Wireless Network Design: Optimization Models and Solution ...

328 Nitin Salodkar and Abhay Karandikar
number of users having independent and diverse channel states, there exists a user
having good channel state with a high probability. Moreover, this probability increases
with the number of users. The implications of opportunistic scheduling have
been investigated in further detail in Section 14.4.
Let Cg(x,P(x)) denote the set of achievable rates under a policy allocation policy
P(x). It can be expressed as:
Cg(x,P) =
�
R : ∑ R
i∈S
i ≤ W log
�
1 + ∑i∈S x i P i
WN0
�
�
∀S ∈ {1,...,N} . (14.18)
A power allocation policy P is feasible if it satisfies the power constraints of all
users, i.e., E[P(X)] = ¯P. Let F be the set of all feasible power allocation policies.
The throughput capacity region is defined as the union of the set of rates achievable
under all power control policies P ∈ F, i.e.,
C( ¯P) = �
E[Cg(X,P(X))]. (14.19)
P∈F
In a general case of asymmetric channels and power constraints, weighted rate
maximization is a more appropriate metric. Let γ = [γ 1 ,...,γ N ] T be a vector of
weights assigned to the users. The weighted rate maximization problem can be expressed
as:
max γ · R, (14.20)
subject to the constraint that the rate vector lies in the capacity region:
R ∈ C( ¯P). (14.21)
Using a Lagrangian formulation [8], it can be shown that the optimal power allocation
policy can be computed by solving, for each channel state vector X = x, the
following optimization problem:
subject to:
maxγ
· R − λ · P, (14.22)
R,P
R ∈ Cg(X,P). (14.23)
The optimal solution to (14.22) thus provides a power allocation P(x) and a rate
allocation R(x) at channel state vector X = x. If the choice of λ = [λ 1 ,...,λ N ] T
ensures that the power constraint is met then R ∗ = E[R(X)] is an optimal solution
to (14.22). It can be shown that the optimal solution to (14.22) is a greedy successive
decoding scheme where the users are decoded in an order that is dependent on the
interference experienced by them.