#### 2.1. System Model

In the case of multi-cell WLAN scenarios, there is a set of cells

$\mathbb{G}=\{{g}_{1},{g}_{2},\dots ,{g}_{L}\}$. A cell

${g}_{l}$ contains an AP and

${n}_{l}$ users. An AP consist of multiple directional beams. Furthermore, a schedule set for the cell

g is defined as

${\mathbb{S}}^{g}=\{{S}_{1}^{g},{S}_{2}^{g},\dots ,{S}_{N}^{g}\}$, where a schedule includes two users for a single beam in a time slot, which is presented as

${S}_{i}^{g}=\{{u}_{i,1}^{g},{u}_{i,2}^{g}\}$. When more than two users are allocated a schedule, finding a rate-region can be highly time consuming due to the complex calculation of multiple variables for optimization. Even if more than two users can be supported, two user scheduling is experimentally proven to be the most cost-efficient scheduling compared to other numbers of users in a schedule [

16]. Thus, we assume the number of users in a schedule is two for a single beam in this scenario.

When we consider a schedule

${S}_{i}^{g}$ that is scheduled by beam

b with power

${p}_{b}^{g}$ in the cell

g, the power allocated to two users in the schedule is expressed as Equation (

1):

where

$\alpha $ is the proportion of the beam power allocated to

${u}_{i,1}^{g}$.

$\mathbf{w}\left({u}_{i,\xb7}^{g}\right)$ is the interfering signal from the other beams and from other APs, and

$n\left({u}_{i,\xb7}^{g}\right)$ is additive white Gaussian noise at

${u}_{i,\xb7}^{g}$. Then, the received signals of the users scheduled to beam

b in a cell

g are described as Equation (

2):

where

${h}_{b,{u}_{i,\xb7}^{g}}$ is the channel gain between an AP of cell

g and node

${u}_{i,\xb7}$, and

$\mathbf{w}\left({u}_{i,\xb7}^{g}\right)$ and

$n\left({u}_{i,\xb7}^{g}\right)$ are interference and white Gaussian noise with variance

${\sigma}^{2}$ for node

${u}_{i,\xb7}$, respectively.

We assume the user

${u}_{i,1}^{g}$ is closer to the direction of beam

b of an AP in the cell

g than

${u}_{i,2}^{g}$ and

${u}_{i,1}^{g}$ can completely decode and cancel the packets for

${u}_{i,2}^{g}$ because the power allocated to

${u}_{i,2}^{g}$ is larger than the power allocated to

${u}_{i,1}^{g}$. When we consider collected interference at users

${u}_{i,1}^{g},{u}_{i,2}^{g}$ in AP

g from the other APs as

${I}_{{u}_{i,1}^{g}}^{m},{I}_{{u}_{i,2}^{g}}^{m}$ and the intra-cell interference at the users, which is allocated to beam

b as

${I}_{b,{u}_{i,1}^{g}},{I}_{b,{u}_{i,2}^{g}}$, then we can obtain the throughput of users as Equations (

3) and (

4):

where

${\mathbb{B}}_{g}$ is the set of beams in a cell

g,

${\mathbf{p}}_{g}$ is the beam power allocation vector for the beams in a cell

g, and

W is the bandwidth of the channel.

Now, we define the normalized throughput for the schedule

${S}_{i}^{g}$ as:

Here, both

${C}_{{u}_{i,1}^{g}}(1,{\mathbf{p}}_{g})$ and

${C}_{{u}_{i,2}^{g}}(0,{\mathbf{p}}_{g})$ represent the throughput of users

${u}_{i,1}^{g}$ and

${u}_{i,2}^{g}$ when all the beam power of cell

g,

${p}_{b}^{g}$ is entirely allocated to those users, respectively. This normalized throughput measures the sum of rationality of the throughput of users when NOMA is applied compared to the throughput of users when NOMA is not applied, which evaluates the effectiveness of the NOMA scheme. Now, the optimal value of

$\alpha $ that maximizes the normalized throughput can be easily obtained by the value of

$\alpha $, which makes the derivatives of Equation (

5) equal to zero, because normalized the throughput in Equation (

5) is a concave function.

where

${I}_{{u}_{i,1}^{g}}^{mn}={I}_{b,{u}_{i,1}^{g}}+{I}_{{u}_{i,1}^{g}}^{m}+{\sigma}^{2}$ means the interference plus noise power at user

${u}_{i,1}^{g}$, which is measured by the cumulative sum of interference from other beams in a single cell, interference from the other cells, and the noise power.

Finally, we can derive the explicit solution form of the maximum normalized capacity of the schedule

${S}_{i}^{g}$ as the following Equation (7) using the optimal

$\alpha $ value as depicted in Equation (

6).

#### 2.2. Problem Formulation

In this section, an optimization problem that maximize the fairness among schedules considering users’ average throughput is presented. The proportional fairness (PF) scheduler for NOMA maximizes

${\sum}_{{S}_{i}^{g}\in {\mathbb{S}}_{g}}log\left(\overline{R({u}_{i,1}^{g};t)}+\overline{R({u}_{i,2}^{g};t)}\right)$ where

${\mathbb{S}}_{g}$ is defined as a set of schedules and

$\overline{R(k;t)}$ is the average throughput of user

k at time

t [

17]. Then, we can define the average throughput of user

k and the instant throughput of the user at time

t as

$r(k;t)$. When the averaging time period is

${\tau}_{c}$, the averaged throughput of user

k can be recursively derived as Equation (

8).

where:

To perform the proportional fairness algorithm among users in the cells [

17], we derive the objective function

$f(\mathbf{p},\mathbf{e})$, which is proportional to the instant normalized throughput and inversely proportional to the average user throughput of the users in a schedule. Then, we build an optimization problem that maximizes the sum of user throughput as Equation (11):

Here, ${P}_{max}$ is the maximum power budget of an AP, ${p}_{b}^{g}$ is the allocated power of beam b in a cell g, and ${e}_{{S}_{i},b}^{g}$ is the scheduling indicator, which is equal to one if a schedule ${S}_{i}$ in a cell g is selected by beam b, otherwise zero.