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The efficiency and heavy traffic properties of the score function method in sensitivity analysis of queueing models

Part of: Markov processes Special processes Limit theorems Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  01 July 2016

Søren Asmussen  and
Reuven Y. Rubinstein
Søren Asmussen*
Affiliation:
Aalborg University
Reuven Y. Rubinstein*
Affiliation:
Technion—Israel Institute of Technology
*
∗Postal address: Institute of Electronic Systems, Aalborg University, Fr. Bajersvej. 7, DK-9220 Aalborg, Denmark.
∗∗Postal address: Faculty of Industrial Engineering and Management, Technion, Haifa 32000, Israel.
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Abstract

This paper studies computer simulation methods for estimating the sensitivities (gradient, Hessian etc.) of the expected steady-state performance of a queueing model with respect to the vector of parameters of the underlying distribution (an example is the gradient of the expected steady-state waiting time of a customer at a particular node in a queueing network with respect to its service rate). It is shown that such a sensitivity can be represented as the covariance between two processes, the standard output process (say the waiting time process) and what we call the score function process which is based on the score function. Simulation procedures based upon such representations are discussed, and in particular a control variate method is presented. The estimators and the score function process are then studied under heavy traffic conditions. The score function process, when properly normalized, is shown to have a heavy traffic limit involving a certain variant of two-dimensional Brownian motion for which we describe the stationary distribution. From this, heavy traffic (diffusion) approximations for the variance constants in the large sample theory can be computed and are used as a basis for comparing different simulation estimators. Finally, the theory is supported by numerical results.

Keywords

GRADIENT ESTIMATIONMONTE CARLO METHODREGENERATIVE SIMULATIONCONTROL VARIATEHEAVY TRAFFICBIVARIATE BROWNIAN MOTIONPERFORMANCE EVALUATIONQUEUEING NETWORKS

MSC classification

Primary: 65C05: Monte Carlo methods
Secondary: 60K25: Queueing theory 60J65: Brownian motion 60F17: Functional limit theorems; invariance principles
Type
Research Article
Information
Advances in Applied Probability , Volume 24 , Issue 1 , March 1992 , pp. 172 - 201
Copyright
Copyright © Applied Probability Trust 1992 

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Footnotes

Research supported by the L. Edelstein Research Fund of the Technion, Israel Institute of Technology, and by the Danish Natural Science Research Council.

References

[1] Arjas, E. and De Smit, J. H. A. (1973) On the total waiting time during a busy period of the single server queue. CORE Discussion Paper 7312, Catholic University, Leuven.Google Scholar
[2] Arsham, H., Feuerverger, A., Kreimer, J., Mcleish, D. L. and Rubinstein, R. Y. (1989) Sensitivity analysis and the 'what if problem in simulation analysis. Math. Comput. Modelling 12, 193–219.CrossRefGoogle Scholar
[3] Asmussen, S. (1985) Conjugate processes and the simulation of ruin problems. Stoch. Proc. Appl. 20, 213–229.CrossRefGoogle Scholar
[4] Asmussen, S. (1987) Applied Probability and Queues. Wiley, Chichester.Google Scholar
[5] Asmussen, S. (1989) Validating the heavy-traffic performance of regenerative simulation. Stoch. Models 5, 617–628.Google Scholar
[6] Asmussen, S. (1990) Exponential families and regression in the Monte Carlo study of queues and random walks. Ann. Statist. 18, 1851–1867.CrossRefGoogle Scholar
[7] Asmussen, S. (1991) Queueing simulation in heavy traffic. Math. Operat. Res. Google Scholar
[8] Asmussen, S. and Rubinstein, R. Y. (1990) The performance of likelihood ratio estimators using the score function. Stoch. Models. Google Scholar
[9] Cao, X. R. (1987) Sensitivity estimates based on one realization of stochastic system. J. Statist. Comp. Sim. 27, 211–232.CrossRefGoogle Scholar
[10] L'Ecuyer, P. (1989) A unified view of infinitesimal perturbation analysis and likelihood ratios. Management Sci. Google Scholar
[11] Feuerverger, A., Mcleish, D. L. and Rubinstein, R. Y. (1986) A cross-spectral method for sensitivity analysis of computer simulation models. C . R. Math. Rep. Acad. Sci. R. Soc. Can. VIII, N5, 335–339.Google Scholar
[12] Fishman, G. S. (1978) Principles of Discrete Events Simulation. Wiley, New York.Google Scholar
[13] Gadrich, T. (1989) Application of Radon-Nikodym derivatives to sensitivity analysis of stochastic models. Math. Programming Study 19, 101–119.Google Scholar
[14] Glasserman, P. (1990) Stochastic monotonicity, total positivity, and conditional Monte Carlo for likelihood ratios. Manuscript, AT & T Bell Laboratories, Holmdel, NJ.Google Scholar
[15] Glynn, P. W. (1986a) Sensitivity analysis for stationary probabilities of a Markov chain. Proc. 4th Conf. Applied Mathematics and Computers. Google Scholar
[16] Glynn, P. W. (1986b) Stochastic approximation for Monte Carlo optimization. Proc. 1986 Winter Simulation Conference. Google Scholar
[17] Glynn, P. W. (1986C) Optimization of stochastic systems. Proc. 1986 Winter Simulation Conference. Google Scholar
[18] Glynn, P. W. (1987) Likelihood ratio gradient estimation: an overview. Proc. 1987 Winter Simulation Conference , 366–375.Google Scholar
[19] Glynn, P. W. and Sanders, J. L. (1986) Monte Carlo optimization of stochastic systems: two new approaches. Proc. 1986 ASME Computers in Engineering Conference. Google Scholar
[20] Kreimer, J. (1984) Stochastic Optimization—an Adaptive Approcah. D.Sc. Thesis, Technion, Haifa, Israel.Google Scholar
[21] Law, A. M. and Kelton, W. D. (1982) Simulation Modeling and Analysis. McGraw-Hill, New York.Google Scholar
[22] Meketon, M. S. (1987) Optimization in simulation: a survey of recent results. Proc. 1987 Winter Simulation Conference , 58–67.CrossRefGoogle Scholar
[23] Pflug, G. Ch. (1988) Sensitivity analysis of semi-Markov processes. Manuscript, University of Giessen, West Germany.Google Scholar
[24] Rao, C. R. (1968) Linear Statistical Inference and its Applications. Wiley, New York.Google Scholar
[25] Reiman, M. I. and Weiss, A. (1989) Sensitivity analysis for simulation via likelihood ratios. Operat. Res. CrossRefGoogle Scholar
[26] Rief, H. (1988) Monte Carlo uncertainty analysis. In CRC-Handbook on Uncertainty Analysis , ed. Ronen, Y. CRC Press, Boca Raton, FL.Google Scholar
[27] Rubinstein, R. Y. (1976) A Monte Carlo method for estimating the gradient in a stochastic network. Unpublished manuscript, Technion, Haifa, Israel.Google Scholar
[28] Rubinstein, R. Y. (1986a) The score function method for sensitivity analysis of computer simulation models. Math. Comput. Simul. 28, 1–29.Google Scholar
[29] Rubinstein, R. Y. (1986b) Monte Carlo Optimization, Simulation and Sensitivity of Queueing Networks. Wiley, New York.Google Scholar
[30] Rubinstein, R. Y. (1987a) Performance evaluation and the 'what if problem via the score function method. Manuscript, Technion, Haifa, Israel.Google Scholar
[31] Rubinstein, R. (1987b) Modified importance sampling for performance evaluation and sensitivity analysis of computer simulation models. Manuscript, George Washington University.Google Scholar
[32] Rubinstein, R. Y. (1989a) Sensitivity of computer simulation models via the efficient score. Operat. Res. 37, 72–81.Google Scholar
[33] Rubinstein, R. Y. (1989b) Monte Carlo methods for performance evaluation, sensitivity analysis and optimization of stochastic systems. In Encyclopedia of Computer Science and Technology , ed. Kent, J. To appear.Google Scholar
[34] Rubinstein, R. Y. and Shapiro, A. (1989) Optimization of simulation models by the score function method. Operations Research Mimeograph Series 397, Faculty of Industrial Engineering and Management, Technion, Haifa, Israel.Google Scholar
[35] Suri, R. and Zazanis, M. A. (1988) Perturbation analysis gives strongly consistent sensitivity estimates for the GI/G/1 queue. Management Sci. 34, 39–63.CrossRefGoogle Scholar
[36] Whitt, W. (1989) Planning queueing simulations. Management Sci. 35, 1341–1366.Google Scholar
[37] Williams, D. (1979) Markov Processes, Martingales and Diffusions. Wiley, Chichester.Google Scholar