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

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Control Techniques for Complex Networks using .net vs 2010 todraw barcode 3 of 9 in web,windows application iReport Draft copy April 22, 2007. Figure 5.1: Velocity space for the processor-sharing model with parameters de ned in (4.31), and its relaxation with velocity space V.

Example 5.3.1.

Processor sharing model Consider the model shown in Figure 2.6 with the parameters speci ed in Example 4.2.

2. Figure 5.1 compares the velocity space V for the uid model previously shown in Figure 4.

2, with the velocity space V for a one-dimensional relaxation. It follows from the geometry illustrated in Figure 4.2 that T is a linear function of x R2 .

In fact, for the numerical values used in Example 4.2.2 the left hand sloping + face of V is given by {v : 1 , v = (1 1 )} = {v : 3v1 + 4v2 = 1}, so that T (x) = 3x1 +4x2 for x R2 .

The minimal draining time T for the one-dimensional + relaxation coincides with T in this example.. 5.3.1 Minimal process The most bas ic control solution for a relaxation is a generalization of the non-idling condition. De nition 5.3.

2. Minimal solution Suppose that R W Rn is a closed convex set containing the origin, and w W is + any given initial condition. An R-minimal solution w starting from w is any feasible solution satisfying the differential constraints (5.

23) and, (i) w (t) R for all t > 0. (ii) If w is any other solution with identical initial condition w and satisfying w(t) R for all t > 0 then,. ws (t) ws (t),. t 0, s {1, . . .

, n}.. When R = W t Code 39 Extended for .NET hen w is called minimal, or point-wise minimal. In this case the process is said to be controlled using the work-conserving policy.

. Control Techniques for Complex Networks Draft copy April 22, 2007. An R-minimal solution exists in one or two dimensions since R is a convex subset of the positive orthant. In higher dimensions we show by example that such a strong form of minimality may not be feasible. Be forewarned that point-wise minimality on W is not necessarily a desirable property.

While the work-conserving policy is typically optimal for a one dimensional relaxation, in Section 5.3.2 we nd that optimal solutions for a relaxation are frequently not work-conserving for dimensions two or higher.

In the one-dimensional relaxation there is a minimal solution on W = R+ , de ned by the non-idling policy: (t) = 0 when w(t) > 0. Similarly, an R-minimal solution always exists for the two dimensional relaxation, regardless of R. For each w0 R2 de ne, 0 (5.

26) Rw0 = w W : wi wi , for all i , We denote by [w]R the projection of w onto the set Rw in the standard 2 norm. Proposition 5.3.

2 follows from Theorem 5.3.3 that follows.

Proposition 5.3.2.

Consider a two-dimensional relaxation of the uid scheduling model, and suppose that R W is a closed convex set containing the origin. Then, for each initial condition w W there exists an R-minimal solution w that can be expressed, w (t; w) = [[w]R t]R , t > 0, w W. (5.

27) The existence of a minimal solution is not guaranteed when the dimension is three or greater.. 1 1. Station 1 Station 2 Station 3 Figure 5.2: Three-station network Example 5.3. 2.

Minimal solution for a three-station network Consider the three-station network shown in Figure 5.2. It is assumed that all service rates are equal to unity, so that the 3 6 workload matrix can be expressed, 1 1 0 1 0 1 = 1 0 1 1 0 1 1 0 1 0 1 1 For simplicity consider the arrival-free model where 1 = 6 = 0, so that = 0.

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