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...ink). Similar approaches have been used to achieve fairness in data generation [13], [26] and in session rate allocations [12]. The network utility maximization framework is also well-developed [29], =-=[33]-=-, but mostly for fairness among nodes, not across the different time slots. To apply it, α−fair functions are used under certain objective functions that will be described in Sections V and VI. α−fair...

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...], · · · ,SK [l]) with Ul(·)’s denoting individual utility functions. Since both the ATPC and AIPC involve L transmit covariance matrices, the Lagrange dual decomposition (see, e.g., a tutorial paper =-=[44]-=-) is a general method to deal with this type of average constraints for optimization over a number of parallel dimensions, which is explained as follows. By introducing a set of dual variables, νk’s, ...

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...then be obtained from these KKT optimality conditions: 3 Note that this method has been recently applied broadly to resource allocation problems for communication systems (see, e.g., a tutorial paper =-=[39]-=- and references therein). In particular, this method has been applied to multiuser resources allocation over fading or parallel Gaussian channels in, e.g., [28], [31], [40]-[42].12 Lemma 3.1: The opt...

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...cal shares, rather than a more complicated weight vector. 4. Pisces Optimizations Pisces’s algorithms and its four part decomposition can be conceptually derived as a distributed optimization problem =-=[24]-=-. In designing Pisces, we consider a multitimescale decomposition of the fair-sharing problem into the corresponding Pisces mechanisms. Such a formulation allows us to craft the Pisces algorithms in a...

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...allows each link to perform the optimization in (P3) autonomously until convergence. The Gauss-Seidel iteration, the Jacobi iteration, or an entirely asynchronous iteration may be adopted to this end =-=[18]-=-. Table I lists a Gauss-Seidel version of the iterative algorithm. The ensuing proposition links the solution of the original problem (P2) with that of its first-order approximation (P3). Proposition ...

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...ibuted Rate Control Algorithm The above sUnicast problem is a linear program with size proportional to the number of nodes in V , and thus can be solved in polynomial time. Optimization decomposition =-=[9]-=- is well-suited for solving such a problem in a decentralized manner. Peculiar to the sUnicast problem, we propose the following dual-decomposition procedure [9] that leads to a distributed rate contr...

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...ion [15], [16]. The mathematical tools widely used in this new approach are essentially optimization problem decomposition by Lagrange relaxation, sub gradient algorithms and Lyapunov stability [17], =-=[18]-=-. The work presented in this paper is based on the decomposition of an optimization problem defined over the multi-channel network model. The solution is related to the general scheduling algorithm pr...

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... technique has been extensively used in network resource allocation for elastic data traffic (e.g., [24]–[27]). Here we will briefly review the main results, and details can be found in, for example, =-=[31]-=-. First, we relax the constraint in (1) with a dual variable and obtain the following Lagrangian where . The variable can be interpreted as the shadow price for the constrained resource . Then Problem...