# Outage Performance of Wireless Powered Decode-and-Forward Relaying Networks in Rician Fading
## 3 Outage Probability and Throughput Analysis
 \tag{10}\]  
where \(P_{\text{out\_R}}\) is the outage probability of the first hop link, and \(P_{\text{out\_D}}\) is the outage probability of the second hop link. According to Equation (7), we have that  
\[ \begin{split} P_{\text{out\_R}}&=F_{\text{Y}}\Bigg{(} \frac{\gamma_{\text{th}}}{\gamma_{0}d_{\text{SR}}^{-m}}\Bigg{)}\\ &=1-\sum_{f=0}^{\Theta_{1}}\sum_{g=0}^{\Theta_{1}+N_{\text{S}}-1} \frac{(N_{\text{S}}K_{\text{SR}})^{f}\big{(}\Psi/d_{\text{SR}}^{-m}\big{)}^{S} }{\text{e}^{N_{\text{S}}K_{\text{SR}}+\Psi/d_{\text{SR}}^{-m}}f!g!},\end{split}  \tag{11}\]  
where \(\Psi=(1+K_{\text{SR}})\gamma_{\text{th}}/(\lambda_{\text{SR}}\gamma_{0})\). We can obtain the probability density function (PDF) of \(X\), i.e., the PDF of the link channel gain from \(R\) to \(D\)[7], as  
\[f(x)=a\sum_{l=0}^{\Theta_{2}}\frac{(bK)^{l}x^{l}e^{-bx}}{(l!)^{2}}, \tag{12}\]  
where \(a=(1+K)e^{-K}/\lambda\), \(b=(1+K)/\lambda\), and \(\lambda\) is the mean value of random variable \(x\). It is noticed that the theoretical value of \(\Theta_{2}\) is infinite, but in practical, a relatively large value can be selected for calculating. According to Equations (7) and (12), the outage probability of the second hop can be given by  
\[ \begin{split}& P_{\text{out\_D}}=P_{\text{r}}\left(\sum_{j=1}^{N_{ \text{S}}}\abs{h_{\text{S}_{\text{S}}}}^{2}<\frac{\gamma_{\text{th}}}{\varphi \gamma_{0}Xd_{\text{SR}}^{-m}d_{\text{RD}}^{-m}}\right)\\ &=1-\int_{0}^{\Theta_{1}}\sum_{f=0}^{\Theta_{1}}\sum_{g=0}^{f+N_{ \text{S}}-1}\sum_{l=0}^{\Theta_{2}}\frac{a(bK_{\text{RD}})^{l}(N_{\text{S}}K_{ \text{SR}})^{f}}{f!g!(l!)^{2}e^{N_{\text{S}}K_{\text{SR}}}\varrho^{\text{S}}} \times\left(\frac{\Psi}{d_{\text{SR}}^{-m}d_{\text{RD}}^{-m}}\right)^{g}x^{l- g}e^{-\frac{\Psi}{\varphi\sigma d_{\text{SR}}^{-m}d_{\text{RD}}^{-m}}-bx}dx,\end{split}  \tag{13}\]  
where \(K_{\text{RD}}\) is the relay \(R\) rice factor to destination node \(D\), and \(\lambda_{\text{RD}}\) is the mean value of random variable \(X\). \(a=(1+K_{\text{RD}})e^{-K_{\text{RD}}}/\lambda_{\text{RD}}\), \(and\quad b=(1+K_{\text{RD}})/\lambda_{\text{RD}}\). See Appendix A for detailed derivation. Using the equation \(\int_{0}^{\Theta_{1}}x^{n-1}e^{-\beta/x-\gamma x}=2(\beta/\gamma)^{v/2}K_{v}(2 \sqrt{\beta\gamma})\) from [39] and \((x+y)^{m}=\sum_{n=0}^{m}(m!x^{m-n}y^{n})/((m-n)!n! )\), the outage probability of the second hop link can be re-expressed by  
\[ \begin{split}& P_{\text{out\_D}}=1-2\sum_{f=0}^{\Theta_{1}}\sum_{g=0}^{f+N_{ \text{S}}-1}\sum_{l=0}^{\Theta_{2}}\frac{a(bK_{\text{RD}})^{l}(N_{\text{S}}K_{ \text{SR}})^{f}}{f!g!(l!)^{2}e^{N_{\text{S}}K_{\text{SR}}}\varrho^{\text{S}}} \\ &\qquad\times\left(\frac{\Psi}{d_{\text{SR}}^{-m}d_{\text{RD}}^{ -m}}\right)^{g}\left(\frac{\Psi}{b\varphi d_{\text{SR}}^{-m}d_{\text{RD}}^{-m }}\right)^{\frac{l-g+1}{2}}\times K_{l-g+1}\left(2\sqrt{\frac{\Psi b}{\varphi d _{\text{SR}}^{-m}d_{\text{RD}}^{-m}}}\right).\end{split}  \tag{14}\]  
with Equations (5), (8), (9), (10), (11) and (14), the system end-to-end outage probability can be given by  
\[ \begin{split}& P_{\text{out}}^{\text{(e2e)}}=1-\left(\sum_{f=0}^{ \Theta_{1}}\sum_{g=0}^{f+N_{\text{S}}-1}\frac{(N_{\text{S}}K_{\text{SR}})^{f} (\Psi/d_{\text{SR}}^{-m})^{g}}{e^{N_{\text{S}}K_{\text{SR}}+\Psi/d_{\text{SR}}^ {-m}}f!g!}\right)\\ &\qquad\times\left(\sum_{f=0}^{\Theta_{1}}\sum_{g=0}^{f+N_{\text{ S}}-1}\frac{(N_{\text{S}}K_{\text{SR}})^{f}D^{g}}{f!g!e^{N_{\text{S}}K_{\text{SR}}+D}}\right)\\ &\qquad\times\sum_{f=0}^{\Theta_{1}}\sum_{g=0}^{f+N_{\text{S}}-1} \sum_{l=0}^{\Theta_{2}}\frac{2a(bK_{\text{RD}})^{l}(N_{\text{S}}K_{\text{SR}}) ^{f}\Psi^{g}e^{-N_{\text{S}}K_{\text{SR}}}}{f!g!(l!)^{2}\varrho^{g}\left(d_{ \text{SR}}^{-m}d_{\text{RD}}^{-m}\right)^{g}}\\ &\qquad\times\left(\frac{\Psi}{b\varphi d_{\text{SR}}^{-m}d_{\text{ RD}}^{-m}}\right)^{\frac{1-g+1}{2}}K_{l-g+1}\left(2\sqrt{\frac{\Psi b}{ \varphi d_{\text{SR}}^{-m}d_{\text{RD}}^{-m}}}\right).\end{split}  \tag{15}\]  
Following Equation (15), the system throughput can be given by  
\[T_{h}=R\Big{(}1-P_{\text{out}}^{\text{(e2e)}}\Big{)}(1-\rho)/2 \tag{16}\]