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Control Techniques for Complex Networks in .NET Printer barcode 3/9 in .NET Control Techniques for Complex Networks

Control Techniques for Complex Networks use .net framework barcode 39 generator toincoporate 3 of 9 in .net Microsoft .NET Draft copy April 22, 2007. 2 D = 18. 2 D = 6. 2 D = 24. - 20 - 10 0. - 20. - 10. - 20. - 10. CRW model CBM model q p 9.2 q a 2.3 q p = 9.2 q a = 2.3 q p 26 qa 5 q p = 27.6 q a = 6.9 q p 32 q a 8.3 q p = 36.8 q a = 9.2 Figure 7.22: Optimal policies in a power distribution system. The policy for the CRW model was computed using value iteration.

The grey region indicates those states for which ancillary service ramps up at maximum rate, and the constant qp is the value such that primary ramps up at maximum rate when Q(t) < q p . The optimal policy for the CBM model closely matches the optimal policy for the discrete-time model in each case..

2 (a) { 3, 0, 3}, a = 6 2 (b) { 6, 3, 0, 3, 6}, b = 18 2 (c) { 6, 0, 6}, c = 24. The marginal distribution has zero mean since the support is symmetric in each case. The cost parameters are taken to be cbo + v = 100, ca = 10, cp = 1, and we take p+ = 1, a+ = 2. The state process X is restricted to an integer lattice to facilitate computation of an optimal policy using value iteration.

Optimal policies are illustrated in Figure 7.22: The constant q p is de ned as the maximum of q 0 such that U p (t) = 1 when X(t) = (q, 0)T . The grey region represents Ra , and the constant q a is an approximation of the value of q for x on the right-hand boundary of Ra .

Also shown in Figure 7.22 is a representation of the optimal policy for the CBM model with rst and second order statistics consistent with the CRW model. That is, 2 the demand process D was taken to be a drift-less Brownian motion with variance D equal to 6, 18, or 24 as shown in the gure.

The constants q p , q a indicated in the gure are the optimal parameters for the CBM model obtained using the diffusion heuristic (7.71). The optimal policy for the CBM model closely matches the optimal policy for the discrete-time model in each case.

. Theorem 7.5.1 ANSI/AIM Code 39 for .

NET and the approximations obtained through the diffusion heuristic predict the numerical results previously described in Section 5.6.2.

Example 7.5.3.

Simple re-entrant line without demand. Control Techniques for Complex Networks Draft copy April 22, 2007. Consider the three-dimensional CRW model (4.10) for the simple re-entrant line shown in Figure 2.9.

We maintain the assumptions of Section 5.6.2: the model is homogeneous, and c is the 1 norm.

The effective cost (5.88) is of the form required in Theorem 7.5.

1: The function c(y) = max(y2 , y1 y2 ) is monotone in y1 , and the monotone 1 region Y+ := {y R2 : 2 y1 y2 y1 } has non-empty interior. + Recall that the effective cost was categorized according to four cases listed in Section 5.6.

In this example Case I cannot hold since c is never monotone. The relaxation may satisfy the conditions of Case II or Case III, depending upon the speci c values of { i , 1 }. Based on the lower boundary of Y+ we de ne the height process, H(t) = Y2 (t) 1 Y1 (t), 2 t 0.

. Applying (5.8 visual .net Code 3 of 9 7), we nd that while 1 (t) = 0 it evolves according to the recursion, H(t + 1) = H(t) S2 (t + 1) + S2 (t + 1) 2 (t) + 1 S1 (t + 1), 2 The cost function for the height process is de ned by cH (r) = max( c r, c+ r) H H r R.

t 0.. We have c+ = c+ = 1 and c = c . = 1, giving cH (r) = . r. . 2 2 H H T he height process H for the unconstrained process Y satis es the same re cursion. Under an af ne policy for Y de ned by the switching curve (5.

89), we have 1 0 and 2 is de ned so that the height process evolves as a re ected random walk, H (t + 1) = max(H (t), y 2 ) S2 (t + 1) + 1 S1 (t + 1), 2 We have under any policy, c (Y (t)) = cH (H (t)) + (c+ + c+ /m )Y1 (t), 1 2 (7.73) t 0. (7.

72). and applying Code-39 for .NET (5.88) we have c+ + c+ /m = 1/m = 2 with m = 1 .

The discounted1 2 2 cost optimal policy for Y is obtained on solving the discounted cost optimal control problem for the height process H since Y 1 is uncontrolled. To estimate the threshold y that solves the average-cost optimal control problem 2 in Case II* we take two different paths. First we compute the invariant measure for H when y 2 = 0 and apply the formula (7.

42). This is only a heuristic since the density assumption imposed to obtain (7.42) is not satis ed for H .

Next, we construct a CBM model with identical rst and second order statistics and apply the diffusion heuristic De nition 7.4.1 to obtain a second estimate.

Based on these two approaches we obtain the two approximations, y (CRW) 2 y (CBM) 2.
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