EE 685 presentation

Utility-Optimal Random-Access ControlBy Jang-Won Lee, Mung Chiang and A. Robert Calderbank

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.

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

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 Us(xs) that is concave increasing in its transmission rate xs

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 ϵ S Us(xs) over xs subject to capacity constraints.

S

Problem Framework

s1 s2 s3 ss

..........

DESTINATION NODES

SOURCE NODES

link l4 : S(l4)={s1,s3}

l1

l2

l3 l5

l6

The optimization problem :Primal problem

Utility functions :Relationship to fairness

System Model and Notation :

System Model and Notation :

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

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

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

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

The optimization problem :Lagrangian for primal problem

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)

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

The optimization problem :Dual problem solution

The optimization problem :Dual problem solution

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 ∈Lout(n), through the following iterations indexed by t

The optimization problem :Distributed Algorithm 1

The optimization problem :Distributed Algorithm 1

The optimization problem :Distributed Algorithm 2

The optimization problem :Distributed Algorithm 2

Remarks :Remark 2

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, M1 and M2, are obtained as

Remarks :Remark 5

HeuristicsHeuristics decreasing message passing

HeuristicsHeuristics decreasing message passing

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*,

THEOREM 1 proof (optimality and convergence)

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