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EE 685 presentation Utility-Optimal Random-Access Control By Jang-Won Lee, Mung Chiang and A. Robert Calderbank
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Objective of the paper Aims to design medium access control (MAC) protocols for wireless networks through the network utility maximization (NUM) framework. Problem formulation through a collision/persistence probabilistic model and aligning selfish utility with total social welfare. Controlling the tradeoff between efficiency and fairness of radio resource allocation. Proposing distributed algorithms to solve the utility-optimal random- access control problem, which lead to more message passing overhead than the current protocols, but significant potential for efficiency and fairness improvement.
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Motivation and basic approach Due to the inadequate feedback mechanism in the BEB protocol, neither convergence nor social welfare optimality can be assured Need for new distributed algorithms convergent to the global optimum of total network utility is obvious A probabilistic-modeled NUM problem for wireless MAC will be solved by optimal algorithms that will be converted to random access MAC protocols. Therefore, optimality with respect to prescribed user utilities, which determine protocol efficiency and fairness, is guaranteed
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Problem Framework The problem is formulated for A network that consists of a set L of unidirectional links of capacities c l, where l is element of L. The network is shared by a set S of sources, where source s is characterized by a utility function U s (x s ) that is concave increasing in its transmission rate x s Each link l is shared by a set S(l) of sources. The goal is to calculate source rates that maximize the sum of the utilities ∑ s S U s (x s ) over x s subject to capacity constraints.
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S Problem Framework s1s1 s2s2 s3s3s.......... DESTINATION NODES SOURCE NODES link l 4 : S(l 4 )={s 1,s 3 } l1l1 l2l2 l3l3 l5l5 l6l6
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The optimization problem : Primal problem
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Utility functions : Relationship to fairness
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System Model and Notation :
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Optimization problem : in terms of probabilistic link capacities The objective of this problem is to obtain the optimal data rate x and the optimal persistence probabilities p for links, and P for nodes so as to maximize the network utility
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Optimization problem : take log of the constraint and log change of variables
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Lemma 1 : Concavity after variable change Lets define a new function g l (x l ) as follows Note that the curvature should be bounded away from 0 as much as So the traffic should be elastic enough for the concavity of utility function after the variable change
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Lemma 2 : Concavity after variable change Hence, if α > 1, g l (x l ) 0. So in this type of utility functions, if α > 1, U’ l (x l ) becomes a strictly concave function as desired. So throughout the paper α > 1 has been assumed
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The optimization problem : Lagrangian for primal problem
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The optimization problem : Dual problem Note that in this Lagrangian, we do not need to relax the second constraint in problem (5). By definition, the Lagrange dual function is Dual problem typically formulated as the minimization of upper boundary for the Lagrangian The maximization of Lagrangian (equation 7) can be independently made in each node in parallel (over x’,p,P)
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The optimization problem : Dual problem solution Since Lagrangian function has two components that can be separately maximized in terms of x’ and (p,P) pair, we have
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The optimization problem : Dual problem solution
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We can now solve the dual problem (8) by using a subgradient projection algorithm 4 at each link l, i.e., at each node n such that l ∈ L out (n), through the following iterations indexed by t
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The optimization problem : Distributed Algorithm 1
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The optimization problem : Distributed Algorithm 2
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Remarks : Remark 2
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Remarks Remark 4 The number of message passing required in each of the above two algorithms depends on the network topology. The average numbers of message passing in each iteration for Algorithm 1 and Algorithm 2, M 1 and M 2, are obtained as
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Remarks : Remark 5
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Heuristics Heuristics decreasing message passing
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THEOREM 1 (optimality and convergence) Proceeding to prove the optimality and convergence of Algorithms1 and 2. For a rigorous proof, we first need the following technical condition to have a unique solution to problem (10) at the optimal dual solution. At the optimal dual solution λ*,
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THEOREM 1 proof (optimality and convergence)
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Performance results The performances of proposed protocols have been compared with those of the deterministic approximation protocol and the standard BEB protocol, showing that both protocols can provide not only a higher network utility and a larger fairness index, but also a wider dynamic range of the tradeoff curve between efficiency and fairness. Performance guarantee of convergence to the global optimum of the NUM formulation is rigorously proved for the proposed algorithms, and simplifying heuristics are then developed based on the optimal algorithmst
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