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Perturbation Analysis Gives Strongly Consistent Sensitivity Estimates for the M/G/1 Queue

Author

Listed:
  • Rajan Suri

    (Department of industrial Engineering, University of Wisconsin, Madison, Wisconsin 53706)

  • Michael A. Zazanis

    (Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, Illinois 60201)

Abstract

The technique of perturbation analysis has recently been introduced as an efficient way to compute parameter sensitivities for discrete event systems. Thus far, the statistical properties of perturbation analysis have been validated mainly through experiments. This paper considers, for an M/G/1 queueing system, the sensitivity of mean system time of a customer to a parameter of the arrival or service distribution. It shows analytically that (i) the steady state value of the perturbation analysis estimate of this sensitivity is unbiased, and (ii) a perturbation analysis algorithm implemented on a single sample path of the system gives asymptotically unbiased and strongly consistent estimates of this sensitivity. (No previous knowledge of perturbation analysis is assumed, so the paper also serves to introduce this technique to the unfamiliar reader.) Numerical extensions to GI/G/1 queues, and applications to optimization problems, are also illustrated.

Suggested Citation

  • Rajan Suri & Michael A. Zazanis, 1988. "Perturbation Analysis Gives Strongly Consistent Sensitivity Estimates for the M/G/1 Queue," Management Science, INFORMS, vol. 34(1), pages 39-64, January.
    • Handle: RePEc:inm:ormnsc:v:34:y:1988:i:1:p:39-64
    DOI: 10.1287/mnsc.34.1.39
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    Cited by:

    1. Nan Chen & Yanchu Liu, 2014. "American Option Sensitivities Estimation via a Generalized Infinitesimal Perturbation Analysis Approach," Operations Research, INFORMS, vol. 62(3), pages 616-632, June.
    2. M. Hossein Safizadeh, 1990. "Optimization in simulation: Current issues and the future outlook," Naval Research Logistics (NRL), John Wiley & Sons, vol. 37(6), pages 807-825, December.
    3. L. Jeff Hong, 2009. "Estimating Quantile Sensitivities," Operations Research, INFORMS, vol. 57(1), pages 118-130, February.
    4. Zhenyu Cui & Michael C. Fu & Jian-Qiang Hu & Yanchu Liu & Yijie Peng & Lingjiong Zhu, 2020. "On the Variance of Single-Run Unbiased Stochastic Derivative Estimators," INFORMS Journal on Computing, INFORMS, vol. 32(2), pages 390-407, April.
    5. Peter W. Glynn & Yijie Peng & Michael C. Fu & Jian-Qiang Hu, 2021. "Computing Sensitivities for Distortion Risk Measures," INFORMS Journal on Computing, INFORMS, vol. 33(4), pages 1520-1532, October.
    6. Schumacher, J.M., 1988. "Discrete events : Perspectives from system theory," Other publications TiSEM 655c7240-4f86-4a73-a59b-a, Tilburg University, School of Economics and Management.
    7. Barry L. Nelson, 2004. "50th Anniversary Article: Stochastic Simulation Research in Management Science," Management Science, INFORMS, vol. 50(7), pages 855-868, July.
    8. Mark J. Cathcart & Steven Morrison & Alexander J. McNeil, 2011. "Calculating Variable Annuity Liability 'Greeks' Using Monte Carlo Simulation," Papers 1110.4516, arXiv.org.
    9. Rubinstein, Reuven Y. & Shapiro, Alexander, 1990. "Optimization of static simulation models by the score function method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 32(4), pages 373-392.
    10. Dinah W. Cheng, 1994. "On the design of a tandem queue with blocking: Modeling, analysis, and gradient estimation," Naval Research Logistics (NRL), John Wiley & Sons, vol. 41(6), pages 759-770, October.
    11. Michael C. Fu, 2008. "What you should know about simulation and derivatives," Naval Research Logistics (NRL), John Wiley & Sons, vol. 55(8), pages 723-736, December.
    12. Wai Kin Victor Chan, 2016. "Linear Programming Formulation of Idle Times for Single-Server Discrete-Event Simulation Models," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 33(05), pages 1-17, October.
    13. Yongqiang Wang & Michael C. Fu & Steven I. Marcus, 2012. "A New Stochastic Derivative Estimator for Discontinuous Payoff Functions with Application to Financial Derivatives," Operations Research, INFORMS, vol. 60(2), pages 447-460, April.
    14. Yijie Peng & Michael C. Fu & Bernd Heidergott & Henry Lam, 2020. "Maximum Likelihood Estimation by Monte Carlo Simulation: Toward Data-Driven Stochastic Modeling," Operations Research, INFORMS, vol. 68(6), pages 1896-1912, November.
    15. Cao, Xi-Ren, 1996. "Perturbation analysis of discrete event systems: Concepts, algorithms, and applications," European Journal of Operational Research, Elsevier, vol. 91(1), pages 1-13, May.
    16. Wai Kin (Victor) Chan & Lee Schruben, 2008. "Optimization Models of Discrete-Event System Dynamics," Operations Research, INFORMS, vol. 56(5), pages 1218-1237, October.
    17. Michael C. Fu & Huashuai Qu, 2014. "Regression Models Augmented with Direct Stochastic Gradient Estimators," INFORMS Journal on Computing, INFORMS, vol. 26(3), pages 484-499, August.
    18. Bernd Heidergott & Warren Volk-Makarewicz, 2016. "A Measure-Valued Differentiation Approach to Sensitivities of Quantiles," Mathematics of Operations Research, INFORMS, vol. 41(1), pages 293-317, February.
    19. Guangxin Jiang & Michael C. Fu, 2015. "Technical Note—On Estimating Quantile Sensitivities via Infinitesimal Perturbation Analysis," Operations Research, INFORMS, vol. 63(2), pages 435-441, April.

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