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...sets of input variables and to identify the optimal operating conditions for the input variables. For example, consider computer network with several Input/Output (I/O) devices. We are interested in how the steady-state average queue length (response) at the bottleneck I/O device is influenced by the service rates (input variables) of the I/O devices. If the network is not mathematically tractable, simulation experiments can be performed and the corresponding metamodel is constructed.
We often express a metamodel as
Y = X[beta] + E,
where Y is a vector of simulation responses, X is a design matrix used to observe Y, E is the dependence structure of the simulation random error vector, and [beta] is the vector of metamodel coefficients. A natural estimator of [beta] is the ordinary least squares estimator:
[^.[beta]] = ([X.sup.t]X)[.sup.-1][X.sup.t]Y,
where [X.sup.t] is the transpose of X; and the associated dispersion matrix is
[[SIGMA].sub.[^.[beta]]] = ([X.sup.t]X)[.sup.-1][X.sup.t][[SIGMA].sub.Y]X([X.sup.t]X)[.sup.-1],
where [[SIGMA].sub.Y] denotes the dispersion matrix of Y.
A special characteristic of a simulation experiment is that it provides analysts with the ability to assign the Pseudo-Random Number (PRN) streams to the experiments. It is possible to assign PRN streams to estimate a metamodel so as to affect the dependence structure of the simulation error vector E to better estimate the unknown metamodel coefficients [beta]. For general articles discussing the topic of metamodeling, the reader can consult Racite and Lawlor (1972), Montgomery and Evans (1975), Barton (1992, 1997), Sargent (1992), Caughlin (1995), Tew (1995), Chen and Kleijnen (1999), Santos and Nova (1999), Kleijnen and Sargent (2000), Law and Kalton (2000), Kleijnen et al. (2001), Lamb and Cheng (2002) and Batmaz and Tunali (2003).
Suppose we consider a [2.sup.k] factorial design in which the mean effect, main effects, and some interaction effects need to be estimated. Through appropriate assignment of PRN streams to design points, we can systematically allocate the variances of the effects estimators. Schruben (1979) showed that the sum of the variances of independent estimators is a constant no matter what assignment of PRN streams is used. That is, if one decreases the variances of some estimators, the variances of certain other estimators will increase simultaneously. The variances are "swapped" rather than "reduced" for the important estimators, such as the estimators of the mean effect and the main effects, in such a technique. The methodology of assigning PRNs is called a PRN assignment strategy or correlation induction strategy, or a Variance Swapping Rule (VSR). We adopt a VSR to refer to such techniques throughout this paper.
Existing VSRs include the Independent Random-Number (IRN) streams rule, Common Random-Number (CRN) streams rule, Assignment Rule (AR) (Schruben and Margolin, 1978), Multiple Blocks (MB) rule (Hussey et al., 1987a, 1987b), and Extended Multiple Blocks (E-MB) rule (Song and Su, 1996). In the IRN rule, independent streams are used for each design point and produce the same variance for all estimators. That is, there is only one class of variance for all effects if the IRN rule is applied. In the CRN rule, the same common streams are used for all design points and produce the same variance for all estimators except for that of the mean effect. That is, there are two classes of variance for all effects if the CRN rule is applied. Schruben and Margolin (1978) first developed the widely accepted AR rule, which allows for the simultaneous re-use of PRN streams in both direct and antithetic forms in or-thogonally blockable designs. In AR, the design points are divided into two orthogonal blocks. The same PRN streams are used for the design points in the first block, while their antithetic streams are used for the design points in the second block. MB is an extension of AR, and is used in [2.sup.k] factorial designs to decrease the variance of the main effects estimators as much as possible. The idea is to divide the design points into as many blocks as possible with the constraint that no main effects are confounded with blocks. Song and Su (1996) extended the MB rule to a general linear metamodel in [2.sup.k] factorial designs. Song et al. (2005) again extended the MB rule to a quadratic metamodel in [3.sup.k] factorial designs.
In the original papers that proposed AR and MB, the authors differentiated all effects into effects of interest and no interest. Therefore, the goal of these papers was to pursue the lowest variances for estimators of the effects of interest. Song and Su (1998) discovered that there are three classes of variances among all estimators when applying the AR and MB strategies to estimate metamodels. For example, the first class includes all main effects (naturally considered the most-important effects); the second class contains the mean effect and some two-factor interaction effects; and the third class contains the higher-factor interactions. Specifically, when applying three-class VSRs in simulation experiments, practitioners can specify two requirement sets, say [S.sub.1] and [S.sub.2], where [S.sub.1] contains the most-important effects and [S.sub.2] contains the second-most-important effects, including the mean effect. Let [S.sub.3] be an additional set containing all estimators not in either [S.sub.1] or [S.sub.2]. All estimators in the same class have a common variance. The three-class VSRs attain the following two goals:
Goal 1. all estimators of effects in [S.sub.1] have the lowest possible variances; and
Goal 2. all estimators of effects in [S.sub.2] have smaller variances than estimators of the remaining effects.
We observe that in applying the three-class VSRs we can not allow for variance swapping among effects in the third class. Specifically, if [S.sub.3] contains the highest-interaction effect and some lower-interaction effects, goal 1 and goal 2 do not ensure that the estimators of the lower-interaction effects have smaller variances than that for the highest-interaction, which is considered to be the least important effect. This observation motivates us to add a new goal to our problem.
Goal 3. Allow variance swapping among effects estimators in [S.sub.3].
To attain goal 3 as well as goals 1 and 2, we propose a five-class VSR that induces correlations among all blocks in which all design points have a special correlation structure. We therefore name the proposed rule the Correlated-Blocks (CB) VSR. The CB rule allows one to make a finer distinction among the levels of interest...
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