## 5 Numerical Results and Discussion
In addition, the outage probability of case 4 is approximately \(10^{-5}\) when \(\sigma^{2}_{Sv}\) is small and slightly increases with an increase in \(\sigma^{2}_{Sv}\). Moreover, with the increase in \(\sigma^{2}_{Sv}\), the outage probability curve of case 4 closely aligns with that of case 1. However, the outage performance remains unsatisfactory at a level of \(10^{-4}\) due to significant velocity variances of the AWACS and mobile GS. Compared with Figure 4b, a UAV relay is added in Figure 4c. It is evident that the additional relay significantly enhances the system's overall outage performance. The outage probability of cases 1 and 3 decreases to approximately \(10^{-11}\) when \(\sigma^{2}_{Sv}\) is below 2, and the outage probability of case 4 reduces to around \(10^{-8}\). However, there is no significant change in case 2. The outage probability of case 1 remains stable at about \(10^{-5}\) as \(\sigma^{2}_{Sv}\) increases, whereas case 4 maintains stability at around \(10^{-4}\), which is similar to \(N=3\). Figure 5 shows the outage probability versus the \(\sigma^{2}_{Dv}\). It can be observed that the outage performance becomes worse the larger the \(\sigma^{2}_{Dv}\). Similarly, a large \(\sigma^{2}_{Dv}\) will result in a large pointing error. The performance of different cases when \(\sigma^{2}_{Sv}=2\) is analyzed in Figure 5a. When \(\sigma^{2}_{Dv}\) exceeds 3, the outage probability curve of case 4 coincides with that of case 1, indicating that case 4 can serve as a viable replacement for the complex case 1. Figure 3: Outage probability of the proposed four different relay selection modes for different numbers of UAV relays when \(\sigma^{2}_{Sv}=\sigma^{2}_{Dv}=1\) and \(H_{UAV}=0.5\) km: (**a**) case 1; (**b**) case 2; (**c**) case 3; (**d**) case 4. Furthermore, when \(\sigma_{Dv}^{2}\) surpasses 6, the curves of case 3 align with those of cases 1 and 4, suggesting that case 3 can be adopted as a simpler alternative relay mode. As depicted in Figure 5b, the value of \(\sigma_{Sv}^{2}\) increases to 10, and the simultaneous increase in the AWACS and mobile GS results in a significant deterioration of outage performance. It is observed from the figure that the outage probability of case 1 and case 2 exhibits the lowest values compared to other cases, about \(10^{-9}\). However, case 2 experiences a rapid deterioration in performance as \(\sigma_{Dv}^{2}\) increases, which differs from case 4 which exhibits a minimum outage probability of about \(10^{-6}\) but changes relatively slowly with increasing \(\sigma_{Dv}^{2}\). In addition, the outage probability of case 3 consistently remains at approximately \(10^{-3}\) without significant variation as \(\sigma_{Dv}^{2}\) increases. Furthermore, the outage probability performance of cases 1, 2, and 4 is improved by the increase in the number of relay nodes, as depicted in Figure 5c. However, the outage probability of case 3 remains relatively unchanged compared to Figure 5b. In Figure 6, we compare the outage performance of the proposed cases with respect to the \(H_{UAV}\). As can be observed, with the increase in \(H_{UAV}\), the outage probabilities of cases 1, 2 and 4 increase sharply and then reach a stable state. In case 3, the outage probability does not change significantly when \(H_{UAV}\) is less than 0.5 km. This is because the utilization of diversity technology in the UAV-to-GS links enhances the outage performance of the second hop significantly; thereby, the overall end-to-end outage performance of the system is limited by the first hop from the AWACS to the UAV.