## 4 Adaptive Transmission Strategy Design and Outage Probability Analysis
### Outage Probability Analysis  
According to [31], when \(\{\min(\gamma_{1},\gamma_{2})<\gamma_{th}\}\), the network will be outage. \(\gamma_{1}\) and \(\gamma_{2}\) are the SNRs at SU\({}_{1}\) and SU\({}_{2}\), respectively, and \(\gamma_{th}\) is the SNR threshold to avoid a network outage. Therefore, the network outage probability can be given as  
\[P_{\text{out}}=P_{r}\{\min(\gamma_{1},\gamma_{2})<\gamma_{th}\} \tag{13}\]  
Assuming \(\gamma_{1}\leq\gamma_{2}\), with the found \(P_{th}\), the analysis of the outage probability can be presented as follows. 1. When \(P_{T}>P_{th}\), from (8), the SNR at SU\({}_{1}\) can be expressed as \(\gamma_{1}^{s}=\frac{(1-\rho)\rho\eta P_{2}|f_{b}|^{2}|g_{b}|^{2}}{X+N_{1}}\), where \(X=(1-\rho)\rho\eta P_{p}|f_{b}|^{2}|l_{r}|^{2}+(1-\rho)P_{P}|l_{1}|^{2}\). Based on (13) and \(\gamma_{1}^{s}\), the outage probability at SU\({}_{1}\) can be expressed as  
\[ \begin{split} P_{out_{1}}^{s}&=P_{r}\{\gamma_{1}^{ s}<\gamma_{th}\}\\ &=P_{r}\Bigg{\{}\frac{(1-\rho)\rho\eta P_{2}|f_{b}|^{2}|g_{b}|^{2}}{(1 -\rho)\rho\eta P_{p}|f_{b}|^{2}|{l_{r}}|^{2}+(1-\rho)P_{p}|{l_{1}}|^{2}+N_{1}} <\gamma_{th}\Bigg{\}}\\ &=P_{r}\bigg{\{}w<\frac{\gamma_{th}}{(1-\rho)\rho\eta P_{2}}\Big{[} (1-\rho)P_{p}|{l_{1}}|^{2}+N_{1}\Big{]}\bigg{\}}\end{split}  \tag{14}\]  
where \(w=|f_{b}|^{2}\bigg{(}|g_{b}|^{2}-\frac{\gamma_{th}P_{p}|{l_{r}}|^{2}}{P_{2}} \bigg{)}\). Meanwhile, the probability density function of the random variable \(w\) can be given as  
\[ \begin{split} p_{w}(w)=\Bigg{\{}\frac{1}{\sigma_{1}^{2}\big{(} \sigma_{2}^{2}+a\sigma_{2}^{2}\big{)}}\int_{0}^{\infty}\frac{1}{v}e^{-\frac{w} {\sigma_{1}^{2}}-\frac{w}{\sigma_{2}^{2}}}dv,w>0\\ \frac{1}{\sigma_{1}^{2}\big{(}\sigma_{2}^{2}+a\sigma_{2}^{2}\big{)} \big{)}}\int_{0}^{\infty}\frac{1}{v}e^{-\frac{w}{\sigma_{2}^{2}}-\frac{w}{ \sigma_{1}^{2}}}dv,w\leq 0\end{split}  \tag{15}\]  
Let \(u=|g_{b}|^{2}-\frac{\gamma_{th}P_{p}|{l_{r}}|^{2}}{P_{2}}\), and when \(u\leq 0\), we can obtain  
\[P_{r}\{\gamma_{1}^{s}<\gamma_{th},u\leq 0\}=\frac{a\sigma_{r}^{2}}{\sigma_{2}^{2} +a\sigma_{r}^{2}} \tag{16}\]where \(a=\frac{\gamma_{th}P_{p}}{P_{2}}\). When \(u>0\), we can obtain  
\[ \begin{split} P_{r}\{\gamma_{1}^{\text{s}}<\gamma_{th},u>0\}=& \mu_{\sigma}-\mu_{\sigma}\sqrt{\frac{4c}{\sigma_{1}^{2}\sigma_{2}^{2}}}K_{1} \Bigg{(}\sqrt{\frac{4c}{\sigma_{1}^{2}\sigma_{2}^{2}}}\Bigg{)}\\ &+\frac{\mu_{\sigma}}{\sigma_{1}^{2}}\int_{0}^{\infty}\frac{b}{v +b}e^{-\frac{c}{\omega_{2}}-\frac{c}{\sigma_{1}^{2}}}dv\end{split}  \tag{17}\]  
where \(K_{1}\) is the Bessel function of the second kind with \(K_{1}(z)=\frac{z}{4}\int_{0}^{\infty}\frac{e^{-t-\frac{z^{2}}{4t}}}{t^{2}}dt\), \(\mu_{\sigma}=\frac{\sigma_{2}^{2}}{\sigma_{2}^{2}+a\sigma_{1}^{2}}\), \(b=\frac{\gamma_{th}P_{p}\sigma_{1}^{2}}{\rho\eta P_{2}\sigma_{2}^{2}}\), and \(c=\frac{\gamma_{th}N_{1}}{(1-\rho)\rho\eta P_{2}}\). The integral part of (17) can be calculated according to [29]. Then, combining (16) and (17), the outage probability at SU\({}_{1}\) can be obtained as  
\[ \begin{split} P_{out}^{\text{s}}=&\mu_{\sigma}-\mu_ {\sigma}\sqrt{\frac{4c}{\sigma_{1}^{2}\sigma_{2}^{2}}}K_{1}\Bigg{(}\sqrt{ \frac{4c}{\sigma_{1}^{2}\sigma_{2}^{2}}}\Bigg{)}+\frac{a\sigma_{r}^{2}}{\sigma _{2}^{2}+a\sigma_{r}^{2}}\\ &+\frac{\mu_{\sigma}}{\sigma_{1}^{2}}\int_{0}^{\infty}\frac{b}{v +b}e^{-\frac{c}{\omega_{2}}-\frac{c}{\omega_{1}^{2}}}dv\end{split}  \tag{18}\]  
2. When \(P_{T}\leq P_{th}\), from (11), the SNR at SU\({}_{1}\) can be expressed as \(\gamma_{1}^{d}=\frac{P_{th}h^{2}}{P_{p}l_{1}^{2}+\sigma_{1}^{2}}\). Then, based on (13) and \(\gamma_{1}^{d}\), the outage probability at SU\({}_{1}\) can be obtained as  
\[ \begin{split} P_{out_{1}}^{\text{d}}&=P_{r}\bigg{\{} \gamma_{1}^{d}<\gamma_{th}\Big{\}}=P_{r}\Bigg{\{}\frac{P_{2}h^{2}}{P_{p}l_{1}^ {2}+\sigma_{1}^{2}}<\gamma_{th}\Bigg{\}}\\ &=P_{r}\Bigg{\{}h^{2}<\frac{\gamma_{th}\big{(}P_{p}l_{1}^{2}+ \sigma_{1}^{2}\big{)}}{P_{2}}\Bigg{\}}\\ &=1-\frac{\sigma_{h}^{2}}{d\sigma_{p_{1}}^{2}+\sigma_{h}^{2}}e^{ -\frac{\gamma_{th}\sigma_{2}^{2}}{P_{2}\sigma_{h}^{2}}}\end{split}