# 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
\begin{array}{cccccccc}-\lambda_{1}&\lambda_{1}&0&0&0&0&0\\ 0&-\lambda_{2}&\lambda_{2}&0&0&0&0\\ 0&0&-\lambda_{3}&\lambda_{3}&0&0&0\\ 0&0&0&-\lambda_{4}-\lambda_{6}&\lambda_{4}&\lambda_{6}&0\\ 0&0&0&-\lambda_{5}&\lambda_{5}&0&0\\ 0&0&0&0&0&-\lambda_{7}&\lambda_{7}\\ \lambda_{8}&0&0&0&0&0&-\lambda_{8}\end{array} \right]\]  
Similarly, let \(P=(p_{0},p_{1},p_{2},p_{3},p_{4},p_{5},p_{6})\) be the row vector corresponding to the steady-state probability of each marking, and according to Equation (6), the steady-state probability of each marking is obtained as: \begin{table}
\begin{tabular}{c c c} \hline
**Place/Transition** & **Marking/Rate** & **Meaning** \\ \hline \(P_{randy}\) & 1 & Ready \\ \(P_{syn}\) & 0 & Synchronization \\ \(P_{unit}\) & 0 & Waiting for transmission link \\ \(P_{randc}\) & 0 & Communication/Connection \\ \(P_{swap}\) & 0 & Migration / Service switching \\ \(P_{rec1}\) & 0 & Message arriving to satellite L1 \\ \(P_{rec2}\) & 0 & Message arriving to satellite L2 \\ \(P_{band}\) & 1 & On-Star Bandwidth Resources \\ \(t_{span}\) & \(\lambda_{1}\) & Requesting synchronization \\ \(t_{request}\) & \(\lambda_{2}\) & Requesting service \\ \(t_{start}\) & \(\lambda_{3}\) & Starting service \\ \(t_{startsup}\) & \(\lambda_{4}\) & Staring swap \\ \(t_{endswap}\) & \(\lambda_{5}\) & Ending swap \\ \(t_{transup}\) & \(\lambda_{6}\) & 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 \end{tabular}
\end{table} 
Table 4: List of SPN objects in Figure 8.  
Figure 9: Isomorphic Markov chain. \[ \begin{array}{l}P(M_{0})=p_{0}=\lambda_{1}^{-1}/\left(\lambda_{1}^{-1}+\lambda_{2} ^{-1}+\lambda_{3}^{-1}+\lambda_{1}^{-1}+\lambda_{5}^{-1}\lambda_{4}\lambda_{6}^ {-1}+\lambda_{7}^{-1}+\lambda_{8}^{-1}\right)\\ P(M_{1})=p_{1}=\lambda_{2}^{-1}/\left(\lambda_{1}^{-1}+\lambda_{2}^{-1}+ \lambda_{3}^{-1}+\lambda_{6}^{-1}+\lambda_{5}^{-1}\lambda_{4}\lambda_{6}^{-1}+ \lambda_{7}^{-1}+\lambda_{8}^{-1}\right)\\ P(M_{2})=p_{2}=\lambda_{3}^{-1}/\left(\lambda_{1}^{-1}+\lambda_{2}^{-1}+ \lambda_{3}^{-1}+\lambda_{6}^{-1}+\lambda_{5}^{-1}\lambda_{4}\lambda_{6}^{-1}+ \lambda_{7}^{-1}+\lambda_{8}^{-1}\right)\\ P(M_{3})=p_{3}=\lambda_{6}^{-1}/\left(\lambda_{1}^{-1}+\lambda_{2}^{-1}+ \lambda_{3}^{-1}+\lambda_{6}^{-1}+\lambda_{5}^{-1}\lambda_{4}\lambda_{6}^{-1}+ \lambda_{7}^{-1}+\lambda_{8}^{-1}\right)\\ P(M_{4})=\lambda_{5}^{-1}\lambda_{4}\lambda_{6}^{-1}/\left(\lambda_{1}^{-1}+ \lambda_{2}^{-1}+\lambda_{3}^{-1}+\lambda_{6}^{-1}+\lambda_{5}^{-1}\lambda_{4 }\lambda_{6}^{-1}+\lambda_{7}^{-1}+\lambda_{8}^{-1}\right)\\ P(M_{5})=p_{5}=\lambda_{7}^{-1}/\left(\lambda_{1}^{-1}+\lambda_{2}^{-1}+ \lambda_{3}^{-1}+\lambda_{6}^{-1}+\lambda_{5}^{-1}\lambda_{4}\lambda_{6}^{-1}+ \lambda_{7}^{-1}+\lambda_{8}^{-1}\right)\\ P(M_{6})=p_{6}=\lambda_{8}^{-1}/\left(\lambda_{1}^{-1}+\lambda_{2}^{-1}+ \lambda_{3}^{-1}+\lambda_{6}^{-1}+\lambda_{5}^{-1}\lambda_{4}\lambda_{6}^{-1}+ \lambda_{7}^{-1}+\lambda_{8}^{-1}\right)\\ \end{array} \]  
1. Token density function in each place is as follows: \[ \begin{array}{l}P(M(P_{ready})=1)=P(M_{0})=p_{0}\\ P(M(P_{syn})=1)=P(M_{1})=p_{1}\\ P(M(P_{wait})=1)=P(M_{2})=p_{2}\\ P(M(P_{connect})=1)=P(M_{3})=p_{3}\\ P(M(P_{swap})=1)=P(M_{4})=p_{4}\\ P(M(P_{rc1})=1)=P(M_{5})=p_{5}\\ P(M(P_{rc2})=1)=P(M_{6})=p_{6}\\ P(M(P_{band})=1)=P(M_{0})+P(M_{1})+p(M_{2})+p(M_{3})=p_{0}+p_{1}+p_{2}+p_{3}\\ \end{array} \]
2. The average number of tokens on a place in the steady-state can be calculated as: \[ \begin{array}{l}\bar{u}_{ready}=p_{0},\ \bar{u}_{synt}=p_{1}\\ \bar{u}_{wait}=p_{2},\ \bar{u}_{connect}=p_{3}\\ \bar{u}_{swap}=p_{4},\ \bar{u}_{rc1}=p_{5}\\ \bar{u}_{re2}=p_{6},\ \bar{u}_{band}=p_{0}+p_{1}+p_{2}+p_{3}\\ \end{array}