# 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 Active Defense-Based Satellite Network with SPN
\] The average number of tokens contained in the set of all places from the request made by the client to the completion of the service is calculated as: \[\overline{N}=\bar{u}_{wait}+\bar{u}_{swap}+\bar{u}_{connect}+\bar{u}_{rec1}+ \bar{u}_{rec2}+\bar{u}_{band}=1+p_{1}+p_{2}+p_{3}\]
3. The utilization rate of \(t_{request}\) is: \[U(t_{request})=P(M_{1})=p_{1}\]
4. The rate from \(t_{request}\) to \(P_{wait}\) is: \[R(t_{request},P_{wait})=W(t_{request},P_{wait})\times U(t_{request})\times\lambda_ {2}=\lambda_{2}p_{1}\]  
Therefore, the average latency of the satellite network security protection system based on the Moving Target Defense technique is:  
\[T=\overline{N}/R(t_{request},P_{wait})=\lambda_{1}^{-1}+\lambda_{2}^{-1}+2 \lambda_{3}^{-1}+2\lambda_{6}^{-1}+\lambda_{5}^{-1}\lambda_{4}\lambda_{6}^{-1}+ \lambda_{7}^{-1}+\lambda_{8}^{-1}\] (12) Average throughput is: \[O=P(M_{6})\times\lambda_{8}=\lambda_{8}p_{6}=\frac{1}{\lambda_{1}^{-1}+ \lambda_{2}^{-1}+\lambda_{3}^{-1}+\lambda_{6}^{-1}+\lambda_{5}^{-1}\lambda_{4} \lambda_{6}^{-1}+\lambda_{7}^{-1}+\lambda_{8}^{-1}}\] (13) Utilization of on-board bandwidth resources is: \[U=P(M(P_{band})=0)=\frac{\lambda_{5}^{-1}\lambda_{4}\lambda_{6}^{-1}+ \lambda_{7}^{-1}+\lambda_{8}^{-1}}{\lambda_{1}^{-1}+\lambda_{2}^{-1}+\lambda_{ 3}^{-1}+\lambda_{6}^{-1}+\lambda_{5}^{-1}\lambda_{4}\lambda_{6}^{-1}+\lambda_{ 7}^{-1}+\lambda_{8}^{-1}}\] (14) \[\lambda_{2}^{-1},\lambda_{3}^{-1},\lambda_{6}^{-1},\lambda_{7}^{-1}, \lambda_{8}^{-1}\] have the same meaning as described in Section 4.1, \(\lambda_{1}^{-1}\) denotes the synchronous authentication delay i.e., \(\tau_{1}\), \(\lambda_{4}^{-1}\) denotes the single service hopping time slot i.e., \(\tau_{4}\), and \(\lambda_{5}^{-1}\) denotes the data migration delay i.e., \(\tau_{5}\). From the theoretical reasoning, we can see that the network time delay, throughput, and bandwidth utilization of the satellite network moving target defense system are not only related to the link transmission duration, request delay, and service delay, but also affected by synchronization delay, hopping rate, and data migration delay introduced by MTD strategy. Compared with traditional satellite networks, the MTD-based satellite network has the following relationship in terms of average delay, average throughput, and bandwidth utilization:  
\[ \begin{split}\lambda_{1}^{-1}+\lambda_{2}^{-1}+2\lambda_{3}^{-1}+2 \lambda_{6}^{-1}+\lambda_{5}^{-1}\lambda_{4}\lambda_{6}^{-1}+\lambda_{7}^{-1} +\lambda_{8}^{-1}\geq\lambda_{2}^{-1}+2\lambda_{3}^{-1}+2\lambda_{6}^{-1}+ \lambda_{7}^{-1}+\lambda_{8}^{-1}\\ \overline{\lambda_{1}^{-1}+\lambda_{2}^{-1}+2\lambda_{3}^{-1}+2 \lambda_{6}^{-1}+\lambda_{5}^{-1}\lambda_{4}\lambda_{6}^{-1}+\lambda_{7}^{-1}+ \lambda_{8}^{-1}}\leq\frac{1}{\lambda_{2}^{-1}+2\lambda_{3}^{-1}+2\lambda_{6} ^{-1}+\lambda_{7}^{-1}+\lambda_{8}^{-1}}\\ \frac{\lambda_{5}^{-1}\lambda_{4}\lambda_{6}^{-1}+\lambda_{7}^{-1} +\lambda_{8}^{-1}}{\lambda_{1}^{-1}+\lambda_{2}^{-1}+2\lambda_{3}^{-1}+2 \lambda_{6}^{-1}+\lambda_{5}^{-1}\lambda_{4}\lambda_{6}^{-1}+\lambda_{7}^{-1}+ \lambda_{8}^{-1}}\leq\frac{\lambda_{7}^{-1}+\lambda_{8}^{-1}}{\lambda_{2}^{-1} +2\lambda_{3}^{-1}+2\lambda_{6}^{-1}+\lambda_{7}^{-1}+\lambda_{8}^{-1}}\end{split}  \tag{15}\]  
Consequently, the synchronous authentication technology, service hopping, and data migration introduced by MTD will reduce the overall performance of the satellite network, specifically, increase the average network latency, reduce the average throughput, and increase the consumption of bandwidth resources on the satellite network.