## 4 Performance Probability Analysis
### Coverage Probability  
The coverage probability is a measure of the service provided by FlyMesh, as shown in Figure 4. Due to the motion invariance of GPP, we can assume that the coverage probability of a typical user \(b_{o}\) (\(b_{o}\in\Phi\)) is the same at any location. Each typical user is connected to its closest FBS, i.e., the typical user is served by only one closest FBS. Therefore, the coverage probability when SINR is greater than the threshold \(\
u\) is denoted as:  
\[ \begin{split}\mathbf{P}_{cov}&=\mathbf{P}( SINR>\
u,|b_{o}|\leq|b_{i}|)\\ &=\mathbf{P}\Bigg{\{}\frac{P_{h}h_{b_{o}}^{\prime}|b_{o}|^{-a}G_{ b_{o}}}{r_{2}I_{2}+\sigma^{2}}>\
u,|b_{o}|\leq|b_{i}|\Bigg{\}},\end{split}  \tag{29}\]  
where \(P_{t}\) is the transmit power of the UAVs. \(h_{i}^{{}^{\prime}}\) indicates that the mobile user only experiences Rayleigh fading, which is an exponential random variable with a mean of 1. \(I=\sum_{b_{k}\in\Phi\setminus b_{o}}P_{t}h_{b_{k}}^{\prime}|b_{k}|^{-a^{\prime} }G_{b_{k}}\) is the total interference received at the typical user, and \(r_{2}\) is the interference impact factor. \(\sigma_{0}\) denotes the background noise. \(|\cdot|\) is the Euclidean distance from the FBS to the user. We usually assume that the path loss exponent \(a^{{}^{\prime}}>2\). \(G_{b_{o}/b_{k}}\) can be obtained by (8). Therefore, the coverage probability can be further expressed as:  
\[ \begin{split}\mathbf{P}_{cov}&=\sum_{b_{o}=0}^{ \infty}P(SINR>\
u\mid b_{o},|b_{o}|\leq|b_{i}|)\\ &=\int_{|b_{o}|=0}^{\infty}\Bigg{\{}h_{b_{o}}^{\prime}>\frac{ \
u\big{(}r_{2}I_{2}+\sigma^{2}\big{)}}{P_{t}|b_{o}|^{-a^{\prime}}G_{b_{o}}},| b_{o}|\leq|b_{i}|\Bigg{\}}f(b_{o})db_{o}\\ &=\int_{|b_{o}|=0}^{\infty}\mathbb{E}\Bigg{\{}\frac{-\
u\sigma^{2 }}{P_{t}|b_{o}|^{-a^{\prime}}G_{b_{o}}}\Bigg{\}}\exp\Bigg{(}\frac{-\
u r_{2}I_{ 2}}{P_{t}|b_{o}|^{-a^{\prime}}G_{b_{o}}}\Bigg{)}\Bigg{\}}f(b_{o})db_{o}\\ &=\int_{|b_{o}|=0}^{\infty}\Bigg{\{}\exp\Bigg{(}\frac{-\
u\sigma^{ 2}}{P_{t}|b_{o}|^{-a^{\prime}}G_{b_{o}}}\Bigg{)}\mathcal{L}_{B}(s),|b_{o}|\leq |b_{i}|\Bigg{\}}f(b_{o})db_{o},\end{split}  \tag{30}\]  
where \(s=\frac{\
u r_{2}}{P_{t}|b_{o}|^{-a^{\prime}}G_{b_{o}}}\), \(f(b_{o})\) can be simplified for \(\beta\)-GPP after applying a diagonal approximation as \(f(b_{o})=2 cb_{o}e^{ct_{o}^{2}}\). Therefore, (30) can be written as:  
\[\mathbf{P}_{cov}=\int_{|b_{o}|=0}^{\infty}\exp\Bigg{(}\frac{-\
u\sigma^{2}}{P _{t}|b_{o}|^{-a^{\prime}}G_{b_{o}}}\Bigg{)}\mathcal{L}_{B}(s)2 cb_{o}e^{ct_{o}^ {2}}db_{o}. \tag{31}\]The authors of [31] gave the following expression for the Laplace transform of the interference of a general motion-invariant DPP under the recent FBS correlation scheme, i.e.,  
\[ \begin{split}\mathcal{L}_{l_{2}}(s)=&\mathbb{E}_{b_{ 0}}[e^{-sl}]\\ =&\frac{\sum_{0}^{\infty}\frac{(-1)^{n}}{n!}\int_{( \mathbb{R}^{2})^{n}}\det[\mathcal{K}_{b_{0}}(b_{i},b_{j})]}{\sum_{0}^{\infty} \frac{(-1)^{n}}{n!}\int_{B(\Theta_{b},b_{n})^{n}}\det[\mathcal{K}_{b_{0}}(b_{i },b_{j})]}\cdots\\ &\frac{\prod_{i=1}^{n}\left[1-\frac{|b_{i}|\geq|b_{0}|}{1+G_{b_{i }}\
u P_{i}|b_{i}|^{-\alpha}}\right]db_{1}\cdots db_{n}}{db_{1}\cdots db_{n}}. \end{split}  \tag{32}\]  
We used the approximation for the determinant of kernel matrix \(\mathcal{K}_{b_{0}}(b_{i},b_{j})\)  
\[\det[\mathcal{K}_{b_{0}}(b_{i},b_{j})]_{1_{1\leq i\leq n}}\approx\prod_{i=1}^ {n}\mathcal{K}_{b_{0}}(b_{i},b_{i}), \tag{33}\]  
and we know \(\sum_{n=0}^{\infty}\frac{-x^{n}}{n!}=e^{-x}\). Combined (4) and (5), the (32) can be rewritten as:  
\[ \begin{split}\mathbb{E}_{b_{0}}^{t}[e^{-sl}]&=\exp \left(-\frac{c}{\pi}\int\mathcal{B}(o,b_{0})^{c}\left(\frac{\
u G_{b_{0}}P_{i }|b|^{-\alpha^{\prime}}}{\
u G_{b_{0}}P_{i}|b|^{-\alpha^{\prime}}+1}\right. \right.\\ &\left.\left.+\frac{c}{\psi G_{b_{0}}P_{i}|b|^{-\alpha^{\prime}} +1}-e^{-\frac{c}{\beta}|b-b_{i}|^{2}}\right)db\right).\end{split}