# Modeling and Performance Analysis of Satellite Network Moving Target Defense System with Petri Nets
## 2 Background
### Petri Nets  
Petri Nets is a graphical and mathematical modeling tool that can be applied to many systems, and it is also a promising tool to describe and study information processing systems with concurrent, asynchronous, distributed, parallel, uncertain, or random characteristics [10]. Petri Nets has been widely focused on by researchers as soon as it is proposed, and has been widely used in a variety of fields, especially in the analysis and processing of large-scale complex systems. More and more experts and scholars use Petri Net for research. In this paper, we will use this technology to evaluate the performance. In order to make it easier to understand our follow-up work, we present here the relevant knowledge of Petri Nets. **Definition 1**. : _A Petri Net is represented by a five-tuple, \(PN=(P,T,F,W,M_{0})\), where:_  
\[\begin{array}{l}P=(P_{1},P_{2},...,P_{m})\\ T=(t_{1},t_{2},...,t_{n})\\ F\subseteq(P\times T)\cup(T\times P)\\ W:F\rightarrow\{1,2,3,...\}\\ M_{0}:P\rightarrow\{0,1,2,3,...\}\\ P\cup T=\oslash\) _and \(P\cap T=\oslash\)_  
\(P\) represents the finite set of places, which means the possible local state of the system; \(T\) represents the finite set of transitions, which describes the event that can trigger a change in the state of the system; \(F\) represents the finite set of directed arcs, which denotes the connection between the state of the system and the event, with the direction either pointing from place to transition or from transition to place. If there exists an arc that goes from a place \(P\) to a transition \(t\), then \(P\) is termed as an input place of \(t\). Conversely, if there exists an arc that goes from \(t\) to \(P\), then \(P\) is termed as an output place of \(t\). A transition may have one or more input and output places; \(W\) is called weight, which is the weight (positive integer) carried on the arc, and is 1 by default; \(M_{0}\) represents the initial marking, understood as the initial state of the whole system, is an m-dimensional vector, m denotes the total number of place \(P\), and the p th content of \(M\), denoted \(M(P)\), represents the number of tokens in the p th place; Tokens are usually contained in places, and can be transferred over different places as the transition occurs. In order to simulate the dynamic behavior of the system, the state or tokens in the Petri Net change according to the following transition (trigger) rules:  
1. If the input position \(P\) of each transition \(t\) contains at least \(W(p,t)\) tokens, where \(W(p,t)\) is the weight of the arc from \(P\) to \(t\), then the transition \(t\) is said to be enforceable;
2. The trigger of an enforceable transition \(t\) will result in the removal of \(W(p,t)\) tokens from each input place of \(t\) and the addition of \(W(t,p)\) tokens to each output place of \(t\), where \(W(t,p)\) is the weight of the arc from \(t\) to \(P\). In the Petri Nets model, places are drawn in the form of circles, transitions are drawn as bars or boxes, and arcs are represented by arcs with arrows; a token is represented by a solid black dot. They are shown in Figure 2.  
With the continuous study of Petri Nets, researchers have found that the classical PN has many defects. For example, without considering the time factor, the transition will be triggered as soon as the trigger condition is met, i.e., there is no delay, so the time-related performance metrics cannot be obtained. In addition, the scale of the model is easy to become very large, so it is difficult to reason and analyze the model. Therefore, to overcome these shortcomings and describe complex systems more conveniently, many high-level Petri Nets have emerged, including Colored Petri Nets (CPN), Time Petri Nets (TPN), Stochastic Petri Nets (SPN), and so on. Among them, SPN introduces the concept of time into traditional Petri Nets, which is well known for its capability and flexibility in modeling complex systems.