# A Fairness Index Based on Rate Variance for Downlink Non-Orthogonal Multiple Access System
## 5 Simulation and Analysis  
In this section, simulations based on MATLAB are used to verify the effectiveness of the proposed fairness index and joint power allocation algorithm. We suppose that BS is located in the center of the cellular area, and users are randomly distributed around BS. The channels between users and BS are Rayleigh channels [22], the mean value of channel coefficients being 0 and the variance being 1. Simulation parameters are shown in Table 2.  
Figure 2 shows the curves of the proposed fairness index, Jain index, and GUI index with the power allocation factor for strong channel gain user (\(\alpha_{s}\)). 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. As can be seen from Figure 2, the proposed fairness index and Jain index [24] are convex curves, while GUI index is a concave curve. This is because the proposed fairness index and Jain index are based on the rate, that is, the closer the user rate in the system, the greater the fairness. The GUI fairness index is related to user power allocation in the system, that is, the greater the rate of users with more power allocation, the greater the fairness. From the perspective of fairness value range, the lower bound of Jain index is affected by the number of users, which is suitable for analyzing systems with a fixed number of users. The lower bound of GUI fairness index is affected by the number of users and channel conditions, which is suitable for analyzing systems with fixed number of users and channel conditions. The range of the proposed fairness index is not affected by the number of users and channel conditions. Thus, it is more sensitive to the changes of resource allocation in the system and suitable for analyzing systems with changing number of users and channel conditions. Figure 3 compares the throughput of the proposed algorithm, the algorithm in reference [30] (the algorithm of [PERSON] 2021), and the fractional transmit power allocation (FTPA) algorithm under different user numbers. The algorithm in reference [30] is a power \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline BS transmit power \(P_{t}\) & 40 dBm \\ total bandwidth \(W\) & 1 MHz \\ cell radius \(D\) & 500 m \\ path loss exponent \(\lambda\) & 5 \\ noise unilateral power spectral density & \(-174\ dBm/Hz\) \\ error tolerance \(\epsilon\) & 0.001 \\ lower bound of intra cluster fairness & 0.7 \\ lower bound of inter cluster fairness & 0.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation parameters. Figure 2: Comparison of fairness indexes (\(M=2\)). allocation algorithm based on channel gain of paired users, the power allocation factor of strong channel gain users in the \(n\)-th cluster is \(\alpha_{n}^{s}=0.5e^{-3\mu}\), where \(\mu=|h_{n}^{w}|/|h_{n}^{s}|\). As can be seen from Figure 3, with the increase of the number of users, the throughput advantage of the proposed algorithm becomes more and more obvious. When the number of users \(M=60\), the throughput of the proposed algorithm is \(1.9135\times 10^{4}\) bps, the throughput of the algorithm in reference [30] is \(1.5044\times 10^{4}\) bps, and the throughput of FTPA algorithm is \(1.7512\times 10^{4}\) 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. Figure 4 compares the system fairness of the proposed algorithm, the algorithm in reference [30], and the FTPA algorithm under different user numbers. It can be seen that with the increase of the number of users, the fairness of the system gradually tends to be stable. The reason is that as the number of users in the system gradually increases, the channel gain difference between users gradually decreases.