## 3 Outage Probabilities
### Outage Probabilities
The signal power from the source at the relay is written as:  
\[P_{SR}^{RX}=K_{1}P_{S}^{TX}d_{SR}{}^{-\alpha}, \tag{12}\]  
and the interference power from the other-channel interferer \(1,m\) at the relay can be expressed as:  
\[P_{O1R,m}^{RX}=K_{1}\lambda_{m,m}P_{O1,m}^{TX}d_{O1R,m}{}^{-\alpha}. \tag{13}\]The SIR at the relay only considering the \(m^{th}\)-channel interferer, denoted as \(SIR_{SR,m}\), can be expressed as follows:  
\[SIR_{SR,m}=\frac{P_{SR}^{RX}}{P_{O1R,m}^{RX}}=\frac{P_{S}^{TX}{d_{SR}}^{-\alpha} }{\lambda_{m,n}{P_{O1,m}^{TX}{d_{O1R,m}}}^{-\alpha}}. \tag{14}\]  
The interference range, denoted as \(r_{O1R,m}\), can be defined as the minimum \(d_{O1R,m}\) satisfying \(SIR_{SR,m}\geq\Gamma_{R}\), where \(\Gamma_{R}\) is the required SIR at the receiver. \[r_{O1R,m}=d_{SR}\left(\frac{\Gamma_{R}\lambda_{m,n}{P_{O1,m}^{TX}}}{P_{S}^{TX }}\right)^{\frac{1}{\alpha}}. \tag{15}\]  
The outage probability for the source-relay link from the \(m^{th}\)-channel interferer can be approximately calculated as follows:  
\[ \begin{array}{l}P_{SR,m}^{Outage}\approx\min\Big{\{}1,\iint_{Interference_{ SR}}f_{O,m}(x,y)dxdy\Big{\}}\\ \approx\min\Big{\{}1,\pi\,r_{O1R,m}2\,\widehat{f}_{O,m}^{SR}\left(x_{R},y_{R} \right)\Big{\}}\\ =\min\Bigg{\{}1,\pi\,{d_{SR}}^{2}\bigg{(}\frac{\Gamma_{R}\lambda_{m,n}{P_{O1,m}^{TX}}}{P_{S}^{TX}}\bigg{)}^{\frac{2}{\alpha}}\widehat{f}_{O,m}^{SR}\left(x _{R},y_{R}\right)\Bigg{\}}\end{array}  \tag{16}\]  
where \(Interference_{SR}\) is the circular area of radius \(r_{O1R,m}\) centered at the relay position \((x_{R},y_{R})\), and \(\widehat{f}_{O,m}^{SR}(x_{R},y_{R})\) is the average interferer density of \(Interference_{SR}\), defined as:  
\[\widehat{f}_{O,m}^{SR}(x_{R},y_{R})\equiv\frac{1}{\pi\,r_{O1R,m}}\iint\limits _{Interference_{SR}}f_{O,m}(x,y)dxdy. \tag{17}\]  
If interferers are distributed in a uniform distribution over the cell area, in other words, Equation (9) holds, then Equation (16) can be rewritten as follows:  
\[P_{SR,m}^{Outage}\approx\min\bigg{(}1,\frac{r_{O1R,m}}{R^{2}}\bigg{)}=\min \Bigg{\{}1,\frac{{d_{SR}}^{2}}{R^{2}}\bigg{(}\frac{\Gamma_{R}\lambda_{m,n}{P_ {O1,m}^{TX}}}{P_{S}^{TX}}\bigg{)}^{\frac{2}{\alpha}}\Bigg{\}}. \tag{18}\]  
Hence, the outage probability for the source-relay link considering all \(M-1\) sub-channels producing the interference to the \(n^{th}\) sub-channel can be written as follows:  
\[P_{SR}^{Outage}=1-\prod_{m=1,m\
eq n}^{M}\left(1-P_{SR,m}^{Outage}\right) \approx 1-\prod_{m=1,m\
eq n}^{M}\left(1-\min\Bigg{\{}1,\frac{{d_{SR}}^{2}}{R^ {2}}\bigg{(}\frac{\Gamma_{R}\lambda_{m,n}{P_{O1,m}^{TX}}}{P_{S}^{TX}}\bigg{)} ^{\frac{2}{\alpha}}\Bigg{\}}\Bigg{\}}\right). \tag{19}\]  
Similarly, the interference range at the destination for the second transmission from the relay, expressed as \(r_{O2D,m}\), can be defined as:  
\[r_{O2D,m}=d_{RD}\Bigg{(}\frac{\Gamma_{D}\lambda_{m,n}{P_{O2,m}^{TX}}}{P_{R}^{ TX}}\Bigg{)}^{\frac{1}{\alpha}}. \tag{20}\]  
The outage probability for the relay-destination link from the other-channel interferer \(2,m\) can be approximately calculated as follows:\[ \begin{array}{l}P_{RD,m}^{Outage}\approx\min\left\{1,\iint_{Interference_{RD}}f_{ O,m}(x,y)dxdy\right\}\\ =\min\left\{1,\pi\,r_{O2D,m}\widetilde{f}_{O,m}^{RD}(x_{D},y_{D})\right\}\\ =\min\left\{1,\pi\,d_{RD}2^{2}\left(\frac{r_{D}\lambda_{m,n}P_{O2,m}^{TX}}{P_ {R}^{TX}}\right)^{\frac{2}{\alpha}}\widetilde{f}_{O,m}^{RD}(x_{D},y_{D})\right\} \end{array}