# 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
Use the uncertain satellite network evolution model in Section 4.2 and the delay bounded routing algorithm in Section 5.1 to simulate the end-to-end delay of each application for every time interval \(\triangle t\) under a specific disruption (i.e., with satellite failure) and in a normal state (i.e., without satellite failure);Using Equations (11) and (17), measure the behavior of the LEO-SCS after the \(j\)th disruption,

\[\frac{\int\limits_{\Psi}^{t_{0j}+T_{a}}A_{\text{B}}(t|d)dt}{\int\limits_{t_{0j}+T _{a}}^{t_{0j}+T_{a}}A_{\text{B}}(t|ns)dt}=\frac{\sum\limits_{k=\left[\frac{t_{0 j}}{2}\right]}^{\left[\frac{t_{0j}+T_{a}}{2}\right]}A_{\text{B}}(k\triangle t |d)\triangle t}{\sum\limits_{k=\left[\frac{t_{0j}}{2}\right]}^{\left[\frac{t_{0 j}+T_{a}}{2}\right]}A_{\text{B}}(k\triangle t|ns)\triangle t}, \tag{18}\]

where \(t_{0j}\) is the time at which the \(j\)th disruption started;
4. Repeat steps (2) and (3) \(N\) times, and calculate that there are \(N_{r}\) behaviors after a disruption that are greater than \(r_{eth}\). The belief resilience of the LEO-SCS is \[\widehat{R}_{E}=\frac{N_{r}}{N}.\] (19)

## 6 Case Study

### Overview

#### 6.1.1 Topology

We consider a Walker LEO-SCS with eight orbital planes at a height of 1300 km. The inclination of each orbital plane is 70 degrees, and six satellites are evenly distributed on each plane. As shown in Figure 4, each satellite has four adjacent satellites, two intraplanes and two inter-planes. There are four gateways in the system-in Beijing, Haikou, Kashgar, and Jiamusi. These connect the terrestrial network in China with the satellite network, and all can provide satellite connections for terrestrial users. According to the virtual node strategy in Section 4.1, we divide the Earth's surface into 104 ground grids, each with an area of about \(4.9\times 10^{6}\) square kilometers. A virtual node is set at the center of each grid, to provide service for all terminal users in the grid. Figure 5 shows the 104 virtual nodes, which are marked with plus (\({}^{\prime}+^{\prime}\)) signs. Figure 4: Walker constellation topology with 8 \(\times\) 6 nodes. As described in Section 4.2.1, the processing delay of each node in the system is considered an uncertain variable. To the best of our knowledge, processing delay of communication networks has not yet been characterized as uncertain variables. When the delay is a random variable, it can be represented by the uniform distribution [37; 38]. Considering that the delay in an LEO-SCS depends on the propagation delay and processing delay and the latter one is a deterministic variable, we assume that the processing delays of satellites and virtual nodes obey a linear uncertainty distribution, the processing delays of gateways follow a zigzag uncertainty distribution, and their related parameters are shown in Table 4.

#### 6.1.2 Application

We consider transmissions between user terminals and terrestrial users in China, where all applications have the same priority. The delay bounded routing algorithm in Section 5.2 is used, and the delay thresholds of all applications are 200 ms. The traffic requirement from zone \(Z_{sn}\) to zone \(Z_{dn}\) at hour \(H\) can be calculated as [39]

\[T(Z_{dn},Z_{sn},a_{H})=\frac{(u_{sn}\times h_{dn})^{0.5}}{(dist(dn,sn))^{1.5}} \times a_{H}, \tag{20}\]

where \(u_{sn}\) is the user density in \(Z_{sn}\), \(h_{dn}\) is the host density in \(Z_{dn}\), \(dist(dn,sn)\) is the distance between \(Z_{dn}\) and \(Z_{sn}\), \(a_{H}\) is the activity percentage at hour \(H\) in \(Z_{sn}\). According to the activity percentage per hour of users around the world [39], the number of Internet users, the number of hosts, and other statistical data [40; 41], the real-time traffic in the LEO-SCS can be calculated using Equation (20). #### 6.1.3 Failure and Recovery

We assume that the failure time of satellites in an LEO-SCS are independently and identically distributed with an exponential distribution, with a failure rate of \(\lambda=1/T_{s}\), where \(T_{s}=5\) years is the design lifetime of satellites.