# A Fairness Index Based on Rate Variance for Downlink Non-Orthogonal Multiple Access System
## 2 System Model  
Consider a downlink NOMA system, a single antenna base station (BS) is deployed in the regional center with a radius of \(R\). BS provides services for \(M\big{(}M=2\times N,N\in\mathbb{N}^{+}\big{)}\) active users in the area, where \(N\) represents the number of user clusters, and the set of active users is represented as \(\mathbf{U}=\{U_{1},U_{2},\ldots,U_{M}\}\). Assuming that the BS completely knows the channel state information (CSI), \(h_{m}\) represents the channel gain between the  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Fairness Index** & **Value Range** & **Fairness Criteria** \\ \hline proposed fairness index & \([0,1]\) & rate \\ Jain index & \(\left[\frac{1}{K},1\right]\) & rate \\ GUI index & \([0,1]\) & channel state and power allocation \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison of our proposed fairness index with other indexes.  
BS and the user \(m\), and the set of user channel gains is expressed as \(\textbf{{H}}=\{h_{1},h_{2},\ldots,h_{M}\}\). A distributed user clustering algorithm is adopted among users [35]. First, the users are sorted according to the channel gain from large to small, \(\textbf{{Us}}=\{U_{1}^{s},U_{2}^{s},\ldots,U_{N}^{s}\}\) represents the strong channel gain user set, and \(\textbf{{U}}\textbf{{w}}=\{U_{1}^{w},U_{2}^{w},\ldots,U_{N}^{w}\}\) represents the weak channel gain user set, \(U_{n}^{s}\) and \(U_{n}^{w}\) will be paired into a cluster, where \(n\in[1,N]\). The clustered downlink NOMA system is shown in Figure 1.  
Assuming that the transmit power of the BS is \(P_{t}\), 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:  
\[y_{n}^{s}=h_{n}^{s}\sqrt{p_{n}^{s}}x_{n}^{s}+h_{n}^{s}\sqrt{p_{n}^{w}}x_{n}^{w }+\eta_{n}^{s} \tag{1}\]  
\[y_{n}^{w}=h_{n}^{w}\sqrt{p_{n}^{s}}x_{n}^{s}+h_{n}^{w}\sqrt{p_{n}^{w}}x_{n}^{w }+\eta_{n}^{w} \tag{2}\]  
where, \(h_{n}^{s}\) represents the channel gain between BS and \(U_{n}^{s}\) in the \(n\)-th cluster, \(h_{n}^{w}\) represents the channel gain between BS and \(U_{n}^{w}\) in the \(n\)-th cluster. \(p_{n}^{s}\) and \(p_{n}^{w}\) represent the transmit power of \(U_{n}^{s}\) and \(U_{n}^{w}\) allocated by BS, respectively. \(x_{n}^{s}\) and \(x_{n}^{w}\) represent the received signal of \(U_{n}^{s}\) and \(U_{n}^{w}\) transmitted by BS, respectively. \(\eta_{n}^{s}\) and \(\eta_{n}^{w}\) are Gaussian white noise with unilateral power spectral density \(n_{0}\). Using the successful interference cancellation (SIC) technology to decode the received signal, \(U_{n}^{w}\) can obtain a transmission rate as:  
\[R_{c-n}^{w}=W_{n}\log_{2}\biggl{(}1+\frac{|h_{n}^{w}|2p_{n}^{w}}{|h_{n}^{w}|2p _{n}^{s}+W_{n}n_{0}}\biggr{)} \tag{3}\]  
\(U_{n}^{s}\) can obtain a transmission rate as:  
\[R_{c-n}^{s}=W_{n}\log_{2}(1+\frac{|h_{n}^{s}|2p_{n}^{s}}{W_{n}n_{0}}) \tag{4}\]  
The throughput of the \(n\)-th cluster is:  
\[R_{c-n}=R_{c-n}^{s}+R_{c-n}^{w} \tag{5}\]