3.2 Fairness property
To investigate the fairness property of the formulated problem, let us denote by
R
u
n
the sum rate of each user
u in each cell
n,
the set of users in cell
n,
the set of users with the minimum rate in cell
n,
the set of other users (the complimentary set of
), and
(
) the set of subcarriers allocated to MS
(
), where
(
). Note that
means that
(13)
Then, for each subcarrier k, there exists three states considering which one of the two user sets does the allocated user belong to in each cell. Specifically, state 1 corresponds to , meaning that in each cell, subcarrier k is allocated to a user with the minimum rate of that cell. State 2 corresponds to , meaning that in each cell, subcarrier k is allocated to a user with a rate larger than the minimum rate of that cell. State 3 corresponds to ∃ n1,n2 such that and , meaning other possible allocation results for subcarrier k in all cells. Note that when all subcarriers are in state 1, per-cell user fairness is reached, meaning all users inside the same cell obtain the same rate.
We first consider the following lemma, which will be used later in this section.
Lemma 1. Given any feasible values of variables A, B and P, if there exists n and k where and (meaning that there exists subcarrier k in state 2 or state 3), then part of the feasible powers at this subcarrier k in each cell( and ) can be saved while keeping the WSMR of the system non-decreased.
Proof. Considering the subcarrier k in cell n where and , we now introduce two sets of cells and . Note that when k is in state 2, and . Also, when k is in state 3, and .
Let us consider the case when
k is in state 3. We define
(14)
where s
k
∈(0,1], , and . Please note that s
k
is the same for all cells.
,
can be denoted as
(15)
which can be further written as
(16)
It is obvious that is an increasing function of s
k
. Similarly, , , and are also increasing functions of s
k
. Thus, is an increasing function of s
k
.
,
can be denoted as
(17)
It is obvious that is a decreasing function of s
k
. Similarly, , , and are also decreasing functions of s
k
. Thus, is a decreasing function of s
k
.
Note that , where the inequality should be understood as elementwise. Thus, , such that and . Note that the WSMR of the system is non-decreased, while the power is saved from subcarrier k in cell .
Let us introduce
. After substituting
with
, we now have
and
. Let us define
where
c
k
∈(0,1). Thus,
can be denoted as
(18)
It is obvious that is an increasing function of c
k
. Similarly, , , and are also increasing functions of c
k
. Thus, is an increasing function of c
k
. There exists a such that . The power can be saved from subcarrier k in each cell n while keeping the WSMR of the system non-decreased.
We now consider the case when k is in state 2, meaning . By using the previous derivations, we can still find , such that the power can be saved from subcarrier k in each cell n while keeping the WSMR of the system non-decreased. This concludes the proof of Lemma 1.
We now state the main result of this section. Based on Lemma 1, the following theorem is proposed, which sheds light on the fairness property of the formulated problem.
Theorem 1. With any feasible values A=A0 and B=B0, the optimum power allocation Popt of the formulated problem results in the same transmission rates for all users in the same cell.
Proof. When A=A0 and B=B0, we assume at the optimum, ∃ n0 where and a subcarrier in state 2 or state 3. Then, where , subcarrier k can be in either state 3 or state 1.
When subcarrier k is in state 3, as discussed in the proof of Lemma 1, we can still improve by decreasing the interfering powers of subcarrier k in cell n′ while keeping the minimum user rate of cell n′ non-decreased.
When subcarrier
k is in state 1, as at the optimum ∃
k0≠
k that is in state 2 or state 3, we can save power from
in each cell
n while keeping the WSMR of the system non-decreased. Let us denote the saved power in each cell
n as
Δ P
n
and define
where
t
k
>1. Thus,
can be denoted as
(19)
It is obvious that is an increasing function of t
k
. Similarly, , , and are also increasing functions of t
k
. Thus, for the subcarrier k in state 1, is an increasing function of t
k
. Thus, there exists a such that and . Here, denotes the set of subcarriers in state 1.
Therefore, in cell n, can still be improved. Thus, the minimum user rate of each cell can still be improved by adjusting power allocation. Specifically, we can find matrices S,C,T,P1=S C T Popt, according to the previous discussions, such that R(P1,A0,B0)>R(Popt,A0,B0). Here, each element of S,C, and T takes value from (0,1],(0,1], and [1,∞), respectively. This obviously contradicts the optimum assumption and concludes the proof of Theorem 1.