## 4 System Performance Analysis
### System Channel Capacity Analysis  
In a communication system, for a given signal-to-noise ratio (\(SNR\)) channel, according to [PERSON]'s theorem, the normalized channel capacity of the system can be expressed by the following Equation (29) [19]:  
\[C/B=\log_{2}^{(1+SNR)} \tag{29}\]  
In an FSO communication system, the relationship between the instantaneous signal-to-noise ratio of a channel \(SNR\) and the average signal-to-noise ratio of a channel is \(SNR=\gamma_{0}h^{2}\). Affected by atmospheric turbulence, under the condition of fully correlated atmospheric fading, the channel capacity value fluctuates randomly, so the expected value of the channel capacity is used to describe the average channel capacity of the channel; its expression is \(Cerg=E(C(h))\). According to a previous study [20], the average channel capacity of a channel can be expressed as:  
\[Cerg=\int_{0}^{\infty}B\log_{2}\Big{(}1+\gamma_{0}h^{2}\Big{)}f(h)dh \tag{30}\]  
In the formula, \(B\) is the channel bandwidth and \(\gamma_{0}\) is the average signal-to-noise ratio of the channel. In the system shown in Figure 1, the signal-to-noise ratio of the source node to the relay node link, transmitting high priority data and low priority data, can be expressed as:  
\[\gamma_{high}=\frac{|h_{NR}|^{2}\lambda P_{tr}}{(1-\lambda)|h_{NR}|^{2}P_{tr}+ N_{0}} \tag{31}\]  
\[\gamma_{low}=\frac{|h_{NR}|^{2}(1-\lambda)P_{tr}}{N_{0}} \tag{32}\]  
where \(h_{NR}\) is the channel status information of the source node to the relay node link, \(N\in\{S_{1},S_{2},S_{3},S_{4}\}\). \(N_{0}\) is the variance of the noise at the relay node \(R\), and \(P_{tr}\) is the signal transmission power. Similarly, the signal-to-noise ratio of sending high-priority data between adjacent source nodes can be expressed by Equation (33):  
\[\gamma^{{}^{\prime}}_{high}=\frac{\left|h_{ij}\right|^{2}\lambda P_{tr}}{\left(1- \lambda\right)\left|h_{ij}\right|^{2}P_{tr}+N^{{}^{\prime}}_{0}} \tag{33}\]  
where \(h_{ij}\) is the channel status information of the link between the adjacent source nodes, \(i,j\in\{S_{1},S_{2},S_{3},S_{4}\}\). \(N^{{}^{\prime}}_{0}\) is the variance of the noise at source node \(j\). Equation (1) is substituted (30), and the normalized average channel capacity of the channel can be obtained as:  
\[Cerg/B=\frac{(\alpha\beta)^{(\alpha+\beta)/2}}{\Gamma(\alpha)\Gamma(\beta)} \int_{0}^{\infty}\log_{2}\left(1+\gamma_{0}h^{2}\right)h^{(\frac{\alpha+\beta} {2})-1}G_{0,2}^{2,0}\left[\alpha\beta h\left|\frac{-\alpha-\beta}{2},-\frac{ \alpha-\beta}{2}\right.\right]dh \tag{34}\]  
Since the system operates in half-duplex mode, the normalized average channel capacity from the source node to the relay node can be expressed by the following equation:  
\[Cerg_{NR}/B=\frac{(\alpha\beta)^{(\alpha+\beta)/2}}{\Gamma(\alpha)\Gamma(\beta )}\left( \begin{array}{c}\int_{0}^{\infty}\log_{2}\left(1+\gamma_{high}h_{NR} ^{2}\right)h_{NR}^{(\frac{\alpha+\beta}{2})-1}G_{0,2}^{2,0}\left[\alpha\beta h _{NR}\right]\frac{-\alpha-\beta}{2}\left[\alpha\beta h_{NR}\right]-\frac{ \alpha-\beta}{2}\left[\alpha\beta h_{NR}\right]\\ +\int_{0}^{\infty}\log_{2}\left(1+\gamma_{low}h_{NR}^{2}\right)h_{NR}^{(\frac {\alpha+\beta}{2})-1}G_{0,2}^{2,0}\left[\alpha\beta h_{NR}\right]\frac{\alpha -\beta}{2},-\frac{\alpha-\beta}{2}\left[\alpha\beta h_{NR}\right]\end{array}