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Asymptotic Formulas for Markov Processes with Applications to Simulation

Author

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  • Ward Whitt

    (AT&T Bell Laboratories, Murray Hill, New Jersey)

Abstract

The simulation run length required to achieve desired statistical precision for a sample mean in a steady-state stochastic simulation experiment is largely determined by the asymptotic variance of the sample mean and, to a lesser extent, by the second-order asymptotics of the variance and the asymptotic bias. The asymptotic variance, the second-order asymptotics of the variance, and the asymptotic bias of the sample mean of a function of an ergodic Markov process can be expressed in terms of solutions of Poisson's equation, as indicated by positive recurrent potential theory. We review this positive recurrent potential theory, giving special attention to continuous-time Markov chains. We provide explicit formulas for birth-and-death processes and diffusion processes, and recursive computational procedures for skip-free chains. These results can be used to help design simulation experiments after approximating the stochastic process of interest by one of the elementary Markov processes considered here.

Suggested Citation

  • Ward Whitt, 1992. "Asymptotic Formulas for Markov Processes with Applications to Simulation," Operations Research, INFORMS, vol. 40(2), pages 279-291, April.
  • Handle: RePEc:inm:oropre:v:40:y:1992:i:2:p:279-291
    DOI: 10.1287/opre.40.2.279
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    Cited by:

    1. Choi, Michael C.H. & Li, Evelyn, 2019. "A Hoeffding’s inequality for uniformly ergodic diffusion process," Statistics & Probability Letters, Elsevier, vol. 150(C), pages 23-28.
    2. Yunan Liu & Ward Whitt & Yao Yu, 2016. "Approximations for heavily loaded G/GI/n + GI queues," Naval Research Logistics (NRL), John Wiley & Sons, vol. 63(3), pages 187-217, April.
    3. Kerry W. Fendick, 2013. "Pricing and Hedging Derivative Securities with Unknown Local Volatilities," Papers 1309.6164, arXiv.org, revised Oct 2013.
    4. Whitt, Ward, 2012. "Fitting birth-and-death queueing models to data," Statistics & Probability Letters, Elsevier, vol. 82(5), pages 998-1004.
    5. Joost Berkhout & Bernd F. Heidergott, 2019. "Analysis of Markov Influence Graphs," Operations Research, INFORMS, vol. 67(3), pages 892-904, May.
    6. Jing Dong, 2022. "Metastability in queues," Queueing Systems: Theory and Applications, Springer, vol. 100(3), pages 413-415, April.
    7. Fen-Ru Shih & Mainak Mazumdar & Jeremy A. Bloom, 1999. "Asymptotic Mean and Variance of Electric Power Generation System Production Costs via Recursive Computation of the Fundamental Matrix of a Markov Chain," Operations Research, INFORMS, vol. 47(5), pages 703-712, October.

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