# Outage Performance of SWIPT-D2D-Based Hybrid Satellite-Terrestrial Networks
## 3 Performance Analysis
### Ts-D2d Scheme
#### 3.2.2 Outage Probability of \(\text{U}_{2}\)  
Firstly, we assume that the target rate at \(U_{2}\) is \(R_{p}\) with only the satellite-terrestrial direct link. Thus, the OP of \(U_{2}\) for the satellite-terrestrial direct link is as follows:\[\mathrm{P}_{\mathrm{out,TS}}^{\mathrm{suz}}(R_{p})=\mathrm{Pr}\big{[}\mathrm{log}_{2 }(1+\Lambda_{\mathrm{suz}2,ts})<R_{p}\big{]}=F_{\Lambda_{\mathrm{suz}2,ts}}({ \gamma^{\prime}}_{p}). \tag{49}\]  
Substituting Equation (22) into Equation (49), we can compute the OP of \(U_{2}\) for the satellite-terrestrial direct link. Secondly, considering that the terrestrial relay link of \(U_{2}\) originates from \(U_{1}\) with the SNR of \(\Lambda_{\mathrm{suz}2,ts}\) in Equation (19), the OP of \(U_{2}\) for the terrestrial relay link is the CDF of \(\Lambda_{\mathrm{suz}2,ts}\). Making use of the PDF of the gamma random variable for Equation (19), we can obtain the PDF of \(\Lambda_{\mathrm{suz}2,ts}\) as follows:  
\[ \begin{split} f_{\Lambda_{\mathrm{suz}2,ts}}(x)& =\int\limits_{0}^{\infty}\frac{1-\tau}{3\omega\tau y}f_{\eta_{r}| sgn_{1}|2}\bigg{(}\frac{x-x\tau}{3\omega\tau y}\bigg{)}f_{[g_{\mathrm{suz}2 }]^{2}}(y)\mathrm{d}y\\ &=\frac{2(\chi_{2})}{2}\frac{m_{r\mathrm{suz}1}+m_{\mathrm{suz}2 }}{2}x\frac{m_{r\mathrm{suz}1}+m_{\mathrm{suz}2}}{2}-1\mathcal{K}_{m_{\mathrm{ suz}2,ts}}(2\sqrt{\chi_{2}x})\\ &=\frac{\chi_{2}}{\Gamma(m_{r\mathrm{suz}1})\Gamma(m_{\mathrm{suz }2})}G_{0,2}^{2,0}\Bigg{[}\chi_{2}x\Bigg{|}\begin{array}{c}-\\ \frac{m_{\mathrm{suz}2}-m_{r\mathrm{suz}1}}{2},-\frac{m_{\mathrm{suz}2}-m_{r \mathrm{suz}1}}{2}\end{array}\Bigg{]},\end{split}  \tag{50}\]  
where \(\chi_{2}=\frac{(1-\tau)m_{r\mathrm{suz}1}m_{\mathrm{suz}2}}{3\omega\tau\Gamma m _{r\mathrm{suz}1}\Lambda_{\mathrm{suz}2}\eta_{r}}\). The corresponding CDF of \(\Lambda_{\mathrm{suz}2,ts}\) can be presented by  
\[F_{\Lambda_{\mathrm{suz}2,ts}}(x)=\frac{\chi_{2}x}{\Gamma(m_{r\mathrm{suz}1}) \Gamma(m_{\mathrm{suz}2})}G_{1,3}^{2,1}\Bigg{[}\chi_{2}x\Bigg{|} \begin{array}[] {c}0,\\ \frac{m_{\mathrm{suz}2}-m_{r\mathrm{suz}1}}{2},-\frac{m_{\mathrm{suz}2}-m_{r \mathrm{suz}1}}{2},-1\end{array} \Bigg{]}. \tag{51}\]  
Thirdly, utilizing Equations (18) and (19) for MRC, we have the OP of \(U_{2}\) for the TS-D2D scheme as follows:  
\[ \begin{split}\mathrm{P}_{\mathrm{out,TS}}^{\mathrm{u2}}(R_{p})& =\mathrm{Pr}\bigg{[}\mathrm{log}_{2}(1+\Lambda_{\mathrm{suz}2,ts})<R_{p}, \frac{1-\tau}{3}\log_{2}(\Lambda_{\mathrm{suz}2,ts}+\Lambda_{\mathrm{suz}1u2,ts })<R_{p}\bigg{]}\\ &=\mathrm{Pr}\Big{[}\Lambda_{\mathrm{suz}2,ts}<{\gamma^{\prime}}_{p}, \Lambda_{\mathrm{suz}2,ts}+\Lambda_{\mathrm{suz}1u2,ts}<\gamma_{p,ts}\Big{]}\\ &=\mathrm{Pr}\Big{[}\Lambda_{\mathrm{suz}2,ts}<\min(\gamma_{p,ts}- \Lambda_{\mathrm{suz}1u2,ts}{\gamma^{\prime}}_{p})\Big{]}\\ &=\varphi_{3}+\varphi_{4},\end{split}  \tag{52}\]  
where \(\varphi_{3}\) denotes the probability of case 3, and \(\varphi_{4}\) denotes the probability of case 4. The probability of case 3 is as follows:  
\[\varphi_{3}=\mathrm{Pr}\Big{[}\Lambda_{\mathrm{suz}2,ts}<{\gamma^{\prime}}_{p ^{\prime}},{\gamma^{\prime}}_{p}<\gamma_{p,ts}-\Lambda_{\mathrm{sru}2,ts} \Big{]}=F_{\Lambda_{\mathrm{suz}2,ts}}({\gamma^{\prime}}_{p})F_{\Lambda_{ \mathrm{suz}2,ts}}(\gamma_{p,ts}-{\gamma^{\prime}}_{p}). \tag{53}\]  
Substituting Equations (49) and (51) into Equation (53), we can obtain \(\varphi_{3}\). The probability of case 4 is as follows:  
\[