on Internet of Things," Mathematics, MDPI, vol. 7(9), pages 1-23, September.
    7. 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.
    8. Lam, Yeh & Zhang, Yuan Lin & Liu, Qun, 2006. "A geometric process model for M/M/1 queueing system with a repairable service station," European Journal of Operational Research, Elsevier, vol. 168(1), pages 100-121, January.
    9. 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.
    10. Miaomiao Yu & Yinghui Tang, 2022. "Analysis of a renewal batch arrival queue with a fault-tolerant server using shift operator method," Operational Research, Springer, vol. 22(3), pages 2831-2858, July.
    11. Freek Verdonck & Herwig Bruneel & Sabine Wittevrongel, 2021. "Delay in a 2-State Discrete-Time Queue with Stochastic State-Period Lengths and State-Dependent Server Availability and Arrivals," Mathematics, MDPI, vol. 9(14), pages 1-17, July.
    12. 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.
    13. Pedram Sahba & Bariş Balciog̃lu & Dragan Banjevic, 2013. "Analysis of the finite‐source multiclass priority queue with an unreliable server and setup time," Naval Research Logistics (NRL), John Wiley & Sons, vol. 60(4), pages 331-342, June.
    14. Dijk, N.M. van & Trapman, F.J.J., 1989. "Exact solutions for central service systems with breakdowns," Serie Research Memoranda 0028, VU University Amsterdam, Faculty of Economics, Business Administration and Econometrics.
    15. Mohammad Firouz & Linda Li & Burcu B. Keskin, 2022. "Managing equipment rentals: Unreliable fleet, impatient customers, and finite commitment capacity," Production and Operations Management, Production and Operations Management Society, vol. 31(11), pages 3963-3981, November.
    16. 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.
    17. Wartenhorst, Pieter, 1995. "N parallel queueing systems with server breakdown and repair," European Journal of Operational Research, Elsevier, vol. 82(2), pages 302-322, April.
    18. Pedram Sahba & Barış Balcıog̃lu & Dragan Banjevic, 2022. "The impact of disruption characteristics on the performance of a server," Annals of Operations Research, Springer, vol. 317(1), pages 239-252, October.

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