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Stochastic Optimization by Simulation: Convergence Proofs for the GI/G/1 Queue in Steady-State

Author

Listed:
  • Pierre L'Ecuyer

    (Département d'I.R.o., Université de Montréal, C.P. 6128, Montréal, Quebec, Canada, H3C 3J7)

  • Peter W. Glynn

    (Operations Research Department, Stanford University, Stanford, California 94305)

Abstract

Approaches like finite differences with common random numbers, infinitesimal perturbation analysis, and the likelihood ratio method have drawn a great deal of attention recently as ways of estimating the gradient of a performance measure with respect to continuous parameters in a dynamic stochastic system. In this paper, we study the use of such estimators in stochastic approximation algorithms, to perform so-called "single-run optimizations" of steady-state systems. Under mild conditions, for an objective function that involves the mean system time in a GI/G/1 queue, we prove that many variant of these algorithms converge to the minimizer. In most cases, however, the simulation length must be increased from iteration to iteration, otherwise the algorithm may converge to the wrong value. One exception is a particular implementation of infinitesimal perturbation analysis, for which the single-run optimization converges to the optimum even with a fixed (and small) number of ends of service per iteration. As a by-product of our convergence proofs, we obtain some properties of the derivative estimators that could be of independent interest. Our analysis exploits the regenerative structure of the system, but our derivative estimation and optimization algorithms do not always take advantage of that regenerative structure. In a companion paper, we report numerical experiments with an M/M/1 queue, which illustrate the basis convergence properties and possible pitfalls of the various techniques.

Suggested Citation

  • Pierre L'Ecuyer & Peter W. Glynn, 1994. "Stochastic Optimization by Simulation: Convergence Proofs for the GI/G/1 Queue in Steady-State," Management Science, INFORMS, vol. 40(11), pages 1562-1578, November.
    • Handle: RePEc:inm:ormnsc:v:40:y:1994:i:11:p:1562-1578
    DOI: 10.1287/mnsc.40.11.1562
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    Citations

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    Cited by:

    1. Gürkan, G., 1997. "Simulation Optimization of Buffer Allocations in Production Lines with Unreliable Machines," Discussion Paper 1997-97, Tilburg University, Center for Economic Research.
    2. Kao, Chiang & Chen, Shih-Pin, 2006. "A stochastic quasi-Newton method for simulation response optimization," European Journal of Operational Research, Elsevier, vol. 173(1), pages 30-46, August.
    3. A. B. Dieker & S. Ghosh & M. S. Squillante, 2017. "Optimal Resource Capacity Management for Stochastic Networks," Operations Research, INFORMS, vol. 65(1), pages 221-241, February.
    4. Sumit Kunnumkal & Huseyin Topaloglu, 2009. "A stochastic approximation method for the single-leg revenue management problem with discrete demand distributions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 70(3), pages 477-504, December.
    5. Sumit Kunnumkal & Huseyin Topaloglu, 2011. "A stochastic approximation algorithm to compute bid prices for joint capacity allocation and overbooking over an airline network," Naval Research Logistics (NRL), John Wiley & Sons, vol. 58(4), pages 323-343, June.
    6. Leyuan Shi & Sigurdur O´lafsson, 2000. "Nested Partitions Method for Stochastic Optimization," Methodology and Computing in Applied Probability, Springer, vol. 2(3), pages 271-291, September.
    7. Huseyin Topaloglu, 2008. "A Stochastic Approximation Method to Compute Bid Prices in Network Revenue Management Problems," INFORMS Journal on Computing, INFORMS, vol. 20(4), pages 596-610, November.
    8. Paul Glasserman & Sridhar Tayur, 1996. "A simple approximation for a multistage capacitated production‐inventory system," Naval Research Logistics (NRL), John Wiley & Sons, vol. 43(1), pages 41-58, February.
    9. Gürkan, G. & Ozge, A.Y., 1996. "Sample-Path Optimization of Buffer Allocations in a Tandem Queue - Part I : Theoretical Issues," Discussion Paper 1996-98, Tilburg University, Center for Economic Research.
    10. Rubinstein, Reuven Y., 1997. "Optimization of computer simulation models with rare events," European Journal of Operational Research, Elsevier, vol. 99(1), pages 89-112, May.
    11. A. B. Dieker & S. Ghosh & M. S. Squillante, 2017. "Optimal Resource Capacity Management for Stochastic Networks," Operations Research, INFORMS, vol. 65(1), pages 221-241, February.
    12. Tito Homem-de-Mello, 2001. "Estimation of Derivatives of Nonsmooth Performance Measures in Regenerative Systems," Mathematics of Operations Research, INFORMS, vol. 26(4), pages 741-768, November.
    13. Sumit Kunnumkal & Huseyin Topaloglu, 2008. "Using Stochastic Approximation Methods to Compute Optimal Base-Stock Levels in Inventory Control Problems," Operations Research, INFORMS, vol. 56(3), pages 646-664, June.

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