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subject to Using Barcode drawer for Java Control to generate, create Interleaved 2 of 5 image in Java applications.ASP.NET bar code for VS .NET dr 0,. Java USPS Confirm Service Barcode for Java r = 1, . . . , R. Visual Studio .NET Data Matrix 2d barcode for Visual C# Note that the objec Uniform Symbology Specification ITF for Java tive function of this problem is the utilitarian social welfare function (cf. 17); it becomes a reasonable objective if we assume that all utilities are measured in the same (monetary) units. Since the objective function is continuous and the feasible region is compact, an optimal solution d = (d1 , .

. . , dR ) exists.

If the functions Ur are strictly concave, then the optimal solution is unique, since the feasible region is convex. In general, the utility functions are not available to the resource manager. As a result, we consider the following pricing scheme for resource allocation, which we refer to as the proportional allocation mechanism.

Each user r gives a payment (also called a bid) of wr to the resource manager; we assume wr 0. Given the vector w = (w1 , . .

. , wr ), the resource manager chooses an allocation d = (d1 , . .

. , dr ). We assume the manager treats all users alike in other words, the network manager does not price discriminate.

Each user is charged the same price > 0, leading to dr = wr / . We further assume that the manager always seeks to allocate the entire resource capacity C; in this case, we expect the price to satisfy wr = C. r The preceding equality can only be satis ed if =.

VS .NET barcode for C# wr > 0, in which case we have Visual Studio .NET Planet for Visual Basic .NET wr . (21.4) C In ot ITF for Java her words, if the manager chooses to allocate the entire resource, and does not price discriminate between users, then for every nonzero w there is a unique price > 0, which must be chosen by the network, given by the previous equation.

We can interpret this mechanism as a market-clearing process by which a price is set so that demand equals supply. To see this interpretation, note that when a user chooses a total payment wr , it is as if the user has chosen a demand function D(p, wr ) = wr /p. Java Industrial 2 of 5 for Java the design of scalable resource allocation mechanisms Visual Studio .NET USD - 8 for C#.NET for p > 0. The d Java ANSI/AIM ITF 25 emand function describes the quantity the user demands at any given price p > 0. The resource manager then chooses a price so that r D( , wr ) = C, i.

e., so that the aggregate demand equals the supply C. For the speci c form of demand functions we consider here, this leads to the expression for given in (21.

4). User r then receives an allocation given by D( , wr ), and makes a payment D( , wr ) = wr . This interpretation will be further explored in Section 21.

3, where we consider other market-clearing mechanisms for allocating a single resource in inelastic supply, with the users choosing demand functions from a family parameterized by a single scalar.. Java ANSI/AIM I-2/5 for Java 21.2.1 Price Taking Users and Competitive Equilibrium In this section, we I-2/5 for Java consider a competitive equilibrium between the users and the resource manager. A central assumption in the de nition of competitive equilibrium is that each user does not anticipate the effect of their payment wr on the price ; i.e.

, each user acts as a price taker. In this case, given a price > 0, user r acts to maximize the following payoff function over wr 0: Pr (wr ; ) = Ur wr wr . (21.

5). The rst term repre sents the utility to user r of receiving a resource allocation equal to wr / ; the second term is the payment wr made to the manager. Observe that this de nition is consistent with the notion that all utilities are measured in monetary units. We now say a pair (w, ) with w 0 and > 0 is a competitive equilibrium if users maximize their payoff as de ned in (21.

5), and the network clears the market by setting the price according to (21.4): (21.6) Pr (wr ; ) Pr (wr ; ) for wr 0, r = 1, .

. . , R; wr (21.

7) = r . C The following theorem shows that under our assumptions, a competitive equilibrium always exists, and any competitive equilibrium maximizes aggregate utility. Theorem 21.

1 There exists a competitive equilibrium (w, ). In this case, the vector d = w/ is an optimal solution to SYSTEM. proof The key idea in the proof is to use Lagrangian techniques to establish that optimality conditions for (21.

6) (21.7) are identical to the optimality conditions for the problem SYSTEM, under the identi cation d = w/ . Observe that under Assumption 1, the payoff (21.

5) is concave in wr for any > 0. Thus considering the rst-order condition for maximization of Pr (wr ; ) over wr 0, we conclude w and are a competitive equilibrium if and only if Ur (dr ) = , Ur (0) , dr = C,. if dr > 0; if dr = 0;. (21.8) (21.9) (21.

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