# Outage Performance of Wireless Powered Decode-and-Forward Relaying Networks in Rician Fading
## 3 Outage Probability and Throughput Analysis  
This section analyzes the outage probability and throughput of the network described in Section 2. As mentioned previously, when the energy collected by \(R\) is less than the triggering threshold of circuit \(E_{\text{Q}}\) (i.e., \(E<E_{\text{Q}}\)), the relay cannot work normally, and in this case, the system will be outage. When \(E\) is greater than the triggering threshold of the circuit, the relay works normally, but the information rate affected by fading may also be less than the decoding requirement threshold. In this case, the system outage may also occur. Therefore, the system outage probability can be described by  
\[P_{\text{out}}^{\text{(e2e)}}=P_{\text{out}}^{\text{(e)}}+P_{\text{suff}}^{ \text{(e)}}*P_{\text{out}}^{\text{(info)}}, \tag{5}\]  
where \(P_{\text{out}}^{\text{(e)}}\) is the probability that the energy collected is less than the circuit threshold \(E_{\text{Q}}\) at \(R\). \(P_{\text{suff}}^{\text{(e)}}\) is the probability that \(E\) is greater than the circuit threshold \(E_{\text{Q}}\), where \(P_{\text{suff}}^{\text{(e)}}=(1-P_{\text{out}}^{\text{(e)}})\) is the probability of sufficient relay energy, and \(P_{\text{out}}^{\text{(info)}}\) represents the probability of link interruption when the relay energy is sufficient. In terms of the energy collection stage described in section 2, we have that  
\[ \begin{split} P_{\text{out}}^{\text{(e)}}&=P_{\text{r}} \big{(}E<E_{Q}\big{)}\\ &=P_{\text{r}}\left(\eta\sum_{j=1}^{N_{\text{S}}}\left|h_{\text{S},\text{R}}\right|^{2}P_{\text{S}}\rho Td_{\text{SR}}^{-m}<E_{Q}\right)\\ &=P_{\text{r}}\left(Y<\frac{E_{Q}}{\eta P_{\text{S}}\rho Td_{ \text{SR}}^{-m}}\right),\end{split}  \tag{6}\]  
where \(Y=\sum_{j=1}^{N_{\text{S}}}\left|h_{\text{S},\text{R}}\right|^{2}\). According to [37], the cumulative distribution function (CDF) of the channel gain for the link from \(S\) to \(R\) in Rice fading can be expressed by  
\[F(x)=1-\sum_{f=0}^{\Theta_{1}}\sum_{g=0}^{f+N_{\text{S}}-1}\frac{(N_{\text{S}} K)^{f}\big{(}(1+K)\frac{x}{\lambda}\big{)}^{S}}{f!g!e^{N_{\text{S}}K_{\text{SR}}+(1 +K)\frac{x}{\lambda}}}, \tag{7}\]  
where \(K\) is the Rice factor of the link, \(\lambda\) is the mean value of random variable \(x\), and \(\Theta_{1}\) with a sufficiently large value. Following Equation (7), the energy outage probability of \(R\) can be re-expressed by  
\[P_{\text{out}}^{\text{(e)}}=1-\sum_{f=0}^{\Theta_{1}}\sum_{g=0}^{f+N_{\text{S} }-1}\frac{(N_{\text{S}}K_{\text{SR}})^{f}Dg}{f!g!e^{N_{\text{S}}K_{\text{SR}}+ D}}, \tag{8}\]  
where \(D=(1+k_{\text{SR}})E_{Q}\big{)}/(\lambda_{\text{SR}}\eta P_{\text{S}}\rho Td_{ \text{SR}}^{-m})\), \(K_{\text{SR}}\) is the Rice factor between for \(S\) and \(R\), and \(\lambda_{\text{SR}}\) is the mean value of random variable \(Y\). As a result, one can obtain the probability as  
\[P_{\text{suff}}^{\text{(e)}}=1-P_{\text{out}}^{\text{(e)}}=\sum_{f=0}^{\Theta_ {1}}\sum_{g=0}^{f+N_{\text{S}}-1}\frac{(N_{\text{S}}K_{\text{SR}})^{f}Dg}{f!g! e^{N_{\text{S}}K_{\text{SR}}+D}}. \tag{9}\]  
Assuming that the minimum required decoding rate of the system is \(R\). Half-duplex relaying is adopted, so that when \(S\) transmits information to \(R\), \(R\) does not send information to \(D\). According to [PERSON]'s theory [38] one has \(R=((1-\rho)/2)\log_{2}(1+\gamma_{\text{th}})\). Thus, the system end-to-end SNR threshold can be expressed by \(\gamma_{\text{th}}=2^{2R/(1-\rho)}-1\). In terms of Equations (3) and (4), the information outage probability when \(R\) has harvested sufficient power is  
\[ \begin{split} P_{\text{out}}^{\text{(info)}}&=\min \{\gamma_{\text{R}},\gamma_{\text{D}}\}<\gamma_{\text{th}}\\ &=P_{\text{r}}(\gamma_{\text{R}}<\gamma_{\text{th}})+P_{\text{r}} (\gamma_{\text{D}}<\gamma_{\text{th}})\ \ \times-P_{\text{r}}(\gamma_{\text{R}}<\gamma_{\text{th}})\times P_{\text{r}}( \gamma_{\text{D}}<\gamma_{\text{th}})\\ &=P_{\text{out\_R}}+P_{\text{out\_D}}-P_{\text{out\_R}}\times P_{ \text{out\_D}},\end{split}