## 3 Performance Evaluation
### Energy Efficiency  
Energy efficiency is an essential index for the HAP relay node, which can improve its endurance time and realize energy efficiency matching with ground networks. The ergodic capacity of FSO and RF links can be calculated by  
\[ \begin{array}{l}\overline{C}=E[W\log_{2}(1+\tau\gamma)]\\ =\frac{W}{\ln{2}}\int_{0}^{\infty}\ln(1+\tau\gamma)f(\gamma)d\gamma\end{array}  \tag{39}\]  
where \(W=W_{f}\) is the link bandwidth of FSO and \(W=W_{r}\) is the bandwidth of RF link. The ergodic capacity of the dual-hop FSO link from OGS to HAP is obtained by using the differential of (28) and (39) expressed as  
\[\begin{array}{l}\overline{C}_{EC}=\exp\left(-\frac{\theta_{2F \omega}^{2}}{2\sigma_{angle,\omega}^{2}}\right)+\left(1-\exp\left(-\frac{ \theta_{2F\omega}^{2}}{2\sigma_{angle,\omega}^{2}}\right)\right)\times\left(F_ {s}+\frac{\theta_{2}^{2}\Gamma^{2}}{\delta\Gamma(\alpha_{1})\Gamma(\beta_{1})} \right.\\ \left.\times G_{1,0,4,3,b,3b+1,0}^{0,1:1,4:3b+1,0}\left[ \begin{array}{c}1\\ -\end{array} \right]\left[ \begin{array}{c}1-\xi_{1}^{2},1-\alpha_{1},1-\beta_ {1},1\\ 1,-\xi_{1}^{2},0\end{array} \right]\left[ \begin{array}{c}1+\xi_{2}^{2}\\ \xi_{2}^{2},\alpha_{2},\beta_{2},0\end{array} \right]\left[ \begin{array}{c} \frac{\xi_{1}^{2}+1}{\alpha_{1}\beta_{1}\xi_{1}^{2}}(\tau\mu_{1})^{\frac{1}{b }},\frac{D_{1}^{2}\Omega}{b^{2}b_{2}}\end{array} \right]\right) \tag{40}\]  
where \(F_{s}\) is given by (16), Equation (E.3)) shown as  
\[F_{s}=\frac{\xi_{1}^{2}b^{\alpha_{1}+\beta_{1}-2}}{(2\pi)^{b-1}\Gamma(\alpha_{ 1})\Gamma(\beta_{1})}G_{b+2,3b+2}^{3b+2,1}\left[ \begin{array}{c}D_{1}^{b}\\ \tau b^{2b}\mu_{1}\end{array} \right]\left[ \begin{array}{c}0,\;1,\;E_{1}\\ E_{2},\;0,\;0\end{array} \right] \tag{41}\]  
The ergodic capacity of the HAP-to-AP single FSO link can be obtained by substituting (13) into (39) and using ((28), Equation (07.34.21.0013.01)) shown as  
\[\overline{C}_{RDf}=\frac{W_{f}\xi_{3}^{2}b^{\alpha_{3}+\beta_{3}-2}}{(2\pi)^{b -1}\Gamma(\alpha_{3})\Gamma(\beta_{3})}G_{b+2,3b+2}^{3b+2,1}\left[ \begin{array} []{c}D_{3}^{b}\\ \tau b^{2b}\mu_{3}\end{array} \right]\left[ \begin{array}{c}0,\;1,\;E_{5}\\ E_{6},\;0,\;0\end{array} \right] \tag{42}\]  
Similarly, the ergodic capacity of the HAP-to-AP single RF link is derived by substituting (15) into (39) and applying ((28), Equation (07.34.21.0013.01)) given by  
\[\overline{C}_{RDr}=\frac{W_{r}}{\Gamma(mN_{t})\ln{2}}G_{3,2}^{1,3}\left(\frac {\tau\overline{\gamma}_{RF}}{m}\left| \begin{array}{c}1,1,1-mN_{t}\\ 1,0\end{array} \right.\right) \tag{43}\]  
Therefore, considering the QoS constraints of ground users inside and outside the hotspot areas, the optimization model with maximizing system energy efficiency \(\eta_{s}\) can be expressed as  
\[ \begin{array}{l}\max_{\rho}\eta_{s}=\frac{\min\{\overline{C}_{EC},\overline{ \overline{c}_{t}}(1-F_{\gamma_{1}}(\gamma_{th}))\}}{P_{th}}\\ \text{s.t.\ }\text{C1}:\ 0\leq\rho\leq 1\\ \text{C2}:F_{C_{RF}}(c\leq C_{th}^{1})\leq\delta_{th1}\\ \text{C3}:F_{C_{t}}(c\leq C_{th}^{2})\leq\delta_{th2}\end{array}  \tag{44}\]  
where \(\overline{C}_{t}=\overline{C}_{RDf}+\overline{C}_{RDr}\) is the total capacity of HAP-to-user parallel FSO/RF links. The objective function in (44) means that the system ergodic capacity is determined by the smaller one between \(\overline{C}_{EC}\) and \(\overline{C}_{t}(1-F_{\gamma_{1}}(\gamma_{th}))\), where \(\overline{C}_{t}(1-F_{\gamma_{1}}(\gamma_{th}))\) is the ergodic capacity of parallel FSO/RF links during the non-outage period. \(C_{1}\) is the range constraint of power distribution coefficient \(\rho\). \(C_{2}\) is the QoS constraint of users outside the hotspot areas, where \(\delta_{th1}\) is the maximum acceptable capacity outage probability and \(C_{th}^{1}\) is the corresponding capacity threshold. \(C_{3}\) is the QoS constraint of users inside the hotspot areas, in which \(\delta_{th2}\) is the maximum acceptable capacity outage probability and \(C_{th}^{2}\) is the corresponding capacity threshold.