×

Correlation-inducing variance reduction in regenerative simulation. (English) Zbl 0872.65123

Let \(\mathcal X = \big\{\mathbf X(t), t\geq0\big\}\) be a multidimensional regenerative process in continuous time, and let \(\big\{T_k\mid k\geq0\big\}\) be a sequence of regenerative points associated with \(\mathcal X\). Let \(\tau_k= T_k-T_{k-1}, k\geq1\), be the length of cycle of the process \(\mathcal X\). Under mild regularity conditions the process \(\mathcal X\) has a limiting distribution; that is \(\mathbf X(t)\to\mathbf X\) as \(t\to\infty\). Suppose that we are interested in estimating the expected value of \(f\big(\mathbf X(t)\big)\) in the long run, where \(f\) is a real-valued integrable function. The authors use (Shelder’s) strongly consistent point estimator of \(r(f)=\lim_{t\to\infty} E\big(f(\mathbf X(t))\big)\) \[ \widehat r\big(n;f\big)={\frac{\overline Y(n)}{\overline \tau(n)}}= {\frac{n^{-1}\sum_{k=1}^{n} Y_k(f)}{n^{-1}\sum_{k=1}^{n} \tau_k}}, \] where \[ Y_k(f)=\int_{T_{k-1}}^{T_k} f\big(\mathbf X(u)\big) du,\qquad 1\leq k\leq n, \] for a regenerative process in continuous time, and \[ Y_k(f)=\sum_{i=T_{k-1}}^{T_k-1} f\big(\mathbf X(i)\big)\qquad 1\leq k\leq n, \] for a regenerative process in discrete time. The main result is a proposal of a variance reduction technique that can be applied to regenerative simulations. The main difference from the classical approach is that the proposed technique induces correlation between consecutive nonoverlapping pairs of regenerations. It is shown analytically, that under mild conditions the proposed technique is superior to conventional techniques which use independent random numbers between regenerations. Two examples are provided to illustrate that the variance reduction is significant.
Reviewer: J.Antoch (Praha)

MSC:

65C99 Probabilistic methods, stochastic differential equations
62F10 Point estimation
62J10 Analysis of variance and covariance (ANOVA)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Glynn, P. W., Regenerative structure of Markov chains simulated via common random numbers, Oper. Res. Lett., 4, 49-53 (1985) · Zbl 0576.60082
[2] Gross, D.; Harris, C. M., Fundamentals of Queuing Theory (1985), Wiley: Wiley New York
[3] Heidelberger, P.; Iglehart, D. L., Comparing stochastic system using regenerative simulation with common random numbers, Adv. Appl. Probab., 11, 804-819 (1979) · Zbl 0434.60076
[4] Hussey, J. R.; Myers, R. H.; Houck, E. C., Correlated simulation experiments in first-order response surface design, Oper. Res., 35, 744-758 (1987)
[5] Iglehart, D. L.; Lewis, P. A.W., Regenerative simulation with internal controls, J. ACM, 26, 2, 271-281 (1979) · Zbl 0395.68084
[6] Iglehart, D. L.; Shelder, G. S., Regenerative Simulation of Response Times in Networks of Queues (1980), Springer: Springer New York · Zbl 0424.90016
[7] Law, A. M.; Kelton, W. D., Simulation Modeling and Analysis (1991), McGraw-Hill: McGraw-Hill New York
[8] Ross, S. M., Introduction to Probability Models (1993), Academic Express: Academic Express New York · Zbl 0781.60001
[9] Schruben, L. W.; Margolin, B. H., Pseudorandom number assignment in statistically designed simulation and distribution sampling experiment, J. Amer. Statist. Assoc., 73, 504-520 (1978) · Zbl 0386.62010
[10] Shedler, G. S., Regeneration and Networks of Queues (1987), Springer, Verlag: Springer, Verlag New York · Zbl 0607.60083
[11] Song, W. T.; Su, C. C., An extension of the multiple-blocks strategy on estimating simulation metamodels, IIE Trans. IE Res., 28, 511-519 (1996)
[12] Sullivan, R. S.; Hayya Jack, C.; Ronny, S., Efficiency of the antithetic variate method for simulating stochastic networks, Management Sci., 28, 536-572 (1982) · Zbl 0484.65089
[13] Tew, J. D.; Wilson, J. R., Estimating simulation metamodels using combined correlation-based variance reduction techniques, IIE Trans., 26, 3, 2-15 (1994)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.