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Upper Bounds on Performance Measures of Heterogeneous M/M/c Queues

Author

Listed:
  • Laoucine Kerbache

    (GREGH - Groupement de Recherche et d'Etudes en Gestion à HEC - HEC Paris - Ecole des Hautes Etudes Commerciales - CNRS - Centre National de la Recherche Scientifique)

  • F. S. Q. Alves
  • H. C. Yehia
  • L.A.C Pedrosa
  • F.R.B. Cruz

    (School of Computer Science and Information Technology - UON - University of Nottingham, UK)

Abstract

In many real-life queueing systems, the servers are often heterogeneous, namely they work at different rates. This paper provides a simple method to compute tight upper bounds on two important performance measures of single-class heterogeneous multi-server Markovian queueing systems, namely the average number in queue and the average waiting time in queue. This method is based on an expansion of the state space that is followed by an approximate reduction of the state space, only considering the most probable states. In most cases tested, we were able to approximate the actual behavior of the system with smaller errors than those obtained from traditional homogeneous multiserver Markovian queues, as shown by GPSS simulations. In addition, we have correlated the quality of the approximation with the degree of heterogeneity of the system, which was evaluated using its Gini index. Finally, we have shown that the bounds are robust and still useful, even considering quite different allocation strategies. A large number of simulation results show the accuracy of the proposed method that is better than that of classical homogeneous multiserver Markovian formulae in many situations.

Suggested Citation

  • 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.
    • Handle: RePEc:hal:journl:hal-00609498
    DOI: 10.1155/2011/702834
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    Citations

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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. Li, Na & Stanford, David A., 2016. "Multi-server accumulating priority queues with heterogeneous servers," European Journal of Operational Research, Elsevier, vol. 252(3), pages 866-878.
    3. F. R. B. Cruz & A. R. Duarte & G. L. Souza, 2018. "Multi-objective performance improvements of general finite single-server queueing networks," Journal of Heuristics, Springer, vol. 24(5), pages 757-781, October.
    4. 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.

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