## 2 State of the Art
### Petri Nets Systems  
This paper corresponds to the evolution of the authors' research. Because of this, some of the next sections are strongly supported in Reference [30]. A Petri net may be defined by 5-tuples N = (P, T, 1, 0, Mo), where  
(1) P = {P1, P2,..., Pm} is a limited set of _places_;  
(2) T = {t1, t2,..., tn} is a limited set of _transitions_, P U T \(\
eq\emptyset\), and P \(\cap\) T = \(\emptyset\);  
(3) I (P, T) \(\rightarrow\) N is an _Input function_ that defines an arc directed from a Place to a Transition, where N is a set of negative integers;  
(4) (T, P) \(\rightarrow\) N is the _Output function_ that defines the arc directed from Transition to Place; and  
(5) Mo: P \(\rightarrow\) N is the _initial marking_. _Marking_ is the assignment of _tokens_ to places of the Petri net. The number and position of tokens may change during the implementation of the Petri network. The simulation example refers to Figure 1, where we have in the Petri net:  
\( \begin{array}{l}\text{P}=\{\text{p1,p2,\ldots,p7}\};\\ \text{T}=\{\text{t1,t2,\ldots,t5}\};\\ \text{I}\ (\text{t1,p1})=\text{2,I}\ (\{\text{t1,pi}\}=0\ \text{for i}=\text{2, 3, \ldots,7};\\ \text{I}\ (\text{t2,p2})=\text{1,I}\ (\text{t2,p7})=\text{1,I}\ (\text{t2,pi})=0\ \text{for i}=\text{1, 3, 4, 5, 6};\\ \text{O}\ (\text{t1,p2})=\text{1,O}\ (\text{t1,p3})=\text{2,O}\ (\text{t1,pi})=0\ \text{for i}=\text{1, 4, 5, 6, 7};\\ \text{O}\ (\text{t2,p4})=\text{1,O}\ (\text{t2,pi})=0\ \text{for i}=\text{1, 2, 3, 5, 6, 7};\\ \text{Mo}=(2\ 0\ 0\ 0\ 0\ 1).\\ \end{array} \)  
According to [PERSON] (1998), \"Petri nets were named after [PERSON], who defined a general-purpose mathematical tool for describing relations existing between conditions and events. This work was done in the years 1960-1962. Since then, Petri nets have resulted in considerable research because they can be used to model properties such as process synchronization, asynchronous events, sequential operations, concurrent operations, and conflicts, or resource sharing. These properties characterize Discrete Event Dynamic Systems (DEDS). This, and other factors, makes Petri nets a promising tool and technology for applying to various types of DEDS. Petri nets provide a powerful communication medium between the user, typically requirements engineer, and the customer as a graphical tool. Instead of using ambiguous textual descriptions or mathematical notation difficult to understand by the customer, complex requirements specifications can be represented graphically using Petri nets. This, combined with computer tools, allows interactive graphical simulation of Petri nets and puts the development engineers a powerful tool to assist in complex engineering systems' development process. The graphical representation also makes Petri nets intuitively very appealing. They are straightforward to understand and grasp--even for people who are not very familiar with Petri nets' details. This is because Petri net diagrams resemble many of the drawings that designers and engineers make while constructing and analysing a system\" [31]. [PERSON] (2003) dealt with \"the dynamic modelling of degrading and repairable complex systems as modularity allows a focus on the needs of a system reliability modelling and tailoring of the modelling formalism accordingly\" [32]. [PERSON] et al. (2007) mentioned that \"Petri Nets provide a logical, easily understood, and compelling way of predicting the reliability of a system or platform\" [33]. [PERSON] (2012) mentions that \"Petri Nets tool is applied to represent the asynchronous and concurrent processing of the order instead of the fault tree analysis\" [34]. [PERSON] and [PERSON] (2016) mentioned that \"the study has aimed to develop a model using Petri Nets to determine the feasibility of adopting this technique to model the maintenance processes efficiently\" [35]. [PERSON] et al.