Aims: Classic Petri nets also known as place transition nets provide many interesting and useful features for system modeling. They are however limited by the place types that are used. A novel approach is presented in this work. A matrix vector transition net model is created and is used to model complex system behavior. This solution extends the modeling power of normal Petri nets.
Proposed Solution: A traditional Petri net is modified to create a matrix vector transition net (MVTN). The idea is to combine the ideas from normal Petri net semantics with a matrix vector approach.
Implementing the Matrix Vector Transition Net: Ordinary Petri net places are replaced with matrices or vectors. The input and output arcs must have a specific function matrix that determines firing. Firing and behavior remain conceptually and functionally similar to that of a Petri net. It is possible to interchange row and column vectors. The behavior of matrix transition nets must elicit similar behavior to that of a place transition net. Instead of normal tokens, matrix elements are used. The matrix vector type of structure increases the modeling power, abstraction capacity and the complexity of the net.
Case Study: To illustrate this work a toy case of an abstract network structure containing processing elements is used to illustrate the use of the matrix vector transition net structure.
Results and Findings: The behavior of matrix transition nets is shown to be similar in principle to that of a place transition net. However instead of tokens, matrix elements are used. It is possible to construct a symbolic marking graph or reach ability graph for the system This type of structure definitely increases the modeling power, abstraction capacity and the complexity of the net. The matrix transition net could be useful for certain types of communication system problems and complex system interfacing. Several other uses can be found for this approach in both computing and modeling.
Keywords: Complex systems; matrices and vectors; matrix vector transition net; modeling; Petri nets; colored Petri nets; symbolic modeling.
Conclusion
The main properties of the MVT nets along with the current availability of Petri net theory, give the possibility for constructing more complex models of modern information and computer systems or structures. Because of the strong influence, matrix theory has on normal Petri net representation and analysis, similar concepts have been considered for the MVTN net. The ability of matrices for expressing system composition and behavior, should not be overlooked. The use of matrices should follow intuitively, because matrices represent complexity using compaction. It might seem that all these concepts can be contained in CPNs however intrinsically these are not focused on matrix representation that would undoubtedly simplify certain expressions of abstract behavior. The models developed can be enhanced and formalized. Extensions of the models can be developed and more complex combinations are possible. The visual expressivity, decomposability, solvability has to be properly understood when constructing representational structures.
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