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Strongly Consistent Steady-State Derivative Estimates

Published online by Cambridge University Press:  27 July 2009

Paul Glasserman ,
Jian-Qiang Hu  and
Stephen G. Strickland
Paul Glasserman
Affiliation:
Graduate School of Business Columbia University New York, New York 10027
Jian-Qiang Hu
Affiliation:
Department of Manufacturing Engineering BostonUniversity Boston, Massachusetts 02215
Stephen G. Strickland
Affiliation:
Department of Systems EngineeringUniversity of Virginia Charlottesville, Virginia 22903
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Abstract

We establish strong consistency (i.e., almost sure convergence) of infinitesimal perturbation analysis (IPA) estimators of derivatives of steady-state means for a broad class of systems. Our results substantially extend previously available results on steady-state derivative estimation via IPA.

Our basic assumption is that the process under study is regenerative, but our analysis uses regenerative structure in an indirect way: IPA estimators are typically biased over regenerative cycles, so straightforward differentiation of the regenerative ratio formula does not necessarily yield a valid estimator of the derivative of a steady-state mean. Instead, we use regeneration to pass from unbiasedness over fixed, finite time horizons to convergence as the time horizon grows. This provides a systematic way of extending results on unbiasedness to strong consistency.

Given that the underlying process regenerates, we provide conditions under which a certain augmented process is also regenerative. The augmented process includes additional information needed to evaluate derivatives; derivatives of time averages of the original process are time averages of the augmented process. Thus, through this augmentation we are able to apply standard renewal theory results to the convergence of derivatives.

Type
Articles
Information
Probability in the Engineering and Informational Sciences , Volume 5 , Issue 4 , October 1991 , pp. 391 - 413
Copyright
Copyright © Cambridge University Press 1991

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References

Cao, X.R. (1987). Realization probability in closed Jackson networks and its applications. Advances in Applied Probability 19: 708–738.CrossRefGoogle Scholar
Cao, X.R. (1988). A sample performance function of closed Jackson queueing networks. operations Research 36(1): 128–136.CrossRefGoogle Scholar
Glasserman, P. (1988). Performance continuity and differentiability in Monte Carlo optimization. In Abrams, M., Haigh, P. & Comfort, J. (eds.), Proceedings of the Winter Simulation Conference. San Diego, CA: Society for Computer Simulation, pp. 518–524.Google Scholar
Glasserman, P. (1991). Structural conditions for perturbation analysis derivative estimation: Finite-time performance indices. Operations Research (to appear).CrossRefGoogle Scholar
Glynn, P. (1986). Stochastic approximation for Monte Carlo optimization. In Wilson, J., Henriksen, J. & Roberts, S. (eds.), Proceedings of the Winter Simulation Conference. San Diego, CA: Society for Computer Simulation, pp. 356–364.Google Scholar
Glynn, P. (1989). A GSMP formalism for discrete-event systems. Proceedings of the IEEE 77(1): 14–23.Google Scholar
Haas, P.J. & Shedler, G.S. (1987). Regenerative generalized semi-Markov processes. Stochastic Models 3: 409–438.CrossRefGoogle Scholar
Heidelberger, P., Cao, X.R., Zazanis, M.A. & Suri, R. (1988). Convergence properties of infinitesimal perturbation analysis estimates. Management Science 34(11): 1281–1301.CrossRefGoogle Scholar
Hu, J.-Q. (1991). Convexity of sample path performance and strong consistency of infinitesimal perturbation analysis estimates. IEEE Transactions on Automatic Control (to appear).Google Scholar
Hu, J.-Q. & Strickland, S.G. (1990). Strong consistency of sample path derivative estimates. Applied Mathematics Letters 3(4): 55–58.CrossRefGoogle Scholar
Reiman, M. & Weiss, A. (1989). Sensitivity analysis for simulations via likelihood ratios. Operations Research 37(5): 830–844.CrossRefGoogle Scholar
Ross, S. (1983). Stochastic processes. New York: Wiley.Google Scholar
Sigman, K. (1990). The stability of open queueing networks. Stochastic Processes and TheirApplications 35(1): 11–25.CrossRefGoogle Scholar
Suri, R. (1987). Perturbation analysis of general discrete event systems. Journal of the ACM 34: 686–717.CrossRefGoogle Scholar
Suri, R. /M (1988). Perturbation analysis gives strongly consistent sensitivity estimates for the M/G/1 queue. Management Science 34(1): 39–64.CrossRefGoogle Scholar
Wardi, Y. & Hu, J.-Q. (1991). Strong consistency of infinitesimal perturbation analysis for tandem queueing networks. Discrete Event Dynamic Systems: Theory and Applications 1(1): 37–59.CrossRefGoogle Scholar
Whitt, W. (1980). Continuity of generalized semi-Markov processes. Mathematics of Operalions Research 5(4): 494–501.CrossRefGoogle Scholar
Zazanis, M. & Suri, R. (1985). Perturbation analysis of the G1/G/1 queue. Technical Report, Industrial Engineering and Management Sciences. Revised version submitted for publication.Google Scholar