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

2. 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.

3. 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

4. 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.

5. 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

6. The optimization problem : Primal problem

7. Utility functions : Relationship to fairness

8. System Model and Notation :

10. 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

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 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

13. 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

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 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)

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.  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

19. The optimization problem : Distributed Algorithm 1

21. The optimization problem : Distributed Algorithm 2

23. Remarks : Remark 2

24. 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

25. Remarks : Remark 5

26. Heuristics Heuristics 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 to problem (10) at the optimal dual solution. At the optimal dual solution λ*,

29. THEOREM 1 proof (optimality and convergence)

30. 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