# A Fairness Index Based on Rate Variance for Downlink Non-Orthogonal Multiple Access System

^{*}

## Abstract

**:**

## 1. Introduction

## 2. System Model

_{t}, the bandwidth is W, and the channel bandwidth of the nth cluster is W

_{n}, BS transmits superimposed coded signals to the n-th cluster, then the signals received by strong channel gain user ${U}_{n}^{s}$ and weak channel gain user ${U}_{n}^{w}$ are, respectively:

_{0}. Using the successful interference cancellation (SIC) technology to decode the received signal, ${U}_{n}^{w}$ can obtain a transmission rate as:

## 3. Fairness Index Based on Rate

**R**= {R

_{1}, …R

_{m}, …, R

_{M}}, and the total throughput of the system can be defined as:

- (1)
- When the transmission rates of all users in the system are equal, the maximum value of F can be obtained (F
_{max}= 1). - (2)
- The minimum value of F can be obtained (F
_{min}= 0) when only one user’s transmission rate is not 0 and all the other users’ rates are 0. - (3)
- The value range of F has nothing to do with the number of users, channel conditions, or transmission power, but only with the distribution of users’ transmission rates.

_{c}

_{−n}and inter cluster fairness F

_{c}are respectively defined as follows:

## 4. Power Allocation under Fairness Constraints

_{n}, then the power allocated to strong channel gain user is ${p}_{n}^{s}={\alpha}_{n}^{s}{p}_{n}$, and the power allocated to weak channel gain user is ${p}_{n}^{w}={\alpha}_{n}^{w}{p}_{n}$, where ${\alpha}_{n}^{s}$ and ${\alpha}_{n}^{w}$ represent the power allocation factor for the strong channel gain user and the weak channel gain, respectively. For convenience, the problem P0 is decoupled into two sub problems: power allocation intra cluster and power allocation inter clusters.

#### 4.1. Power Allocation Intra Cluster

_{c}

_{−n}can be expressed as:

_{n}monotonically increases, and ${F}_{n}\in [0,1]$. If ${\alpha}_{n}^{s}\in \left[{\alpha}_{equal},0.5\right)$, F

_{n}monotonically decreasing, and ${F}_{n}\in \left({\phi}_{mid},1\right]$. According to the principle of PD-NOMA, the greater the power allocated to strong channel gain user, the smaller the fairness and the greater the throughput in the cluster. Combined with the above analysis, if ${\phi}_{c-n}\in \left[0,{\phi}_{mid}\right]$, the throughput in the cluster is the largest when ${\alpha}_{n}^{s}=0.5$; if ${\phi}_{c-n}\in \left({\phi}_{mid},1\right]$, then the throughput in the cluster is the largest when ${F}_{c-n}={\phi}_{c-n}$. According to Equation (16), we can get:

Algorithm 1: Intra cluster power allocation algorithm under fairness constraint |

Input: ${\phi}_{c-n}$, ${p}_{n}$, ${h}_{n}^{s}$, ${h}_{n}^{w}$, ${n}_{0}$ and ${W}_{n}$. Output: ${\alpha}_{n}^{s}$ and ${\alpha}_{n}^{w}$ |

1: if ${\phi}_{c-n}\in [0,{\phi}_{mid}]$,${\alpha}_{n}^{s}=0.5$, turn to step 3; 2: if ${\phi}_{c-n}\in ({\phi}_{mid},1]$, ${x}_{0}=\frac{1}{{\phi}_{c-n}}\times \left[{\left(1-{\phi}_{c-n}{}^{2}\right)}^{\frac{1}{2}}+1\right]$, according to ${R}_{c-n}^{s}/{R}_{c-n}^{w}={x}_{0}$, get ${\alpha}_{n}^{s}$; 3: ${\alpha}_{n}^{w}=1-{\alpha}_{n}^{s}$; |

4: end. |

#### 4.2. Power Allocation Inter Cluster

_{1}, r

_{2}, …, r

_{n}, we can get the optimization problem P3 equivalent to P2:

Algorithm 2: Inter cluster power allocation algorithm under fairness constraint |

Input: ${P}_{t}$, ${\phi}_{c}$, ${\left\{{\alpha}_{n}^{s},{\alpha}_{n}^{w}\right\}}_{n=1}^{N}$, H, W, n_{0}, i |

Output: $\gamma ,\theta ,{\left\{{p}_{n},{r}_{n}\right\}}_{n=1}^{N}$ |

1: if i = 1, initialize ${\left\{{p}_{n}\right\}}_{n}^{N}$$,\text{}{p}_{n}^{i-1}={P}_{t}/N$; else, turn to step 5; |

2: according to ${r}_{n}^{i-1}={R}_{c-n}$, n ∊ [1, N], calculate ${\left\{{r}_{n}\right\}}_{n=1}^{N}$; |

3: initialize θ, ${\theta}^{i-1}={\Vert {\left[\begin{array}{cc}{r}_{1}^{i-1}& {r}_{2}^{i-1}\end{array}\cdots \begin{array}{cc}{r}_{N-1}^{i-1}& {r}_{N}^{i-1}\end{array}\right]}^{T}\Vert}_{2}$; |

4: initialize γ, ${\gamma}^{i-1}=\left({\phi}_{c}+\frac{1}{N-1}\right)\left(N-1\right)$; |

5: using sequential quadratic programming method to solve problems P4 with nonlinear constraints, get $\gamma ,\theta ,{\left\{{p}_{n},{r}_{n}\right\}}_{n=1}^{N}$; |

6: end. |

#### 4.3. Joint Power Allocation Intra and Inter Cluster

Algorithm 3: Joint power allocation inter and intra cluster under fairness constraints |

Input: ${P}_{t}$, ${\phi}_{c}$, ${\left\{{\phi}_{c-n}\right\}}_{n=1}^{N}$, $H$, $W$, ${n}_{0},\epsilon $, $i=0$ Output: ${\left\{{p}_{n},{\alpha}_{n}^{s},{\alpha}_{n}^{w}\right\}}_{n=1}^{N}$ 1: initialize ${\left\{{p}_{n}^{0}={P}_{t}/N\right\}}_{n=1}^{N}$, execute Algorithm 1 and get ${\left\{{\left({\alpha}_{n}^{s}\right)}^{0},{\left({\alpha}_{n}^{w}\right)}^{0}\right\}}_{n=1}^{N}$, calculate ${R}_{sum}^{0}$; 2: i = i + 1; 3: bring ${\left\{{\left({\alpha}_{n}^{s}\right)}^{i-1},{\left({\alpha}_{n}^{w}\right)}^{i-1}\right\}}_{n=1}^{N}$$\text{}\mathrm{into}\text{}\mathrm{Algorithm}\text{}2\text{}\mathrm{and}\text{}\mathrm{get}\text{}{\left\{{p}_{n}^{i}\right\}}_{n=1}^{N}$; 4: for n = 1: N; 5: substitute ${\left\{{p}_{n}^{i}\right\}}_{n=1}^{N}$$\text{}\mathrm{into}\text{}\mathrm{Algorithm}\text{}1\text{}\mathrm{and}\text{}\mathrm{get}\text{}{\left({\alpha}_{n}^{s}\right)}^{i},{\left({\alpha}_{n}^{w}\right)}^{i}$; 6: end; 7: calculate ${R}_{sum}^{i}$; 8: if $\left|{R}_{sum}^{i}-{R}_{sum}^{i-1}\right|>\epsilon $, turn to step 2; 9: output ${\left\{{p}_{n}^{i},{\left({\alpha}_{n}^{s}\right)}^{i},{\left({\alpha}_{n}^{w}\right)}^{i}\right\}}_{n=1}^{N}$; |

10: end. |

## 5. Simulation and Analysis

_{s}). For convenience, it is assumed that the number of users is 2, in which the strong channel gain users are 250 m away from the BS and the weak channel gain users are 350 m away from the BS.

^{4}bps, the throughput of the algorithm in reference [30] is 1.5044 × 10

^{4}bps, and the throughput of FTPA algorithm is 1.7512 × 10

^{4}bps. The throughput of the proposed algorithm increased by about 24% compared with reference [30]. The reason is that the intra cluster user power allocation factor in the algorithm of reference [30] is a fixed coefficient based on the channel coefficient, which limits the range of throughput optimization.

_{c}

_{−n}and φ

_{c}. The value range of φ

_{c}

_{−n}and φ

_{c}is [0.05, 0.9]. As can be seen from Figure 5, the throughput decreases with the increase of the minimum fairness intra and inter clusters, and the downward trend gradually slows down. When φ

_{c}is small, the throughput can be increased by reducing φ

_{c}

_{−n}. When φ

_{c}is large, the reduction of φ

_{c}

_{−n}cannot obtain significant throughput gain. In practical applications, the system throughput can be improved by reasonably adjusting the minimum fairness constraints of one aspect according to the different needs of inter and intra cluster user fairness in different scenarios.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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Fairness Index | Value Range | Fairness Criteria |
---|---|---|

proposed fairness index | [0, 1] | rate |

Jain index | $[\frac{1}{K}$, 1] | rate |

GUI index | [0, 1] | channel state and power allocation |

Parameter | Value |
---|---|

BS transmit power P_{t} | 40 dBm |

total bandwidth W | 1 MHz |

cell radius D | 500 m |

path loss exponent λ | 5 |

noise unilateral power spectral density | −174 dBm/Hz |

error tolerance ε | 0.001 |

lower bound of intra cluster fairness | 0.7 |

lower bound of inter cluster fairness | 0.7 |

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**MDPI and ACS Style**

Yang, J.; Zhu, J.; Pan, Z.
A Fairness Index Based on Rate Variance for Downlink Non-Orthogonal Multiple Access System. *Future Internet* **2022**, *14*, 261.
https://doi.org/10.3390/fi14090261

**AMA Style**

Yang J, Zhu J, Pan Z.
A Fairness Index Based on Rate Variance for Downlink Non-Orthogonal Multiple Access System. *Future Internet*. 2022; 14(9):261.
https://doi.org/10.3390/fi14090261

**Chicago/Turabian Style**

Yang, Jie, Jiajia Zhu, and Ziyu Pan.
2022. "A Fairness Index Based on Rate Variance for Downlink Non-Orthogonal Multiple Access System" *Future Internet* 14, no. 9: 261.
https://doi.org/10.3390/fi14090261