QOS-AWARE RESOURCE ALLOCATION 29
We investigate the problem of optimal rate allocation in the sense of maximizing the aggregated
utility function
( ).
f f
f F
U x
Î
å
U
f
(x
f
), which is also referred to as the social welfare in the literature [13]. Such
an optimization objective achieves Pareto optimality with respect to the resource utilization and also
realizes different fairness models—including proportional and max-min fairness [14]—when appro-
priate utility functions are specified. The problem of optimal resource allocation in wireless ad hoc
networks can be then formulated as a nonlinear optimization problem (primal problem):
P:
maximize
( ).
f f
f F
U x
Î
å
U
f
(x
f
) (2.2)
subject to Rx C (2.3)
over x
f
I
f
(2.4)
This optimization problem maximizes the aggregated utility function in Equation 2.2 with
the resource capacity constraints shown in Equation 2.3 and coming from the shared wireless chan-
nel as discussed in Section 2.2. Note that we are dealing with maximal cliques and clique-flow matrix
R, which represents the “resource usage pattern of each flow as a constraint in (Equation 2.3). If we
can optimize the problem P, we will get both optimal resource utilization and fair resource alloca-
tions among end-to-end flows spanning multiple hops. However, often arriving toward a solution
of the problem P might be complex and difficult since finding the right utility functions is difficult,
hence one might look at the dual problem D to problem P and consider pricing-based description
D of the optimal resource allocation problem rather than utility-based description P.
2.3.2 Price-Based Resource Allocation Problem
To proceed toward the solution of the problem P, so that we achieve optimal resource allocation
(and QoS with respect to rate allocation) in terms of both utilization and fairness, we will transform
the problem P into its dual problem as follows. Owing to Assumption 1 in Section 2.3.1, the ob-
jective aggregated utility function
( ).
f f
f F
U x
Î
å
U
f
(x
f
) in Equation 2.2 is differentiable and strictly concave.
Also, the feasible region of the optimization problem in Equations 2.3 and 2.4 is convex and com-
pact. Hence, by nonlinear optimization theory, there exists a maximizing value of argument x for the
above optimization problem. Let us consider the Lagrangian form of the optimization problem P:
= - +
å å å
( ; ) ( ( ) )
f f f q qf q q
f F q Q q Q
L x U x x R Cµ
Î Î Î
µµ
( ).
f f
f F
U x
Î
å
U
f
(x
f
) - x
f
= - +
å å å
( ; ) ( ( ) )
f f f q qf q q
f F q Q q Q
L x U x x R Cµ
Î Î Î
µµ
+
= - +
å å å
( ; ) ( ( ) )
f f f q qf q
q
f F q Q q Q
L x U x x R Cµ
Î Î Î
µµ
(2.5)
where
µ
q
= (
µ
q
, q Q) is a vector of Lagrange multipliers. Note that Equation 2.5 represents the
method of Lagrange multipliers as we described in Table 2.2, where we have f (x) =
( ).
f f
f F
U x
Î
å
U
f
(x
f
),

Get Quality of Service in Wireless Networks Over Unlicensed Spectrum now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.