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An Introduction to Petri Nets

An Introduction to Petri Nets. Marjan Sirjani Formal Methods Laboratory University of Tehran. Petri Net References. Tadao Murata, Petri nets: properties, analysis and applications , Proceedings of the IEEE, 77(4), 541-80, April 1989.

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An Introduction to Petri Nets

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  1. An Introduction to Petri Nets Marjan Sirjani Formal Methods Laboratory University of Tehran

  2. Petri Net References • Tadao Murata, Petri nets: properties, analysis and applications, Proceedings of the IEEE, 77(4), 541-80, April 1989. • Wil van der Aalst, Process Modeling Using Petri Nets, lecture notes.

  3. Introduction • Petri Nets: Graphical and Mathematical modeling tools • graphical tool • visual communication aid • mathematical tool • state equations, algebraic equations, etc • concurrent, asynchronous, distributed, parallel, nondeterministic and/or stochastic systems

  4. History • 1962: C.A. Petri’s dissertation (U. Darmstadt, W. Germany) • 1970: Project MAC Conf. on Concurrent Systems and Parallel Computation (MIT, USA) • 1975: Conf. on Petri Nets and related Methods (MIT, USA) • 1979: Course on General Net Theory of Processes and Systems (Hamburg, W. Germany) • 1980: First European Workshop on Applications and Theory of Petri Nets (Strasbourg, France) • 1985: First International Workshop on Timed Petri Nets (Torino, Italy)

  5. Informal Definition • The graphical presentation of a Petri net is a bipartite graph • There are two kinds of nodes • Places: usually model resources or partial state of the system • Transitions: model state transition and synchronization • Arcs are directed and always connect nodes of different types

  6. Example p2 t2 t1 p1 p4 t3 p3

  7. State • The state of the system is modeled by marking the places with tokens • A place can be marked with a finite number (possibly zero) of tokens

  8. Fire • A transition t is called enabled in a certain marking, if: • For every arc from a place p to t, there exists a distinct token in the marking • An enabled transition can fire and result in a new marking • Firing of a transition t in a marking is an atomic operation

  9. Fire (cont.) • Firing a transition results in two things: • Subtracting one token from the marking of any place p for every arc connecting p to t • Adding one token to the marking of any place p for every arc connecting t to p

  10. Nondeterminism • Execution of Petri nets is nondeterministic • Multiple transitions can be enabled at the same time, any one of which can fire • None are required to fire - they fire at will, between time 0 and infinity, or not at all

  11. Elevator

  12. One Philosopher

  13. Four Phil

  14. Four Phil (cont.)

  15. Formal Definition • A classical Petri net is a four-tuple (P,T,I,O) satisfying the following conditions: • P is a finite set of places, and T is a finite set of functions • I:P×T→N is a function which determines the input multiplicity for each input arc • O:T×P→N is a function which determines the output multiplicity for each output arc

  16. Exercise • Model a fair traffic light

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