EE 685 presentation

1. Utility-Optimal Random-Access Control By Jang-Won Lee, Mung Chiang and A. Robert Calderbank EE 685 presentation

2. Objective of the paper • Aims to designmedium access control (MAC)protocols for wireless networks through the network utilitymaximization (NUM) framework. • Problem formulation through a collision/persistenceprobabilisticmodel and aligning selfish utility with total socialwelfare. • Controlling the tradeoffbetween efficiency and fairness of radio resource allocation. • Proposing distributed algorithms to solve the utility-optimalrandom-access control problem, which lead to more message passing overheadthan the current protocols, but significantpotential for efficiency and fairness improvement.

3. Motivation and basic approach • Due to the inadequate feedback mechanism in the BEBprotocol, neither convergence nor social welfare optimality canbe assured • Need for new distributed algorithms convergent to the global optimum of total network utility is obvious • A probabilistic-modeled NUM problem forwireless MAC will be solved by optimal algorithms that will be converted to random access MAC protocols. • Therefore, optimality with respect to prescribed user utilities, whichdetermine protocol efficiency and fairness, is guaranteed

4. Problem Framework The problem is formulated for • A network that consists of a set L of unidirectional links of capacities cl, 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 Us(xs) that is concave increasing in its transmission rate xs • Each link l is sharedby a set S(l) of sources. • The goal is to calculate source rates that maximize the sum of the utilities∑sϵ S Us(xs) over xs subject to capacity constraints.

5. Problem Framework DESTINATION NODES link l4 : S(l4)={s1,s3} l3 l5 l2 l1 l6 S SOURCE NODES .......... s1 s2 s3 ss

6. The optimization problem :Primal problem

7. Utility functions :Relationship to fairness

8. System Model and Notation :

9. System Model and Notation :

10. Optimization problem :in terms of probabilistic link capacities • The objective of this problem is to obtain the optimal datarate x and the optimal persistence probabilities p for links,and P for nodes so as to maximize the network utility

11. Optimization problem :take log of the constraint and log change of variables

12. Lemma 1 :Concavity after variable change • Lets define a new function gl(xl) 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

13. Lemma 2 :Concavity after variable change • Hence, if α > 1, gl(xl) < 0 and if α < 1, gl(xl) > 0. So in this type of utility functions, if α > 1, U’l(xl)becomes a strictly concave function as desired. So throughout the paper α > 1 has been assumed

14. The optimization problem :Lagrangian for primal problem

15. The optimization problem :Dual problem • Note that in this Lagrangian, we do not need to relax thesecond constraint in problem (5). By definition, the Lagrangedual 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)

16. 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

17. The optimization problem :Dual problem solution

18. The optimization problem :Dual problem solution • We can now solve the dual problem (8) by using a subgradientprojection algorithm 4 at each link l, i.e., at each node n such that l ∈ Lout(n), through the following iterations indexed byt

19. The optimization problem :Distributed Algorithm 1

20. The optimization problem :Distributed Algorithm 1

21. The optimization problem :Distributed Algorithm 2

22. The optimization problem :Distributed Algorithm 2

23. Remarks :Remark 2

24. RemarksRemark 4 • The number of message passing required ineach of the above twoalgorithms depends on the networktopology. The average numbers of message passing in eachiteration for Algorithm 1 and Algorithm 2, M1 andM2, areobtained as

25. Remarks :Remark 5

26. HeuristicsHeuristics decreasing message passing

27. HeuristicsHeuristics decreasing message passing

28. 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 toproblem (10) at the optimal dual solution. At the optimal dualsolution λ*,

29. THEOREM 1 proof (optimality and convergence)

30. Performance results • The performances of proposed protocols have been compared withthose of the deterministic approximation protocol and thestandard BEB protocol, showing that both protocols canprovide not only a higher network utility and a larger fairnessindex, but also a wider dynamic range of the tradeoff curvebetween efficiency and fairness. • Performance guarantee ofconvergence to the global optimum of the NUM formulation isrigorously proved for the proposed algorithms, and simplifyingheuristics are then developed based on the optimalalgorithmst