# 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
 \tag{6}\]  
By solving the linear equation system from (6), the steady-state probability of each marking can be obtained as follows:  
\[ \begin{split} P(M_{0})&=p_{0}=\lambda_{2}^{-1}/(\lambda_{2}^{-1 }+\lambda_{3}^{-1}+\lambda_{6}^{-1}+\lambda_{7}^{-1}+\lambda_{8}^{-1})\\ P(M_{1})&=p_{1}=\lambda_{3}^{-1}/(\lambda_{2}^{-1}+\lambda_{3}^{-1}+ \lambda_{6}^{-1}+\lambda_{7}^{-1}+\lambda_{8}^{-1})\\ P(M_{2})&=p_{2}=\lambda_{6}^{-1}/(\lambda_{2}^{-1}+\lambda_{3}^{-1}+ \lambda_{6}^{-1}+\lambda_{7}^{-1}+\lambda_{8}^{-1})\\ P(M_{3})&=p_{3}=\lambda_{7}^{-1}/(\lambda_{2}^{-1}+\lambda_{3}^{-1}+ \lambda_{6}^{-1}+\lambda_{7}^{-1}+\lambda_{8}^{-1})\\ P(M_{4})&=p_{4}=\lambda_{8}^{-1}/(\lambda_{2}^{-1}+\lambda_{3}^{-1}+ \lambda_{6}^{-1}+\lambda_{7}^{-1}+\lambda_{8}^{-1})\end{split} \]  
Furthermore, by applying the steady-state probabilities of markings and Formulas (1)-(4) in Section 2.2, the basic performance metrics such as token density function in each place, average number of tokens on a place, utilization rate of the transition, token velocity of the transition, etc. can be derived easily. 1. Token density function in each place is as follows: \[ \begin{split} P(M(P_{\textit{ready}})=1)&=P(M_{0}) =p_{0}\\ P(M(P_{\textit{wait}})=1)&=P(M_{1})=p_{1}\\ P(M(P_{\textit{connect}})=1)&=P(M_{2})=p_{2}\\ P(M(P_{\textit{rec1}})=1)&=P(M_{3})=p_{3}\\ P(M(P_{\textit{rec2}})=1)&=P(M_{4})=p_{4}\\ P(M(P_{\textit{bund}})=1)&=P(M_{0})+P(M_{1})+p(M_{2})=p_{0}+p_{1 }+p_{2}\end{split} \]
2. The average number of tokens on a place in the steady-state can be calculated as:  
Figure 6: Isomorphic Markov chain. \[\bar{u}_{ready} =1\times P(M(P_{ready})=1)=p_{0}\] \[\bar{u}_{wait} =1\times P(M(P_{wait})=1)=p_{1}\] \[\bar{u}_{connect} =1\times P(M(P_{connect})=1)=p_{2}\] \[\bar{u}_{rec1} =1\times P(M(P_{rec1})=1)=p_{3}\] \[\bar{u}_{rec2} =1\times P(M(P_{rec2})=1)=p_{4}\] \[\bar{u}_{band} =1\times P(M(P_{band})=1)=p_{0}+p_{1}+p_{2}\] The average number of tokens contained in the set of all places from the time the service is requested by the user to the time it is completed is calculated as: \[\overline{N}=\bar{u}_{wait}+\bar{u}_{connect}+\bar{u}_{rec1}+\bar{u}_{rec2}+ \bar{u}_{band}=1+p_{1}+p_{2}\]
3. The utilization rate of \(t_{request}\) is: \[U(t_{request})=P(M_{0})=p_{0}\]
4. The rate from \(t_{request}\) to \(P_{wait}\) is: \[R(t_{request},P_{wait})=W(t_{request},P_{wait})\times U(t_{request})\times \lambda_{2}=\lambda_{2}p_{0}\]  
On the basis of all the performance parameters mentioned above, we can further calculate the average time delay by applying [PERSON]'s theorem and principle of balance [41], [PERSON]'s theorem is formulated as:  
\[\bar{u}_{ready} =1\times P(M(P_{ready})=1)=p_{0}\]0  
\(\overline{N}\) represents the average queue length, \(\lambda\) denotes the average arrival rate, and \(T\) means average time delay of the queue. Consequently, the average time delay of the network is:  
\[\bar{u}_{ready} =1\times P(M(P_{ready})=1)=p_{0}\]1  
The average throughput is defined as the average number of tasks completed by the system per unit of time, and it is an important indicator characterizing the performance of the system. The formula for calculating the throughput of each transition \(t\) in the steady-state is:  
\[\bar{u}_{ready} =1\times P(M(P_{ready})=1)=p_{0}\]2  
\(H\) is the set of markings that enable the implementation of transition \(t\), and \(\lambda_{t}\) is the average firing time rate of the transition \(t\) under marking \(M\). The SPN model of satellite network communication completes one data communication service after the implementation of transition \(t_{transdown}\).