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J(W (t)) = J(W (0)) + in .NET Paint barcode code39 in .NET J(W (t)) = J(W (0)) +




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J(W (t)) = J(W (0)) + use .net barcode 3 of 9 generating toassign ansi/aim code 39 with .net Visual Studio 2010 c(W (s)) + bCBM (W (s)) 39 barcode for .NET ds J(W (s)), dN (s) ..

We necessarily have (J ) < since (V ) < . Setting W (0) , taking expectations, and canceling the common terms E [J (W (t))] = E [J (W (0))] gives,. t (c) = E c(W (s)) = E bCBM (W (s)) = t (bCBM ). . 8.7.2 Linear programs Here we consider extensio .net vs 2010 Code 39 ns of the linear programming theory to stochastic workload models. The main complication is that the effective cost c de ned in De nition 5.

3.3 is in general piecewise linear when the cost function for Q is linear. In-spite of this apparent complexity, the linear programming approach has a direct and elegant extension to the CBM model based on Proposition 8.

7.2..

Control Techniques for Complex Networks Draft copy April 22, 2007. Suppose that Assumptions (a) (c) of Theorem 5.3.13 hold, and let {Rj : i = 1, .

. . , O } denote open, connected polyhedral regions satisfying the following: The function bCBM given in (8.

104) is constant on each Rj , c is linear on Rj , and R = closure ( Rj ). We also consider a family of auxiliary functions {cai : 1 i a } that are compatible with c, in the sense that each of these functions is continuous, piecewise linear, and linear on each of the sets {Rj }. Consequently, the assumptions of Theorem 8.

7.3 hold: Letting {J i : 1 i a } denote the associated C 1 value functions, and setting bai = D J i + cai , we obtain the identity (cai ) = (bai ) for each i. These identities CBM CBM are interpreted as equality constraints below.

The variables in the linear program are de ned for 1 i n, 1 j O , by Pj = (Rj ), We have several constraints: (a) Mass constraints: Pj 0 for each j, and Pj = 1. ij = E [Wi (t)1Rj ]..

(b) Region constraints: F Code-39 for .NET or example, 1j 2j if w1 w2 within region Rj . (c) Value function constraints: For some constants {aij } R and vectors { ij : 1 i a , 1 j n} Rn we have the representations for any 1 j O , 1 i a , bai (w) = aij ; CBM cai (w) = ij , w , w Rj .

. Letting j = ( 1j , . . .

, nj )T Rn , 1 j O , we obtain from Theorem 8.7.3, for each i {1, .

. . , a },.

O O ij , j = (cai ) = (bai ) = CBM j=1 j=1 aij Pj . (8.105). (d) Objective function: T here is d Rn O such that := (c) = We illustrate this construction in a two-dimensional example. Example 8.7.

1. KSRS model. dij ij . We return to the two case s introduced in Example 5.3.3 in the setting of Example 5.

3.7. In the workload model we have by the symmetry assumptions imposed in these examples, = ( 1 , 1 )T , and we assume in the CBM model that the covariance matrix satis es 11 = 22 > 0.

Consider rst the policy de ned by the constraint region R = {w W : w1 /3 w2 3w1 }. This coincides with the monotone region W+ = closure (R2 ) shown at right in Figure 5.3 in Case II.

. Control Techniques for Complex Networks Draft copy April 22, 2007. The cost function restric visual .net Code 3/9 ted to R is the same in Cases I and II, and the common value function shown in (5.58) is purely quadratic on R.

Consequently, in this case we have h = J, and 1 (8.106) = (bCBM ) = bCBM = 1 1 (3 11 12 ). 8 Consider now the minimal process on W = R2 in Case I.

The function bCBM is + not constant, so it is not obvious that we can compute exactly using these techniques when R = W. To construct an LP we restrict to the following speci cations: O = 3, with {Ri : i = 1, 2, 3} as shown in Figure 5.4, and a = 2, with ca1 (w) = w1 + w2 and ca2 (w) = 1 max( 1 w1 , 3 w2 , 1 (w1 + w4 )).

3 4 We thus obtain the following constraints: (a) Mass constraints: P1 + P2 + P3 = 1 (b) Region constraints: We have ij 0 for all i, j since W = R2 . Moreover, on + considering the structure of the sets {Ri } we obtain, 3 21 11 , and 3 13 23 . In addition, there are numerous symmetry constraints.

For example, P1 = P3 , and 12 = 22 since 1 = 2 and 11 = 22 . (c) Value function constraints: The value function J 1 is a pure quadratic. In fact, if k(w) = ca1 , w is any linear function on W, then.

1 2 2 J(w) = 1 1 (ca1 w 1 + ca1 w2 ) , 1 2 2.
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