## 4 Performance Probability Analysis
### Coverage Probability
 \tag{34}\]  
For ease of expression, we split (34) by drawing on Reference [32]; it can be written as:  
\[\mathbb{E}_{b_{0}}[e^{-sl}]=\exp\Bigl{(}-\frac{c}{\pi}(\Theta_{1}+\Theta_{2} +\Theta_{3})\Bigr{)}. \tag{35}\]  
Here, we transform the Cartesian coordinate into a polar coordinate system and let \(|b|=r\), \(\Theta_{1}\) can be written as:  
\[\Theta_{1}=\int_{0}^{2\pi}\int_{r_{o}}\frac{\
u G_{b_{0}}P_{t}r^{-\alpha^{ \prime}}}{\
u G_{b_{0}}P_{t}r^{-\alpha^{\prime}}+1}rdrd\theta^{{}^{\prime}}, \tag{36}\]  
where \(r\) denotes the distance to the origin of the polar coordinate system and \(\theta^{{}^{\prime}}\) is the polar angle. Replacing \(\left(\frac{r}{\
u^{\frac{1}{-\alpha^{\prime}}}}\right)^{2}\) with \(u\), we can get  
\[\Theta_{1}=\pi\
u^{\frac{2}{\alpha}}\int\!\!\!\left(\frac{r_{o}}{\
u^{\frac{1} {-\alpha^{\prime}}}}\right)^{2}\frac{G_{b_{0}}P_{t}}{G_{b_{0}}P_{t}+u^{\frac{ r}{\alpha}}}du. \tag{37}\]  
Equation \(\Theta_{2}\) can be expressed in polar coordinate system as:  
\[\Theta_{2}=2\pi\left(\int_{r_{o}}^{\infty}\frac{e^{-\frac{c}{\beta}(r-r_{o})^ {2}}}{\
u P_{t}r^{-\alpha^{\prime}}+1}rdr\right). \tag{38}\]  
According to the same method, \(\Theta_{3}\) can be represented as:  
\[ \begin{split}\Theta_{3}=&\int_{0}^{2\pi}\int_{0}^{ \infty}e^{-\frac{c}{\beta}u^{2}}udud\theta^{{}^{\prime}}-\\ & 2\int_{0}^{\pi/2}\int_{0}^{2r_{o}\cos(\theta^{{}^{\prime}})}e^{- \frac{c}{\beta}u^{2}}udud\theta^{{}^{\prime}}.\end{split}  \tag{39}\]For the first part replacing \(u^{2}\) with \(y\) and the second part replacing \(u^{2}\) with \(x\), we can obtain  
\[\Theta_{3}= \frac{\pi\beta}{c}-\frac{\beta\pi}{2c}-\frac{\beta}{c}\int_{0}^{ \alpha_{o}}\frac{-\frac{c}{\beta}}{\sqrt{4r_{o}-x}}dx. \tag{40}\]  
Bringing (37), (38) and (40) into (34), we are able to obtain the expression for the interference Laplace transform under \(\beta\)-GPP as:  
\[ \begin{split}\mathcal{L}_{I_{2}}(s)=&\exp\left(-c \upsilon^{\frac{2}{\alpha}}\int_{\left(\frac{r_{o}}{\
u-x}\right)}^{\infty} \frac{G_{b_{o}}P_{t}}{G_{b_{o}}P_{t}+u^{\frac{\alpha}{2}}}du\right.\\ &\left.2\pi\left(\int_{r_{o}}^{\infty}\frac{e^{-\frac{\beta}{ \alpha}(r-r_{o})^{2}}}{\
u G_{b_{o}}P_{t}-\alpha^{\prime}+1}rdr\right)+ \right.\\ &\left.\frac{\pi\beta}{c}-\frac{\beta\pi}{2c}-\frac{\beta}{c} \int_{0}^{\alpha_{o}}\frac{-\frac{c}{\beta}}{\sqrt{4r_{o}-x}}dx.\right).\end{split}  \tag{41}\]  
Combining (41) and (31), \(P_{cov}\) is expressed as:  
\[ \begin{split}\mathbf{P}_{cov}=\int_{|b_{o}|=0}^{\infty}\exp \left(\frac{-\
u\sigma^{2}}{P_{t}|b_{o}|-\alpha^{\prime}G_{b_{o}}}\right)\\ &\exp\left(-c\upsilon^{\frac{2}{\alpha^{\prime}}}\int_{\left( \frac{r_{o}}{\
u-x}\right)}^{\infty}\frac{P_{t}}{P_{t}+u^{\frac{\alpha}{2}}} du+\right.\\ &\left.2\pi\left(\int_{r_{o}}^{\infty}\frac{e^{-\frac{c}{\beta}( r-r_{o})^{2}}}{\
u P_{t}r-\alpha^{\prime}+1}rdr\right)+\right.\\ &\left.\left(\frac{\pi\beta}{c}-\frac{\beta\pi}{2c}-\frac{\beta} {c}\int_{0}^{\alpha_{o}}\frac{-\frac{c}{\beta}}{\sqrt{4r_{o}-x}}dx\right) \right)2 cb_{o}e^{\alpha_{o}^{2}}db_{o}.\end{split}  \tag{42}\]  
Coverage probability is the ability of a network to provide a service. The expression shows the effect of the repulsive parameter \(\beta\) and beam width \(\theta\) on the coverage probability. We can adjust the parameters according to the expression to ensure the ability of the UAVs network to provide services.