In this section, we develop user selection schemes with two different objectives: sum rate maximization, and proportional fairness (PF). For each objective, we first derive an optimal user selection criterion and then propose a suboptimal distributed algorithm with low complexity.
Sum rate maximization
We begin with a conventional user selection algorithm for sum rate maximization, which was proposed for a single cell environment. In this case, each BS selects users to maximize only the sum rate of its own cell as
However, this solution is not optimal in a multicell environment due to the intercell interference. In order to maximize the total sum rate of the L cells, we modify the formulation of (12) as
The solution of (13) can only be obtained through centralized optimization among cells, which requires perfect CSI, a lot of signaling overhead among cells, and very high computational complexity. As a more practical solution, we propose a suboptimal distributed user selection algorithm with low complexity. The algorithm is described as in the following steps.
Step 1. Initialization:
Step 2. . Step 3. If , then and go back to the Step 2; otherwise terminate the algorithm.
Each BS independently selects users to be served by using the above algorithm. In Step 1, the set S
of selected users is initialized. In Step 2, the BS chooses one user among the users not in S
so as to maximize the amount of the change in the total sum rate. Note that denotes the amount of the change in the total sum rate when the k th user is added to S
. In Step 3, if the addition of the selected user in Step 2 increases the total sum rate, then the BS adds the user to S
and goes back to Step 2. Otherwise, the algorithm terminates and the final set of selected users is given by S
The most challenging part of the above algorithm is to calculate without sharing information among neighboring cells. We can split into two components as
where denotes the sum rate increment in the i th cell by adding the k th user to S
, and denotes the sum rate decrement in adjacent cells by adding the k th user to S
due to the increased interference. The BS can easily calculate as
However, it is difficult to calculate in the distributed manner, since is dependent on the set of selected users in adjacent cells. Instead of directly calculating , we propose to estimate based on which is fed back from the k th MS in the i th cell. Note that represents the amount of interference caused to adjacent cells by selecting the k th user in the i th cell. The main idea is to estimate by calculating the sum rate decrement in the ith cell to which the BS belongs, with additional interference with the power . Then the estimated sum rate decrement in adjacent cell can be expressed as
where denotes the achievable rate of the k′ th user in the i th cell with additional interference of the power , and it can be calculated as
From (15) and (16), the estimated can be obtained as
The proposed algorithm requires at most KN
computations of per cell, since users are successively selected.
The proportional fairness (PF) scheduling effectively provides a trade-off between the average throughput and fairness among users . The conventional PF scheduling was originally proposed for a single cell environment. In this case, each BS selects users as
where denotes the average throughput estimate of the k th user in the i th cell. We assume that is calculated as
where Tc is the time constant of the averaging window. The solution of (21), however, does not guarantee the system-wide PF due to the intercell interference.
We consider an optimal user selection criterion for the system-wide PF, which can be expressed as
where U1 is the system-wide PF utility function expressed as
As in (13), the optimal solution of (23) needs centralized optimization among cells. Here, we also propose a suboptimal distributed algorithm. Instead of U1 in (23), we use another utility function U2 given as
As provided in the Appendix, the optimization problem (23) remains the same even though U1 is replaced with U2. The use of U2 enables the user selection algorithm to work in a distributed fashion with low computational complexity. Based on the newly defined utility function U2, the proposed algorithm works as follows.
Step 1. Initialization: .
Step 2. .
Step 3. If , then and go back to the Step 2, otherwise terminate the algorithm.
Note that the above algorithm is the same as the distributed algorithm developed in Section 4.1, except that is replaced by , which denotes the amount of the change in U2 when the k th user is added to Si.
As in (14), can be expressed as
where denotes the increment of U2 in the i th cell by adding the k th user to S
, which can be expressed as
in (26) denotes the decrement of U2 in adjacent cells by adding the k th user to S
due to the increased interference. Like the approach used for the total sum rate maximization, we propose to estimate as
Then, by using (27) and (28), the estimation of can be found as
This algorithm also requires at most KN
computations of per cell.