## 4 Performance Analysis
### Outage Probability  
The average probability of outage of the SUE for a frame duration \(T\) is calculated through the condition \(\Gamma_{d-ws,p}^{i}<\bar{R}_{w,p}^{i}\cup\Gamma_{d-s,p}^{i}<\bar{R}_{s,p}^{i}\), where \(\bar{R}_{w,p}^{i}\) and \(\bar{R}_{s,p}^{i}\) are the minimum SE requirements of the WUE and SUE, respectively. \(\bar{R}_{w,p}^{i}\), \(\bar{R}_{s,p}^{i}\in\bar{R}_{u}\). \(\bar{R}_{u}\) is the minimum SE requirement of the \(u\)th UE. The probability of outage at the SUE during direct transmission is \(P_{s,p}^{i}=P(\Gamma_{d-ws,p}^{i}<\tilde{R}_{w,p}^{i},\Gamma_{d-s,p}^{i}<\tilde{R}_{ s,p}^{i})\). The channel condition \(H_{u}=\mathbb{H}_{s,p}^{i}\) for the SUE experiencing outage is given by  
\[P_{s,p}^{i}=P\left(\mathbb{H}_{s,p}^{i}<\sqrt{\frac{\rho_{w,p}^{i}}{\rho\mu_{wi }}}\sqrt{\frac{\mu_{wi}}{\rho_{w,p}^{i}\Big{(}\mu_{si}+\left[\tilde{k}_{g}^{2}+ k_{w}^{2}\right]\Big{)}}}\right)\times P\left(\mathbb{H}_{s,p}^{i}<\sqrt{\frac{ \rho_{s,p}^{i}}{\rho\mu_{si}}}\sqrt{\frac{\mu_{si}}{\rho_{s,p}^{i}\Big{(}\epsilon \mu_{wi}+\left[k_{g}^{2}+k_{w}^{2}\right]\Big{)}}}\right) \tag{11}\]  
where \(\rho_{s,p}^{i}\) and \(\rho_{w,p}^{i}\) are the SNR thresholds for the SUE and WUE. The relation for the SNR threshold and minimum SE requirement is given as \(\rho_{s,p}^{i}=2^{MR_{w,p}^{i}/2}-1\) and \(\rho_{w,p}^{i}=2^{MR_{w,p}^{i}/2}-1\). For a detailed derivation of (11), refer to Appendix A. The probability density function (PDF) for the random variable \(\mathbb{H}_{s,p}^{i}\) is expressed as [33]  
\[f_{\mathbb{H}_{s,p}^{i}}(h_{s,p}^{i})=\frac{h_{s,p}^{i}}{b^{a+1}\Gamma(a+1)}e^ {-\left(h_{s,p}^{i}/b\right)}dh_{s,p}^{i} \tag{12}\]  
where \(a=\frac{\left(m_{w,p}^{i}\right)^{2}}{\sigma_{w,p}^{i}}-1\) and \(b=\frac{\sigma_{w,p}^{i}}{m_{w,p}^{i}}\)[33; 34] with a mean of \(m_{s,p}^{i}=\frac{L_{w,p}^{i}\pi}{2}\) and variance of \(\sigma_{s,p}^{i}=L_{s,p}^{i}\Big{(}1-\frac{\pi^{2}}{16}\Big{)}\)[32]. Here, \(L_{s,p}^{i}\) is given by  
\[L_{s,p}^{i}=\left\{ \begin{array}{cl}L_{2i-1}&&;p=NN-FF\\ \left(\frac{1-\left(-1^{i}\right)}{2}\right)L_{2i-1}+\left(\frac{1+\left(-1^{i }\right)}{2}\right)L_{2i-2}&&;p=OO-EE\\ L_{2i-1}&&;p=SW-SW\\ L_{i}&&;p=SW\\ L_{r_{1}}&&;p=RP\end{array} \right. \tag{13}\]  
\(L_{s,p}^{i}\) is the number of IOS components allocated to the SUE during the \(i\)th sub-frame and \(p\)th pairing. By substituting the PDF of \(\mathbb{H}_{s,p}^{i}\) and solving as in [29; 34], the outage probability at the SUE during direct transmission is determined as  
\[P_{s,p}^{i}=\left(\frac{\gamma\big{(}a+1,\frac{\gamma_{1}}{b}\big{)}}{\Gamma(a +1)}\right)\times\left(\frac{\gamma\big{(}a+1,\frac{\gamma_{2}}{b}\big{)}}{ \Gamma(a+1)}\right) \tag{14}\]  
where \(x_{1}=\sqrt{\frac{\mu_{wi}}{\rho_{w,p}^{i}\big{(}\mu_{si}+\left[\tilde{k}_{g}^ {2}+\tilde{k}_{w}^{2}\right]\big{)}}}\sqrt{\frac{\rho_{w,p}^{i}}{\rho\mu_{wi}}}\) and \(x_{2}=\sqrt{\frac{\mu_{zi}}{\rho_{w,p}^{i}\big{(}\epsilon\mu_{wi}+\left[\tilde{k }_{g}^{2}+\tilde{k}_{w}^{2}\right]\big{)}}}\sqrt{\frac{\rho_{s,p}^{i}}{\rho\mu_ {zi}}}\). Substituting \(a,\ b,\ x_{1}\) and \(x_{2}\) in (14) gives  
\[P_{s,p}^{i}=\left(\frac{\gamma\Bigg{(}\frac{\pi^{2}L_{w,p}^{i}}{16-\pi^{2}}, \frac{2\pi}{16-\pi^{2}}\sqrt{\frac{\mu_{wi}}{\rho_{w,p}^{i}\big{(}\mu_{si}+ \left[\tilde{k}_{g}^{2}+\tilde{k}_{w}^{2}\right]\big{)}}}\sqrt{\frac{\rho_{w,p }^{i}}{\rho\mu_{wi}}}\Bigg{)}}{\Gamma\bigg{(}\frac{\pi^{2}L_{w,p}^{i}}{16-\pi^{ 2}}\bigg{)}}\right)\times\left(\frac{\gamma\Bigg{(}\frac{\pi^{2}L_{w,p}^{i}}{16 -\pi^{2}},\frac{2\pi}{16-\pi^{2}}\sqrt{\frac{\mu_{wi}}{\rho_{w,p}^{i}\big{(} \epsilon\mu_{wi}+\left[\tilde{k}_{g}^{2}+\tilde{k}_{w}^{2}\right]\big{)}}} \sqrt{\frac{\rho_{w,p}^{i}}{\rho\mu_{zi}}}\Bigg{)}}{\Gamma\bigg{(}\frac{\pi^{2 }L_{w,p}^{i}}{16-\pi^{2}}\bigg{)}}\right) \tag{15}\]  
The condition for the WUE experiencing outage is \(\Gamma_{d-w,p}^{i}<\tilde{R}_{w,p}^{i}\). The probability of outage at the WUE during direct transmission is \(P_{w,p}^{i}=P(\Gamma_{d-w,p}^{i}<\tilde{R}_{w,p}^{i})\).