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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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