## 3 Statistics of End-to-End SNR  
Since the HAP relay node employs AF relaying protocol, the equivalent end-to-end CDF of the concerned system can expressed as [33; 42]:  
\[F_{eq}(\gamma)=\int_{0}^{\infty}P\bigg{[}\gamma_{eq}=\frac{\gamma_{1}\gamma_{2 }}{\gamma_{2}+C}<\gamma|\gamma_{1}\bigg{]}f_{\gamma_{1}}(\gamma_{1})d\gamma_{ 1}. \tag{13}\]  
By variable transformation of \(y=\gamma_{1}-\gamma_{7}\), (13) can be further expressed as [33; 43]:  
\[F_{eq}(\gamma)=F_{\gamma_{1}}(\gamma_{1})+\underbrace{\int_{0}^{\infty}F_{ \gamma_{2}}\bigg{(}\frac{C_{\gamma}}{y}\bigg{)}f_{\gamma_{1}}(y+\gamma)dy}_{ \rm Y}. \tag{14}\]  
By putting (2), (3), and (10) into (13), the integral term of (14) can be expressed as:  
\[{\rm Y}=\omega\sum_{q_{1}=0}^{m_{h}-1}\ldots\sum_{q_{N_{u}}=0}^{m_{h}-1}\Xi(N _{u})\Delta_{H}^{-\lambda}\int_{0}^{\infty}G_{1,2}^{1,1}\bigg{(}\frac{\Delta_ {H}C\gamma}{y}\bigg{|} \begin{array}{c}1\\ \lambda,\end{array} \bigg{)}(y+\gamma)^{-1}G_{1,3}^{3,0}\bigg{(}M\bigg{(}\frac{ (y+\gamma)}{\mu_{r}}\bigg{)}^{1/r}\bigg{|} \begin{array}{c}\delta^{2}+1\\ \delta^{2},\alpha,\beta\end{array} \bigg{)}dy. \tag{15}\]By applying the identity defined in Reference [44] (Equation (1.112)), the Meijer-G functions in (15) can be expressed as follows:  
\[ \begin{split}\mathrm{Y}=\frac{\omega}{(2\pi i)^{2}}\sum_{q_{1}=0}^{m _{h}-1}\dots&\sum_{q_{Na}=0}^{m_{h}-1}\Xi(N_{u})\Delta_{H}^{- \lambda}\int_{L_{1}}\int_{L_{2}}\frac{\Gamma(\lambda+s_{1})\Gamma(-s_{1})}{ \Gamma(1-s_{1})}\frac{\Gamma(\xi^{2}+s_{2})\Gamma(\alpha+s_{2})\Gamma(\beta+s_ {2})}{\Gamma(\xi^{2}+1+s_{2})}\times\\ &\qquad\qquad(\Delta_{H}C\gamma)^{-s_{1}}\left(\frac{M}{\mu_{1}^{ 1/r}}\right)^{-s_{2}}\int_{0}^{\infty}y^{s_{1}}(y+\gamma)^{-\frac{s_{2}}{7}-1} dyds_{1}ds_{2}.\end{split}  \tag{16}\]  
By applying the identity in Reference [38] (Equations (3.193(3)) and (8.384(1))), (16) can be solved as:  
\[ \begin{split}\mathrm{Y}=\frac{\omega}{(2\pi i)^{2}}\sum_{q_{1}=0}^ {m_{h}-1}\dots&\sum_{q_{Na}=0}^{m_{h}-1}\Xi(N_{u})\Delta_{H}^{- \lambda}\int_{L_{1}}\int_{L_{2}}\frac{\Gamma(\lambda+s_{1})\Gamma(-s_{1})}{ \Gamma(1-s_{1})}\frac{\Gamma(\xi^{2}+s_{2})\Gamma(\alpha+s_{2})\Gamma(\beta+s _{2})}{\Gamma(\xi^{2}+1+s_{2})}\times\\ &\qquad\qquad\qquad\frac{\Gamma(s_{1}+1)\Gamma(\frac{\gamma}{7}+ 1)}{\Gamma(\frac{\gamma}{7}+1)}(\Delta_{H}C)^{-s_{1}}\left(\frac{2^{1/r}M}{ \mu_{1}^{1/r}}\right)^{-s_{2}}ds_{1}ds_{2}.\end{split}  \tag{17}\]  
By letting \(s=s_{1}\) and \(t=s_{2}\), then, (17) can be rewritten as:  
\[ \begin{split}\mathrm{Y}=\frac{\omega}{(2\pi i)^{2}}\sum_{q_{1}=0 }^{m_{h}-1}\dots&\sum_{q_{Na}=0}^{m_{h}-1}\Xi(N_{u})\Delta_{H}^{- \lambda}\int_{L_{1}}\int_{L_{2}}\Gamma\left(\frac{t}{r}-s\right)\frac{\Gamma( \lambda+s)\Gamma(-s)\Gamma(s+1)}{\Gamma(1-s)}\times\\ &\qquad\qquad\qquad\frac{\Gamma(\xi^{2}+t)\Gamma(\alpha+t)\Gamma( \beta+t)}{\Gamma(\xi^{2}+1+t)\Gamma\left(\frac{t}{7}+1\right)}(\Delta_{H}C)^{ -s}\left(\frac{\gamma^{1/r}M}{\mu_{1}^{1/r}}\right)^{-t}dsdt.\end{split}  \tag{18}\]  
By applying the identity detailed in Reference [45], (18) can be obtained as:  
\[ \begin{split}\mathrm{Y}=\omega\sum_{q_{1}=0}^{m_{h}-1}\dots& \sum_{q_{Na}=0}^{m_{h}-1}\Xi(N_{u})\Delta_{H}^{-\lambda}\\ \times H_{1,0\,3\,1,3\,2}^{0.1;1;2.0}\left(\begin{array}{c} \left(1,-1,\frac{1}{r}\right)\\ -\end{array}\right)\left|\begin{array}{c}\left(1-\lambda,1\right)(0,1)(1,1) \\ \left(0,1\right)\end{array}\right|\begin{array}{c}\left(1-\xi^{2},1\right)(1- \alpha,\,1)(1-\beta,1)\\ \left(-\xi^{2},1\right)\left(0,\,\,\frac{1}{r}\right)\end{array}\right|\begin{array} []{c}\left(1\right.\\ \Delta_{H}C\end{array},\frac{\mu_{1}^{1/r}}{\gamma^{1/r}M}\end{array}\right), \end{split}  \tag{19}\]  
where \(H_{p_{1},\,q_{1};\,p_{2},\,q_{2};\,p_{3},\,q_{3}}^{m_{1},m_{2},m_{3},m_{3},m_{ 3}}\) [.] is the Extended Generalized Bivariate Fox's H-function (EG-BFHF). By substituting (3) and (19) into (13), the concerned system equivalent end-to-end CDF can be obtained as:  
\[