Abstract
In this paper we show that an optimization algorithm using infinitesimal perturbation analysis converges to the minimum of the performance measure. The algorithm is updated every fixed number of customers period rather than every one or two busy periods. The key step of the proof is to verify that the observation noises satisfy the strong law of large numbers. Then applying a stochastic approximation theorem leads to the desired results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chen, H. F., Guo, L., and Gao, A. J. 1988. Convergence and robustness of the Robbins-Monro algorithm truncated at randomly varying bounds.Stoch. Process. Appl. 27, pp. 217–231.
Chong, E. K. P., and Ramadge, P. J, 1992a. Convergence of recursive optimization algorithms using infinitesimal perturbation analysis estimates.Discrete Event Dynam. Syst. 1, pp. 339–372.
Chong, E. K. P., and Ramadge, P. J., 1992b. Stochastic optimization of regenerative systems using infinitesimal perturbation analysis.IEEE Trans. Automat. Control (to appear).
Chong, E. K. P., and Ramadge, P. J., 1993. Optimization of queues using infinitesimal perturbation analysis-based algorithm with general update times.SIAM J. Control Optim. 31, pp. 698–732.
Chow, Y. S., and Teicher, H., 1988.Probability Theory: Independence, Interchangeability, Martingale, 2nd ed. New York: Springer-Verlag.
Fu, M. C., 1990. Convergence of a stochastic approximation algorithm for the GI/G/1 queue using infinitesimal perturbation analysis.J. Optim. Theory Appl. 65, pp. 149–60.
Glasserman, P., Hu, J. Q., and Strickland, S. G., 1991. Strongly consistent steady-state derivative estimates.Probab. Eng. Inform. Sci. 5, pp. 391–413.
Ho, Y. C., 1987. Performance evaluation and perturbation analysis of discrete event dynamic systems.IEEE Trans. Automat. Control 32, pp. 563–572.
Ho, Y. C., and Cao, X. R., 1983. Perturbation analysis and optimization of queueing networks.J. Optim. Theory Appl. 40, pp. 559–582.
Ho, Y. C., and Cao, X. R., 1991.Perturbation Analysis of Discrete Event Dynamic Systems. Boston: Kluwer.
Hu. J. Q., 1992. Convexity of sample path performance and strong consistency of infinitesimal perturbation analysis estimates.IEEE Trans. Automat. Control 37, pp. 258–262.
Kleinrock, L., 1975.Queueing Systems, Vol. I.Theory New York: Wiley.
Meketon, M. S., 1983. A tutorial on optimization in simulations.Proc. Winter Simulation Conf., pp. 58–67.
Robbins, H. and Monro, S., 1951. A stochastic approximation method.Ann. Math. Statist. 22, pp. 400–407.
Stout, W. F., 1974.Almost Sure Convergence. New York: Academic Press.
Suri, R., 1989. Perturbation analysis: the state of the art and research issues explained via the GI/G/1 queue,Proc. IEEE 77, pp. 114–137.
Suri, R., and Leung, Y. T., 1989. Single run optimizations of discrete event simulations—An empirical study using the M/M/1 queue.IIEE Trans. 21, pp. 35–49.
Suri, R., and Zazanis, M. A., 1988. Perturbation analysis gives strongly consistent sensitivity estimates for the M/G/1 queue.Management Sci. 34, pp. 39–64.
Thorisson, H., 1985. The queueGI/G/1: finite moments of the cycle variables and uniform rates of convergence.Stoch. Process. Appl. 19, pp. 85–99.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Tang, QY., Chen, HF. Convergence of perturbation analysis based optimization algorithm with fixed number of customers period. Discrete Event Dyn Syst 4, 359–375 (1994). https://doi.org/10.1007/BF01440234
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01440234