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Workload in general models in .NET Draw barcode 3/9 in .NET Workload in general models




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6.1 Workload in general models using .net vs 2010 toproduce code 3/9 with asp.net web,windows application Web application Here we co .net vs 2010 Code39 nsider notions of workload for the general network model (6.1).

The de nition of workload vectors is more subtle in this general setting in which the matrix B is not square. In particular, we cannot de ne the workload matrix through a matrix inversion as in De nition 4.2.

3. However, a complete generalization of the development of Section 4.2 is possible by re-examining the minimal draining time.

The main results of this section are summarized in Theorem 6.1.1.

The representation (6.7) implies that the workload vectors can be interpreted as Lagrange multipliers since they de ne sensitivity of the minimal draining time with respect to the initial condition x. The representation = W ( ) provides the following interpretation of the network load: The vector is equal to the total amount of exogenous arrivals in one second.

Consequently, the network load is equal to the amount of time required to process these arrivals, given that there are no additional arrivals to the network.. Control Techniques for Complex Networks Draft copy April 22, 2007. Theorem 6. 1.1.

(Geometric Construction of Workload) Suppose that the arrivalfree model is stabilizable. Then, there are vectors { s : 1 s r } R such that the following hold for the model (6.1), (i) The minimal draining time for the arrival-free model de ned in (6.

4) can be expressed as the maximum, W (x) = max s , x ,. 1 s r x R . + (6.7). (ii) The n etwork load de ned in De nition 6.0.1 can be expressed, = W ( ) = max s ,.

1 s r R , + where s = .net vs 2010 barcode 39 s , . (iii) The model (6.

1) is stabilizable if, and only if, < 1. (iv) If the network is stabilizable then the minimal draining time can be expressed, T (x) = max s, x < , 1 s x R . + (6.

8). 1 s r Proof. Par barcode 3 of 9 for .NET ts (i), (iii) and (iv) are contained in Proposition 6.

1.4. The vectors { s } are de ned in De nition 6.

1.1 below. To see (ii) consider the linear program (6.

5) that de nes the network load. This is precisely (6.4) when x = , and hence (ii) follows from (i).

We consider both geometric and algebraic (linear programming) interpretations of workload vectors since both approaches have value for intuition, computation, and improving the theory. 6.1.

0.4 Workload and the velocity space. The proof of Theorem 6.1.1 requires a closer look at the geometry of the velocity space V R , de ned by V := {B + : U}.

(6.9) Letting V0 denote the velocity space for the arrival-free model, V0 := {v = B : U}, (6.10).

one may co .NET Code 3 of 9 nclude from the de nition (6.9) that the set V is expressed as the translation, V = {V0 + } := {v + : v V0 }.

The polyhedron V0 contains the origin in R since by de nition we have 0 U. It follows that this set can be expressed as the intersection of half-spaces, V0 = {v R : s , v os , 1 s v }, (6.11).

Control Techniques for Complex Networks Draft copy April 22, 2007. where the constants {os : 1 s v } take on values zero or one, and s R for each s. We assume that the vectors { s } are minimally speci ed, in the sense that V0 can not be represented via (6.11) using a subset of the vectors { s }.

The set V0 has non-empty interior since the arrival-free model is stabilizable: This follows from Proposition 6.1.3 with = 0, which implies that {v R : < vi < 0} V0 for some > 0.

Hence the set {v V0 : s , v = os } is an ( 1) dimensional face of the polyhedron V0 . This face passes through the origin if, and only if, os = 0. Given this structure, we arrive at a geometric construction of workload vectors: De nition 6.

1.1. Workload In Units Of Time Suppose that the arrival-free model is stabilizable.

Then, (i) The vector s R is called a workload vector if os = 1. The number of distinct workload vectors is denoted r . By reordering, we assume that os = 1 if and only if s r .

(ii) The r -dimensional vector load is given by = ( 1 , . . .

, r )T , where s := s , for s v .. The load p Code 39 for .NET arameters { s } can be used to represent the velocity space V as follows: Proposition 6.1.

2. The velocity space can be expressed,. V = {v R : s , v (os s ), s = 1, . . .

, v }. Proof. We begin with the representation of the velocity space as a translation V = {V0 + }.

Hence, from (6.11), on writing v = v + , V = {v + R : s , v os , s = 1, . .

. , v }. s . = {v R : , v os + s , s = 1, . . .

, v }. A geometric construction of the minimal draining time is possible based on consideration of the translation {V + x0 } for a given initial condition x0 X. If x1 X {V + x0 } then v = x1 x0 V, and it follows that the following trajectory is feasible, and travels from x0 to x1 in exactly one second, q(t) = x0 + tv, 0 t 1.

. = {v R 39 barcode for .NET : s , v os , s = 1, . .

. , v }. (6.12). This shows Visual Studio .NET Code39 that X {V + x0 } is precisely the set of states reachable from x0 in one second or less. Similarly, on scaling V by a real number T > 0 we nd that X {T V + x0 } is identi ed as the set of states reachable from x0 in no more than T seconds.

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