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The BMAP/PH/N retrial queue with Markovian flow of breakdowns

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  • Kim, Chesoong
  • Klimenok, Valentina I.
  • Orlovsky, Dmitry S.

Abstract

We consider a multi-server retrial queue with the Batch Markovian Arrival Process (BMAP). The servers are identical and independent of each other. The service time distribution of a customer by a server is of the phase (PH) type. If a group of primary calls meets idle servers the primary calls occupy the corresponding number of servers. If the number of idle servers is insufficient the rest of calls go to the orbit of unlimited size and repeat their attempts to get service after exponential amount of time independently of each other. Busy servers are subject to breakdowns and repairs. The common flow of breakdowns is the MAP. An event of this flow causes a failure of any busy server with equal probability. When a server fails the repair period starts immediately. This period has PH type distribution and does not depend on the repair time of other broken-down servers and the service time of customers occupying the working servers. A customer whose service was interrupted goes to the orbit with some probability and leaves the system with the supplementary probability. We derive the ergodicity condition and calculate the stationary distribution and the main performance characteristics of the system. Illustrative numerical examples are presented.

Suggested Citation

  • Kim, Chesoong & Klimenok, Valentina I. & Orlovsky, Dmitry S., 2008. "The BMAP/PH/N retrial queue with Markovian flow of breakdowns," European Journal of Operational Research, Elsevier, vol. 189(3), pages 1057-1072, September.
    • Handle: RePEc:eee:ejores:v:189:y:2008:i:3:p:1057-1072
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    References listed on IDEAS

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    1. A. Gómez-Corral, 2006. "A bibliographical guide to the analysis of retrial queues through matrix analytic techniques," Annals of Operations Research, Springer, vol. 141(1), pages 163-191, January.
    2. Quan-Lin Li & Yu Ying & Yiqiang Zhao, 2006. "A BMAP/G/1 Retrial Queue with a Server Subject to Breakdowns and Repairs," Annals of Operations Research, Springer, vol. 141(1), pages 233-270, January.
    3. J. Artalejo, 1999. "A classified bibliography of research on retrial queues: Progress in 1990–1999," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 7(2), pages 187-211, December.
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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. Joanna Rodríguez & Rosa E. Lillo & Pepa Ramírez-Cobo, 2016. "Nonidentifiability of the Two-State BMAP," Methodology and Computing in Applied Probability, Springer, vol. 18(1), pages 81-106, March.
    3. 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.
    4. Falin, G.I., 2010. "A single-server batch arrival queue with returning customers," European Journal of Operational Research, Elsevier, vol. 201(3), pages 786-790, March.
    5. 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.
    6. 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.

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