# Modeling and Performance Analysis of Satellite Network Moving Target Defense System with Petri Nets
## 4 SPN-Based Modeling and Performance Analysis
### Modeling and Analysis of Traditional Satellite Network with SPN
L2 transmits data to B via the downlink, and finally completes the communication between A and B. We use SPN to study the process by which messages are generated from users on the ground, then transmitted through the satellite network and finally returned to the ground. According to the above communication process and referring to the model in [38], we construct an SPN model of the traditional satellite network communication process, as shown in Figure 5.  
Figure 4: Communication process for the satellite network. Figure 3: The composition of the satellite communication system. The meaning of each place and transition in the SPN model above is listed in Table 2. We denote \(\lambda=\{\lambda_{2},\lambda_{3},\lambda_{6},\lambda_{7},\lambda_{8}\}\) as the average transition firing time rate and \(\tau=\{\tau_{2},\tau_{3},\tau_{6},\tau_{7},\tau_{8}\}\) as the average implementation delay. The average transition firing time rate and average implementation delay corresponding to each transition can be clearly found from Table 2.  
According to the performance evaluation process in [40], the performance of the established SPN model can be analyzed by using [PERSON] theory and Markov theory. First, we can get the reachable marking set as \(M=\{\ M_{0},\ M_{1},\ M_{2},\ M_{3},\ M_{4}\}\) of the traditional satellite network SPN model, as shown in Table 3.  
Since the SPN reachable graph is isomorphic to a Continuous Time Markov Chain (CTMC), the isomorphic MC can be obtained by replacing each transition in the reachable graph with its corresponding average firing time rate as shown in Figure 6[40]. The isomorphic MC contains five markings: \(M_{0}(100001)\), \(M_{1}(010001)\), \(M_{2}(001001)\), \(M_{3}(000100)\), \(M_{4}(000010)\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Place/Transition** & **Marking/Rate** & **Meaning** \\ \hline \(P_{randy}\) & 1 & Ready \\ \(P_{init}\) & 0 & Waiting for transmission link \\ \(P_{connect}\) & 0 & Communication/Connection \\ \(P_{rec1}\) & 0 & Message arriving to satellite L1 \\ \(P_{rec2}\) & 0 & Message arriving to satellite L2 \\ \(P_{hand}\) & 1 & On-Star Bandwidth Resources \\ \(t_{request}\) & \(\lambda_{2}\) & Requesting service \\ \(t_{start}\) & \(\lambda_{3}\) & Starting service \\ \(t_{transip}\) & \(\lambda_{6}\) & Transmitting message from ground to satellite L1 via uplink \\ \(t_{trans12}\) & \(\lambda_{7}\) & Transmitting message from L1 to L2 via inter-satellite link \\ \(t_{transdown}\) & \(\lambda_{8}\) & Transmitting message from L2 to ground via downlink \\ \hline \hline \end{tabular}
\end{table} 
Table 2: List of SPN objects in Figure 5.  
Figure 5: SPN model for the traditional satellite network. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \(P_{ready}\) & \(P_{wait}\) & \(P_{connect}\) & \(P_{rec1}\) & \(P_{rec2}\) & \(P_{band}\) \\ \hline \(M_{0}\) & 1 & 0 & 0 & 0 & 0 & 1 \\ \(M_{1}\) & 0 & 1 & 0 & 0 & 0 & 1 \\ \(M_{2}\) & 0 & 0 & 1 & 0 & 0 & 1 \\ \(M_{3}\) & 0 & 0 & 0 & 1 & 0 & 0 \\ \(M_{4}\) & 0 & 0 & 0 & 0 & 1 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Reachable marking set of the SPN model. According to the definition of the transition matrix: \(Q=[q_{i,j}]\), \(i\leq n\)\(j\leq n\), there:  
\[q_{ij}= \begin{cases}\lambda_{k},\textit{the rate on the arc from }M_{i}\textit{ to}M_{j}\textit{ when }i\
eq j\\ 0,\textit{no arc from }M_{i}\textit{ to }M_{j}\textit{ when }i\
eq j\\ -\sum_{k}\lambda_{k},\textit{ }i=j\end{cases}  \tag{5}\]  
We can estimate the SPN model as follows, the transition matrix \(Q\) of the SPN model is:  
\[Q\]  
= \[ \begin{bmatrix}-\lambda_{2}&\lambda_{2}&0&0&0\\ 0&-\lambda_{3}&\lambda_{3}&0&0\\ 0&0&-\lambda_{6}&\lambda_{6}&0\\ \lambda_{2}&0&0&-\lambda_{7}&\lambda_{7}\\ \lambda_{8}&0&0&0&-\lambda_{8}\end{bmatrix} \]  
We assume that \(P=(p_{0},p_{1},p_{2},p_{3},p_{4})\), \(p_{i}\) represents the steady-state probability of the Marking \(M_{i}\). According to Markovian Smooth Distribution theory, we have:  
\[ \begin{cases}P\times Q=0\\ \sum_{0}^{4}p_{i}=1\end{cases}