## 3 Outage Probabilities
### Outage Probabilities
The interference range at the destination resulting from the interference from the \(m^{th}\) sub-channel, expressed as \(r_{O1D,m}\), can be defined as the minimum \(d_{O1D,m}\) that satisfies \(SIR_{SD,m}\geq\Gamma_{D}\), where \(\Gamma_{D}\) is the required SIR value at the destination. \[r_{O1D,m}=d_{SD}\left(\frac{\Gamma_{D}\lambda_{m,n}P_{O1,m}^{TX}}{P_{S}^{TX} }\right)^{\frac{1}{\alpha}}. \tag{4}\]  
Note that the interference range is not the same for all sub-channels and is dependent on \(\lambda_{m,n}P_{O1,m}^{TX}\) for the \(m^{th}\) sub-channel. As shown in Figure 8, the interference range for a sub-channel can be larger than others. Figure 8: Other-channel interference. If the \(m^{th}\) channel interferer is located inside the circle of radius \(r_{O1D,m}\) centered at the destination position \((x_{D},y_{D})\), then the target SIR cannot be satisfied. Hence, the outage probability for the source-destination link resulting from the \(m^{th}\)-channel interferer can be approximately calculated as:  
\[ \begin{array}{l}P_{SD,m}^{Outage}\approx\min\Bigl{\{}1,\iint_{Interference_{ SD}}f_{O,m}(x,y)dxdy\Bigr{\}}\\ =\min\Bigl{\{}1,\pi\,r_{O1D,m}2^{\widetilde{f}_{O,m}^{SD}}(x_{D},y_{D})\Bigr{\}} \\ =\min\Biggl{\{}1,\pi\,d_{SD}2^{\left(\frac{\Gamma_{D}\lambda_{m},p_{O1,m}^{TX} }{P_{S}^{TX}}\right)^{\frac{2}{2}}}\widetilde{f}_{O,m}^{SD}(x_{D},y_{D}) \Biggr{\}}\end{array}  \tag{5}\]  
where \(Interference_{SD}\) is the circular area of radius \(r_{O1D,m}\) centered at the destination position \((x_{D},y_{D})\), \(f_{O,m}(x,y)\) is the interferer density at \((x,y)\), and \(\widetilde{f}_{O,m}^{SD}(x_{D},y_{D})\) is the average interferer density of \(Interference_{SD}\), defined as:  
\[\widetilde{f}_{O,m}^{SD}(x_{D},y_{D})\equiv\frac{1}{\pi\,r_{O1D,m}}\iint_{ Interferences_{SD}}f_{O,m}(x,y)dxdy. \tag{6}\]  
Suppose that there is a hot-spot region for interferers with area \(A_{Hotspot}\), and the interferer density of the hot-spot, denoted as \(D_{O}^{Hotspot}\), is \(\beta\) times of that of other regions, then \(D_{O}^{Hotspot}\) can satisfy  
\[D_{O}^{Hotspot}A_{Hotspot}+\frac{D_{O}^{Hotspot}}{\beta}(\pi\,R^{2}+A_{Hotspot })=1, \tag{7}\]  
and, thus,  
\[D_{O}^{Hotspot}=\frac{\beta}{(\beta-1)A_{Hotspot}+\pi\,R^{2}}. \tag{8}\]  
If interferers are distributed in a uniform distribution across the cell area, in other words, \(\beta=1\) in Equation (8), then:  
\[f_{O,m}(x_{D},y_{D})=D_{O}^{Uniform}\equiv\frac{1}{\pi\,R^{2}}, \tag{9}\]  
and, thus, Equation (5) can be rewritten as follows:  
\[P_{SD,m}^{Outage}\approx\min\biggl{(}1,\frac{r_{O1D,m}^{2}}{R^{2}}\biggr{)}= \min\left\{1,\frac{d_{SD}}{R^{2}}\biggl{(}\frac{\Gamma_{D}\lambda_{m,m}P_{O1, m}^{TX}}{P_{S}^{TX}}\biggr{)}^{\frac{2}{2}}\right\}. \tag{10}\]  
Considering all \(M-1\) sub-channels producing the other-channel interference to the \(n^{th}\) sub-channel, the outage probability for the source-destination link with a uniform distribution of interferers can be expressed as follows:  
\[P_{SD}^{Outage}=1-\prod_{m=1,m\
eq n}^{M}\left(1-P_{SD,m}^{Outage}\right) \approx 1-\prod_{m=1,m\
eq n}^{M}\left(1-\min\left\{1,\frac{d_{SD}}{R^{2}} \biggl{(}\frac{\Gamma_{D}\lambda_{m,m}P_{O1,m}^{TX}}{P_{S}^{TX}}\biggr{)}^{ \frac{2}{2}}\right\}\right). \tag{11}\]  
The outage probability can be reduced by repeated transmission through the relay.