# 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
Set \(sta_{m*,n^{\sharp}}=0\), \(t_{m*,n^{\sharp}}^{o}=t_{m*,n*}^{o}\), \(t_{m*,n*}^{o}=t_{m*,n*}^{o}+t_{m*,n^{\sharp}}^{t}\), and \(sta_{m*,n*}=1\) Else // Orbit \(m*\) has no in-orbit backup satellite  Set \(j=j+1\); End If\(M\times(N_{s}+N_{b})-\sum\limits_{m=1}^{M}\sum\limits_{n=1}^{N_{s}+N_{b}}sta_{m,n}=N_{g} //\) Launch ground backups  Set \(j=1\). Sample \(t_{g}\) according to \(\Phi_{g}\). For\(m=1\) to \(M\) and \(n=1\) to \(N_{s}+N_{b}\) do If\(sta_{m,n}=0\)then  Set \(sta_{m,n}=1\) and \(t_{m,n}^{o}=t_{m*,n*}^{o}+t_{g}\). \[ \begin{split}\text{max}&\alpha_{k}\\ \text{s.t.}&\mathcal{M}\bigg{\{}\sum\limits_{e_{ij}\in p_{k }}d_{i,j}\leq D_{th}\bigg{\}}\geq\alpha_{k}\end{split}  \tag{14}\]

where \(a_{k}\) is the belief degree at the \(k\)th time slice, \(p_{k}\) is any path from the source node to the destination node at the \(k\)th time slice, \(d_{i,j}\) is the delay on edge \(e_{i,j}\), and \(D_{th}\) is the delay threshold of the application. Using an uncertain programming model, the delay minimization path with the maximum belief degree can be obtained for the static uncertain network, and the uncertain distribution of the end-to-end delay on the delay minimization path can be calculated. For details, please refer to Gao [36]. ### Belief Instantaneous Availability Calculation

The belief instantaneous availability can be computed from the uncertain distribution of the end-to-end delay calculated above. For an LEO-SCS, the service requirement differs by application, and the belief instantaneous availability should be computed separately. As mentioned in Section 4, we use the end-to-end delay (an STB parameter) as the key performance parameter for LEO-SCS. The belief instantaneous availability for application \(i\) under a specific disruption and that under a normal state (without disruption) are influenced by epistemic uncertainties and can be respectively computed as

\[A_{B,i}(t)=\mathcal{M}\left\{m_{i}(t|d)>0\right\}=\mathcal{M}\left\{D_{th,i}-D _{i}(t|d)>0\right\} \tag{15}\]

and

\[A_{B,i}(t)=\mathcal{M}\left\{m_{i}(t|ns)>0\right\}=\mathcal{M}\left\{D_{th,i}-D _{i}(t|ns)>0\right\}, \tag{16}\]

where \(D_{th,i}\) is the delay threshold of application \(i\), \(D_{i}(t|d)\) and \(D_{i}(t|ns)\) are its end-to-end delay under one specific disruption and normal state at time \(t\), respectively, and \(m_{i}(t|d)\) and \(m_{i}(t|ns)\) are the delay margin of application \(i\) in one specific disruption and normal state at time \(t\), respectively. Considering the traffic volume of each application, the belief instantaneous availability of an LEO-SCS under one specific disruption and its normal state can be calculated as

\[A_{B}(t|d)=\frac{\sum\limits_{i=1}^{N_{a}}A_{B,i}(t|d)F_{i}(t)}{\sum\limits_{i =1}^{N_{a}}F_{i}(t)}\text{ and }A_{B}(t|ns)=\frac{\sum\limits_{i=1}^{N_{a}}A_{B,i}(t|ns)F_{i}(t)}{\sum\limits_ {i=1}^{N_{a}}F_{i}(t)}, \tag{17}\]

respectively, where \(N_{a}\) is the number of applications in the system, and \(F_{i}(t)\) is the traffic volume of application \(i\) at time \(t\). Combining Equations (11) and (15)-(17), the resilience of an LEO-SCS can be computed. ### Resilience Evaluation

Using the belief instantaneous availability calculation method, the resilience of an LEO-SCS can be evaluated as follows:

1. [label=0., ref=1]
2. Gather the information of applications on the system, including traffic volume \(F_{i}(t)\), source node, destination node, and delay threshold \(D_{th,i}\);
3.