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A Fast Cross-Entropy Method for Estimating Buffer Overflows in Queueing Networks

Author

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  • P. T. de Boer

    (Department of Electrical Engineering, Mathematics, and Computer Science, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands)

  • D. P. Kroese

    (Department of Mathematics, University of Queensland, Brisbane 4072, Australia)

  • R. Y. Rubinstein

    (Faculty of Industrial Engineering and Management, Technion, Haifa, Israel)

Abstract

In this paper, we propose a fast adaptive importance sampling method for the efficient simulation of buffer overflow probabilities in queueing networks. The method comprises three stages. First, we estimate the minimum cross-entropy tilting parameter for a small buffer level; next, we use this as a starting value for the estimation of the optimal tilting parameter for the actual (large) buffer level. Finally, the tilting parameter just found is used to estimate the overflow probability of interest. We study various properties of the method in more detail for the M/M/1 queue and conjecture that similar properties also hold for quite general queueing networks. Numerical results support this conjecture and demonstrate the high efficiency of the proposed algorithm.

Suggested Citation

  • P. T. de Boer & D. P. Kroese & R. Y. Rubinstein, 2004. "A Fast Cross-Entropy Method for Estimating Buffer Overflows in Queueing Networks," Management Science, INFORMS, vol. 50(7), pages 883-895, July.
    • Handle: RePEc:inm:ormnsc:v:50:y:2004:i:7:p:883-895
    DOI: 10.1287/mnsc.1030.0139
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    References listed on IDEAS

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    1. Rubinstein, Reuven Y., 1997. "Optimization of computer simulation models with rare events," European Journal of Operational Research, Elsevier, vol. 99(1), pages 89-112, May.
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    Cited by:

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    2. Joshua C. C. Chan & Eric Eisenstat, 2015. "Marginal Likelihood Estimation with the Cross-Entropy Method," Econometric Reviews, Taylor & Francis Journals, vol. 34(3), pages 256-285, March.
    3. Fatma Başoğlu Kabran & Ali Devin Sezer, 2022. "Approximation of the exit probability of a stable Markov modulated constrained random walk," Annals of Operations Research, Springer, vol. 310(2), pages 431-475, March.
    4. Pieter-Tjerk de Boer, 2005. "Rare-Event Simulation of Non-Markovian Queueing Networks Using a State-Dependent Change of Measure Determined Using Cross-Entropy," Annals of Operations Research, Springer, vol. 134(1), pages 69-100, February.
    5. Bahar Kaynar & Ad Ridder, 2009. "The Cross-Entropy Method with Patching for Rare-Event Simulation of Large Markov Chains," Tinbergen Institute Discussion Papers 09-084/4, Tinbergen Institute.
    6. Ad Ridder, 2005. "Importance Sampling Simulations of Markovian Reliability Systems Using Cross-Entropy," Annals of Operations Research, Springer, vol. 134(1), pages 119-136, February.
    7. Pieter-Tjerk de Boer & Dirk Kroese & Shie Mannor & Reuven Rubinstein, 2005. "A Tutorial on the Cross-Entropy Method," Annals of Operations Research, Springer, vol. 134(1), pages 19-67, February.
    8. K.-P. Hui & N. Bean & M. Kraetzl & Dirk Kroese, 2005. "The Cross-Entropy Method for Network Reliability Estimation," Annals of Operations Research, Springer, vol. 134(1), pages 101-118, February.
    9. Cadini, F. & Santos, F. & Zio, E., 2014. "An improved adaptive kriging-based importance technique for sampling multiple failure regions of low probability," Reliability Engineering and System Safety, Elsevier, vol. 131(C), pages 109-117.
    10. Kaynar, Bahar & Ridder, Ad, 2010. "The cross-entropy method with patching for rare-event simulation of large Markov chains," European Journal of Operational Research, Elsevier, vol. 207(3), pages 1380-1397, December.
    11. Loretta Mastroeni & Giuseppe D'Acquisto & Maurizio Naldi, 2014. "Evaluation of Credit Risk Under Correlated Defaults: The Cross-Entropy Simulation Approach," Departmental Working Papers of Economics - University 'Roma Tre' 0193, Department of Economics - University Roma Tre.
    12. Kamil Demirberk Ünlü & Ali Devin Sezer, 2020. "Excessive backlog probabilities of two parallel queues," Annals of Operations Research, Springer, vol. 293(1), pages 141-174, October.
    13. Joakim Kalvenes & Neil Keon, 2007. "Traffic Estimation and Capacity Assignment in Multimedia Distribution Networks with Guaranteed Quality of Service," Operations Research, INFORMS, vol. 55(3), pages 518-531, June.
    14. Ali Kadhem, Athraa & Abdul Wahab, Noor Izzri & Aris, Ishak & Jasni, Jasronita & Abdalla, Ahmed N., 2017. "Computational techniques for assessing the reliability and sustainability of electrical power systems: A review," Renewable and Sustainable Energy Reviews, Elsevier, vol. 80(C), pages 1175-1186.
    15. M. Garvels, 2011. "A combined splitting—cross entropy method for rare-event probability estimation of queueing networks," Annals of Operations Research, Springer, vol. 189(1), pages 167-185, September.

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