## 4 Performance Probability Analysis
### Connection Probability  
As shown in Figure 3, information was transmitted from the source node \(S\) to the end node \(D\) through the relay node \(R_{i}(i=1,2,\cdots,N)\). UAVs outside the transmission link were regarded as interference nodes. So, the signal-to-interference-to-noise ratio (SINR) between each hop can be expressed as:  
\[\mathrm{Y}_{i}=\frac{P_{i}h_{i}d_{i}^{-\alpha}G_{i}}{r_{1}I+T_{0}}, \tag{9}\]  
where \(I=\sum_{j\in\Phi,\
eq i}P_{i}d_{j}^{-\alpha}h_{j}G_{j}\) is the total interference received by the relay node, \(\Phi\) represents the UAVs node in space. The channel gains \(h_{j}\) and \(h_{i}\) have an exponential distribution with a mean of one. \(P_{i}\) and \(P_{l}\) indicate the transmission power and interference source power at relay node \(R_{i}\), respectively. \(r_{1}\) is the interference impact factor. \(d_{i}\) is the distance from \(R_{i}\) to \(R_{i+1}\), and \(\alpha\) is the path attenuation factor, which depends on the environment. \(T_{0}\) is the average background noise power. \(G_{i/j}\) can be obtained by (8). In a multi-hop node information transmission, SINR exists between each hop node. When operating under the DF protocol, according to (9), the end-to-end SINR is expressed as:  
\[\mathrm{Y}_{DF}=\min(\mathrm{Y}_{1},\mathrm{Y}_{2},\cdots,\mathrm{Y}_{N}). \tag{10}\]  
Then, the connection probability can be expressed as:  
\[\mathbf{P}_{con}=P(\mathrm{Y}_{DF}>Q). \tag{11}\]  
Since \(\mathrm{Y}_{i}\) is independent, from (10) and (11), we can obtain the connection probability \(\mathbf{P}_{con}\) as:  
\[\mathbf{P}_{con}=F_{\mathrm{Y}_{DF}}(Q)=\prod_{i=1}^{N}(1-F_{\mathrm{Y}_{i}}( Q)). \tag{12}\]  
Here, \(F_{X}=\mathbf{P}(X<x)=1-\exp(-ax)\) is an exponential random variable cumulative density function with parameter \(X\). The cumulative density function can be written as:  
Figure 3: Multi-hop UAVs relay system based on \(\beta\)-GPP. \[ \begin{split}\text{F}_{Y_{i}}(Q)&=\textbf{P}\Bigg{(}h_{i}> \frac{Q(r_{1}I+T_{0})}{G_{i}P_{i}d_{i}^{-\alpha}}\Bigg{)}\\ &=\mathbb{E}_{Y_{i}}\Bigg{[}1-\exp\Bigg{(}-\frac{Q(r_{1}I+T_{0}) }{\Theta_{h}G_{i}P_{i}d_{i}^{-\alpha}}\Bigg{)}\Bigg{]}\\ &=\exp(-A_{\text{Y}}T_{0})\mathcal{L}_{I}(A_{\text{Y}}),\end{split}  \tag{13}\]  
where \(h_{i}\) and \(I\) are independent, \(A_{\text{Y}}=\frac{Q}{\Theta_{h}G_{i}P_{i}d_{i}^{-\alpha}}\). \(\mathcal{L}_{I}(X)\) is the Laplace transform of a random variable **x**. Combined with (12), the Laplace transform \(\mathcal{L}_{I}(A_{\text{Y}_{DF}})\) can be obtained as follows:  
\[ \begin{split}\mathcal{L}_{I}(A_{\text{Y}_{DF}})&= \mathbb{E}\Bigg{[}\exp\Bigg{(}-\sum_{\textbf{x}\in\Phi}A_{\text{Y}}P_{i}G_{j} h_{i}(|\textbf{x}|+\epsilon)^{-\beta}\Bigg{)}\Bigg{]}\\ &=\mathbb{E}\Bigg{[}\prod_{\textbf{x}\in\Phi}E\Big{[}\exp(-A_{ \text{Y}}P_{i}G_{j}h_{i}(|\textbf{x}|+\epsilon)^{-\beta})\Big{]}\Bigg{]}\\ &=\mathbb{E}\Bigg{[}\prod_{\textbf{x}\in\Phi}\Bigg{(}1+\frac{ \Theta_{g}A_{\text{Y}}P_{i}G_{j}}{(|\textbf{x}|+\epsilon)^{\beta}}\Bigg{)}^{- 1}\Bigg{]}\\ &=\mathbb{E}\Bigg{[}\exp\Bigg{(}-\sum_{\textbf{x}\in\Phi}\ln \Bigg{(}1+\frac{\Theta_{g}A_{\text{Y}}P_{i}G_{j}}{(|\textbf{x}|+\epsilon)^{ \beta}}\Bigg{)}\Bigg{)}\Bigg{]}.\end{split}  \tag{14}\]  
According to (3), there is  
\[\mathcal{L}_{I}(A_{\text{Y}})=Det(\textbf{I}+\beta\mathcal{K}_{i})^{-\frac{1}{ \beta}}, \tag{15}\]  
and the core matrix \(\mathcal{K}_{i}\) can be interpreted as:  
\[ \begin{split}\mathcal{K}_{i}(\textbf{x},\textbf{y})=& \sqrt{\frac{\Theta_{g}\eta_{\text{Y}}P_{i}G_{j}}{\Theta_{g}A_{ \text{Y}}P_{i}G_{j}+(|\textbf{x}|+\epsilon)^{\alpha}}}\mathcal{K}(\textbf{x}, \textbf{y})\\ &\times\sqrt{\frac{\Theta_{g}A_{\text{Y}}P_{i}G_{j}}{\Theta_{g} \eta_{\text{Y}}P_{i}G_{j}+(|\textbf{y}|+\epsilon)^{\alpha}}}.\end{split}  \tag{16}\]  
We assume  
\[\eth(x)=\sqrt{\frac{\Theta_{g}A_{\text{Y}}P_{i}G_{j}}{\Theta_{g}A_{\text{Y}}P_{ i}G_{j}+(x+\epsilon)^{\alpha}}}. \tag{17}\]  
Denote \(\zeta_{n}\) as an eigenvalue of \(\mathcal{K}_{i}\).