# Uncertainty Theory-Based Resilience Analysis for LEO Satellite Communication Systems
## 4 LEO-SCS and Its Uncertain Evolution Model
### Uncertain Satellite Network Evolution Model
#### 4.2.4 Evolution Rule of Dynamic Uncertain Satellite Network Model
The in-orbit and ground backup strategies in Section 4.2.4 are designed for an LEO-SCS as follows. * In-orbit backup strategy: each operating orbit has a backup orbit that locates directly below at a height of 1000 km, and one backup satellite is deployed in each backup orbit; \begin{table}
\begin{tabular}{c c} \hline \hline
**Node Type** & **Processing Delay (ms)** \\ \hline Virtual node & \(\mathcal{L}(1,15)\) \\ Satellite & \(\mathcal{L}(1,20)\) \\ Gateway & \(\mathcal{Z}(2,3,8)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters of processing delays on nodes. Figure 5: Virtual nodes. * Ground backup strategy: each launch can carry one backup satellite to supplement the in-orbit backup, and the ground transfer time obeys a uniform distribution, \(U(30,60)\) days. The maximum allowable recovery time of the system is determined as 15 days. ### Resilience Analysis and Discussion

We built the uncertain satellite network evolution model for this Walker LEO-SCS according to Section 4.2, and the system resilience is computed according to the resilience estimation model in Section 5. Figure 6 shows the belief instantaneous availability of LEO-SCS under both ideal and actual conditions over 6 years. As shown in Figure 5(a), the belief instantaneous availability of the system fluctuates in the range of 0.994\(\sim\)1. Fluctuation is caused by the dynamic movement of the satellites and internet traffic intensity changes over time. From Figure 5(b), one can see that the belief instantaneous availability of the system degrades several times under actual conditions, and these are caused by satellite failure. When a satellite fails, the link connected with it also fails, the network topology of the system evolves, and ground nodes previously connected through the failed satellite must find another connectable satellite. If none exists, the ground node will lose the satellite connection. Transmissions previously relayed through the failed satellite must select a new path. As the LEO-SCS applies the delay bounded routing algorithm, the new connection or path tends to be longer than the original at the same belief degree. So, satellite failure will cause delay-based availability to decrease, and instantaneous availability will fluctuate at a lower level until a backup satellite replaces the failed one. From Figure 5(b), it can be seen that: (1) some degradations recover quickly and some slowly, depending on whether the in-orbit backup can be used when the satellite fails; (2) availability degradation varies due to the number of satellite failures and the failure time. Two examples are used to illustrate these phenomena. Figure 7 shows the belief instantaneous availability of a system when two co-orbiting satellites fail successively. At time \(t=4980\) min, the first satellite failed, and the system was restored after 193 min using its in-orbit backup. At time \(t=8463\) min, another satellite in the same orbit failed. The only in-orbit backup was used when the first satellite failed, there was no in-orbit backup when the second satellite failed, and the system had to wait for a ground backup. Therefore, the second system recovery consumed a much longer time, and the system was restored at time \(t=68\),952 min. Figure 6: Belief instantaneous availability of a system. (**a**) ideal conditions; (**b**) actual conditions. Figure 8 shows the belief instantaneous availability of a system under dual satellite failures. The first satellite failed at time \(t=100\) h, which slightly reduced the system availability. At time \(t=190\) h, another satellite failed in another orbit, but the first failed satellite was not recovered, as its in-orbit backup had been consumed earlier. At this time, two satellites in the system failed. Compared with a single satellite failure, more ground nodes lose satellite connectivity, and more users choose new longer paths in this case. This led to further availability degradation.