## 4 Performance Analysis
### Outage Probability
Rewriting with respect to the channel condition \(H_{u}=\mathbb{H}_{w,p}^{i}\), the condition for the WUE experiencing outage is given by  
\[P_{w,p}^{i}=P\left(\mathbb{H}_{w,p}^{i}<\sqrt{\frac{\mu_{wi}}{\rho_{w,p}^{i} \Big{(}\mu_{si}+\left[k_{g}^{2}+k_{w}^{2}\right]\Big{)}}}\sqrt{\frac{\rho_{ w,p}^{i}}{\rho\mu_{wi}}}\right) \tag{16}\]For a detailed derivation of (16), refer to Appendix B. Similarly, the PDF for \(\mathbb{H}_{w,p}^{i}\) is expressed as [33]  
\[f_{\mathbb{H}_{w,p}^{i}}(h_{w,p}^{i})=\frac{{h_{w,p}^{i}}^{c}}{d^{c+1}\Gamma(c+1 )}e^{-\left(h_{w,p}^{i}/d\right)}dh_{w,p}^{i} \tag{17}\]  
where \(c=\frac{\left(m_{w,p}^{i}\right)^{2}}{\sigma_{w,p}^{i}}-1\) and \(d=\frac{{c_{w,p}^{i}}^{2}}{m_{w,p}^{i}}\)[33; 34] with a mean of \(m_{w,p}^{i}=\frac{{L_{w,p}^{i}}^{2}}{2}\) and variance of \({\sigma_{w,p}^{i}}^{2}=L_{w,p}^{i}\left(1-\frac{\pi^{2}}{16}\right)\)[32]. Here, \(L_{w,p}^{i}\) is given by  
\[L_{w,p}^{i}=\left\{ \begin{array}{cl}L_{2i}&;p=NN-FF\\ \left(\frac{1-\left(-1^{i}\right)}{2}\right)L_{2i+1}+\left(\frac{1+\left(-1^{ i}\right)}{2}\right)L_{2i}&;p=OO-EE\\ L_{2i}&;p=SW-SW\\ L_{M+1-i}&;p=SW\\ L_{r_{2}}&;p=RP\end{array} \right. \tag{18}\]  
\(L_{w,p}^{i}\) is the number of IOS components allocated to the WUE during the \(i\)th sub-frame and \(p\)th pairing. By substituting the PDF of \(\mathbb{H}_{w,p}^{i}\) and solving as in [29; 34], the outage probability at the WUE during direct transmission is determined as  
\[P_{w,p}^{i}=\left(\frac{\gamma\left(c+1,\frac{x_{3}}{d}\right)}{\Gamma(c+1)}\right) \tag{19}\]  
where \(x_{3}=\sqrt{\frac{{h_{w,i}}}{{\rho_{w,p}}\left({\mu_{w,i}}+\left[k_{3}^{2}+k_{ 6}^{2}\right]\right)}}\sqrt{\frac{{\rho_{w,p}^{i}}}{{\rho_{w,p}}}}\). Substituting \(c,\ d\), and \(x_{3}\) in (19) gives  
\[P_{w,p}^{i}=\frac{\gamma\left(\frac{\pi^{2}L_{w,p}^{i}}{16-\pi^{2}},\frac{2\pi }{16-\pi^{2}}\sqrt{\frac{{\mu_{w,i}}}{{\rho_{w,p}}\left({\mu_{w,i}}+\left[k_{3 }^{2}+k_{6}^{2}\right]\right)}}\right)}{\sqrt{\frac{{\rho_{w,p}^{i}}}{{\rho_{ w,p}}}}}\right)}{\Gamma\left(\frac{\pi^{2}L_{w,p}^{i}}{16-\pi^{2}}\right)} \tag{20}\]  
The condition for outage at the WUE during cooperative relaying is \({\Gamma_{c-w,p}^{i}}<{\bar{R}_{w,p}^{i}}\). The probability of outage at the WUE during cooperative relaying is \({P_{c-w,p}^{i}}=P({\Gamma_{c-w,p}^{i}}<{\bar{R}_{w,p}^{i}})\). Rewriting with respect to \(\mathbb{H}_{c-w,p}^{i}\), the condition for the WUE experiencing outage during cooperative relaying can be expressed as follows:  
\[P_{c-w,p}^{i}=P\left(\mathbb{H}_{c-w,p}^{i}<\sqrt{\frac{1}{{\rho_{w,p}^{i}} \left(k_{g}^{2}+k_{w}^{2}\right)}}\sqrt{\frac{{\rho_{w,p}^{i}}}{{\rho}}}\right) \tag{21}\]  
For a detailed derivation of (21), refer to Appendix C. Similarly, the PDF for the random variable \(\mathbb{H}_{c-w,p}^{i}\) is expressed as [33]  
\[f_{\mathbb{H}_{c-w,p}^{i}}(h_{c-w,p}^{i})=\frac{{h_{c-w,p}^{i}}^{e}}{{f^{e+1} \Gamma(e+1)}}e^{-\left(h_{c-w,p}^{i}/f\right)}d{t_{c-w,p}^{i}} \tag{22}\]  
where \(e=\frac{\left(m_{c-w,p}^{i}\right)^{2}}{\sigma_{c-w,p}^{i}}-1\) and \(f=\frac{{c_{c-w,p}^{i}}^{2}}{m_{c-w,p}^{i}}\), with a mean of \(m_{c-w,p}^{i}=\frac{L\pi}{2}\) and variance of \({\sigma_{c-w,p}^{i}}^{2}=L\left(1-\frac{\pi^{2}}{16}\right)\)[32]. By substituting the PDF of \(\mathbb{H}_{c-w,p}^{i}\) and solving as in [29; 33], the outage probability of the WUE during cooperative relaying is determined as \[P_{c-w,p}^{i}=\left(\frac{\gamma\Big{(}e+1,\frac{x_{4}}{f}\Big{)}}{\Gamma(e+1)}\right) \tag{23}\]  
where \(x_{4}=\sqrt{\frac{1}{\rho_{w,p}^{i}\big{(}k_{s}^{2}+k_{w}^{2}\big{)}}}\sqrt{ \frac{\rho_{w,p}^{i}}{\rho}}\).