# 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
Table 2 shows two types of satellite backup strategies. (2) Evolution algorithm  
To model the uncertain satellite network evolution model, satellite failures and recoveries are added to the dynamic, uncertain satellite network model. The evolution algorithm is summarized as Algorithm 1, whose symbols are explained in Table 3.  
Using Algorithm 1, \(G(V(t),E(t),D(t))\) can be obtained by: (a) deleting the node and all its adjacent edges when it exits; and (b) adding the node and corresponding edges back when it enters. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Backup Strategy** & **Strategy Details** & **Transfer Time** \\ \hline \multirow{6}{*}{In-orbit} & Each orbital plane has a backup & \multirow{2}{*}{The in-orbit backup transfer time} \\  & orbit, where one or more backup & \\  & satellites are deployed. & \\  & An operating satellite that fails & \\  & will de-orbit and a satellite on its & \\  & backup orbit will be transferred to & \\  & the site of failure. & \\ \hline \multirow{6}{*}{Ground} & When in-orbit backups are & \multirow{2}{*}{The ground backup transfer time} \\  & consumed to a certain extent, & \\  & a carrier rocket will convey & \\  & ground backups to supplement & \\  & them. One rocket launch can carry & \\  & one or more satellites. & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Two types of satellite backup strategies. \begin{table}
\begin{tabular}{l l} \hline \hline
**Symbol** & **Meaning** \\ \hline \(G(V,E(t),D(t))\) & Dynamic uncertain network \\ \(G(V(t),E(t),D(t))\) & Uncertain satellite network evolution model \\ \(T\) & Simulation time \\ \(\Phi_{f}\) & Satellite failure time distribution function \\ \(\Phi_{g}\) & Ground backup transfer time distribution function \\ \(M\) & Number of satellite orbital planes \\ \(N_{s}\) & Number of satellites per orbit \\ \(N_{b}\) & Number of in-orbit backup satellites per orbit \\ \(N_{g}\) & Number of satellites carried by one rocket \\  & Satellite number. If \(n\leq N_{s}\), it is the \(n\)th satellite in the \(m\)th orbit. \\ \((m,n)\) & If \(N_{s}<n\leq N_{s}+N_{p}\), it is the \((n-N_{s})\)th backup satellite in the \(m\)th orbit. \\ \(T_{f}\) & Satellite failure time \\ \(T_{r}\) & Satellite recovery time \\ \(t_{m,n}^{o}\) & Operation time of satellite \((m,n)\) \\ \(t_{m,n}^{f}\) & Failure time of satellite \((m,n)\) \\ \(t_{m,n}^{i}\) & Transfer time of backup satellite \((m,n)\) \\ \(t_{g}\) & Ground backup transfer time \\ \(sta_{m,n}\) & Status of satellite \((m,n)\), where \(sta_{m,n}=0\) and \(sta_{m,n}=1\) indicate failure and normal states, respectively, at time \(t_{m,n}^{o}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Symbols in Algorithm 1. ``` Input  LEO-SCS parameters: \(G(V,E(t),D(t))\), \(M\), \(N_{s}\), \(N_{b}\), \(N_{g}\), \(T\);  Failure and recovery parameters: \(\Phi_{f}\), \(\Phi_{g}\) Output  LEO-SCS parameters: \(G(V(t),E(t),D(t))\) Initialization: Let \(t_{m,n}^{o}=0\) and \(sta_{m,n}=1\) for \(m=1,\cdots M,n=1,\cdots,N_{s}+N_{b}\). Set \(j=1\). While\(t_{m,n}^{o}<T\) (\(m=1,\cdots M,n=1,\cdots N_{s}\))  Rank \(t_{m,n}^{o}\) from small to large for \(m=1,\cdots M,n=1,\cdots N_{s}\), and denote the \(j\)th one as satellite \((m*,n*)\). If\(sta_{m*,n*}=1\)then // satellite \((m*,n*)\) is in the normal state  Sample \(t_{m*,n*}^{f}\) according to \(\Phi_{f}\). Set \(t_{m*,n*}^{o}=t_{m*,n*}^{f}=t_{m*,n*}^{o}\) and \(sta_{m*,n*}=0\). Else // satellite \((m*,n*)\) fails  If\(\exists st_{m*,n}=1\) and \(t_{m*,n}^{o}\leq t_{m*,n*}^{o}\) for \(n=N_{s}+1,\cdots,N_{s}+N_{b}\)then // Orbit \(m*\)  has residual in-orbit backups  Calculate \(t_{m*,n}^{t}\) for \(n=N_{s}+1,\cdots,N_{s}+N_{b}\) if \(sta_{m*,n}=1\), and denote the one with the minimal \(t_{m*,n}^{t}\) as backup transfer satellite \((m*,n^{\sharp})\).