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Sufficient stability conditions for multi-class constant retrial rate systems

Author

Listed:
  • K. Avrachenkov

    (Inria Sophia Antipolis)

  • E. Morozov

    (Karelian Research Centre RAS, and Petrozavodsk State University)

  • B. Steyaert

    (Ghent University)

Abstract

We study multi-class retrial queueing systems with Poisson inputs, general service times, and an arbitrary numbers of servers and waiting places. A class-i blocked customer joins orbit i and waits in the orbit for retrial. Orbit i works like a single-server $$\cdot /M/1$$ · / M / 1 queueing system with exponential retrial time regardless of the orbit size. Such retrial systems are referred to as retrial systems with constant retrial rate. Our model is not only motivated by several telecommunication applications, such as wireless multi-access systems, optical networks, and transmission control protocols, but also represents independent theoretical interest. Using a regenerative approach, we provide sufficient stability conditions which have a clear probabilistic interpretation. We show that the provided sufficient conditions are in fact also necessary, in the case of a single-server system without waiting space and in the case of symmetric classes. We also discuss a very interesting case, when one orbit is unstable, whereas the rest of the system is stable.

Suggested Citation

  • K. Avrachenkov & E. Morozov & B. Steyaert, 2016. "Sufficient stability conditions for multi-class constant retrial rate systems," Queueing Systems: Theory and Applications, Springer, vol. 82(1), pages 149-171, February.
    • Handle: RePEc:spr:queues:v:82:y:2016:i:1:d:10.1007_s11134-015-9463-9
    DOI: 10.1007/s11134-015-9463-9
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    References listed on IDEAS

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    1. Langaris, Christos & Dimitriou, Ioannis, 2010. "A queueing system with n-phases of service and (n-1)-types of retrial customers," European Journal of Operational Research, Elsevier, vol. 205(3), pages 638-649, September.
    2. K. Avrachenkov & E. Morozov & R. Nekrasova & B. Steyaert, 2014. "Stability Analysis And Simulation Of N-Class Retrial System With Constant Retrial Rates And Poisson Inputs," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 31(02), pages 1-18.
    3. R. Lillo, 1996. "A G/M/1-queue with exponential retrial," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 4(1), pages 99-120, June.
    4. E. Morozov & B. Steyaert, 2013. "Stability analysis of a two-station cascade queueing network," Annals of Operations Research, Springer, vol. 202(1), pages 135-160, January.
    5. Artalejo, J. R. & Gomez-Corral, A. & Neuts, M. F., 2001. "Analysis of multiserver queues with constant retrial rate," European Journal of Operational Research, Elsevier, vol. 135(3), pages 569-581, December.
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    Cited by:

    1. L. G. Afanasyeva, 2020. "Asymptotic Analysis of Queueing Models Based on Synchronization Method," Methodology and Computing in Applied Probability, Springer, vol. 22(4), pages 1417-1438, December.
    2. Yacov Satin & Evsey Morozov & Ruslana Nekrasova & Alexander Zeifman & Ksenia Kiseleva & Anna Sinitcina & Alexander Sipin & Galina Shilova & Irina Gudkova, 2018. "Upper bounds on the rate of convergence for constant retrial rate queueing model with two servers," Statistical Papers, Springer, vol. 59(4), pages 1271-1282, December.
    3. Murtuza Ali Abidini & Onno Boxma & Bara Kim & Jeongsim Kim & Jacques Resing, 2017. "Performance analysis of polling systems with retrials and glue periods," Queueing Systems: Theory and Applications, Springer, vol. 87(3), pages 293-324, December.
    4. Konstantin Avrachenkov, 2022. "Stability and partial instability of multi-class retrial queues," Queueing Systems: Theory and Applications, Springer, vol. 100(3), pages 177-179, April.

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