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Petri Nets Section 2

Petri Nets Section 2. Roohollah Abdipur. Properties. Reachability “Can we reach one particular state from another?” Boundedness “Will a storage place overflow?” the number of tokens in a place is bounded Safeness - the number of tokens in a place never exceeds one Deadlock-free

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Petri Nets Section 2

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  1. Petri NetsSection 2 RoohollahAbdipur

  2. Properties • Reachability • “Can we reach one particular state from another?” • Boundedness • “Will a storage place overflow?” • the number of tokens in a place is bounded • Safeness - the number of tokens in a place never exceeds one • Deadlock-free - none of markings in R(PN, M0) is a deadlock • Liveness • “Will the system die in a particular state?”

  3. Properties • Reachability • Question: “Can we reach one particular state from another?” • To answer: find R(PN, M0)

  4. t8 p4 t4 M0 = (1,0,0,0,0) p2 M1 = (0,1,0,0,0) t1 M2 = (0,0,1,0,0) p1 M3 = (0,0,0,1,0) t3 t7 t5 M4 = (0,0,0,0,1) p3 t6 p5 Initial marking:M0 t2 t9 t1 t3 t5 t8 t2 t6 M0 M1 M2 M3 M0 M2 M4 Properties • Reachability : Vending machine example • “M2 is reachable from M1 and M4 is reachable from M0.” • In fact, all markings are reachable from every marking.

  5. Properties • Boundedness • “Will a storage place overflow?” • A Petri net is said to be k-bounded or simply bounded if the number of tokens in each place does not exceed a finite number kfor any marking reachable from M0. • The PN for vending machine is 1-bounded. • A 1-bounded Petri net is also safe.

  6. Properties • Safeness • The number of tokens in a place never exceeds one • i.e., a 1-bounded PN is a safe PN • e.g., the PN for vending machine is safe

  7. Properties • Liveness: • A Petri net with initial marking M0 is live if, no matter what marking has been reached from M0, it is possible to ultimately fire any transition by progressing through some further firing sequence. • A live Petri net guarantees deadlock-free operation, no matter what firing sequence is chosen. • E.g., the vending machine is live and the producer-consumer system is also live. • A transition is dead if it can never be fired in any firing sequence

  8. Properties • Liveness: An Example t1 p3 t3 t4 p2 p1 p4 t2 M0 = (1,0,0,1) M1 = (0,1,0,1) M2 = (0,0,1,0) M3 = (0,0,0,1) A bounded but non-live Petri net

  9. M0 = (1, 0, 0, 0, 0) M1 = (0, 1, 1, 0, 0) M2 = (0, 0, 0, 1, 1) M3 = (1, 1, 0, 0, 0) M4 = (0, 2, 1, 0, 0) Properties • Liveness: Another Example p1 t1 p2 p3 t2 t3 p4 p5 t4 An unbounded but live Petri net

  10. Properties • Liveness: • a PN is live if, no matter what marking has been reached, it is possible to fire any transition with an appropriate firing sequence • equivalent to deadlock-free

  11. Properties • Other properties: • Reversibility • Coverability • Persistency • Synchronic distance • Fairness • ….

  12. Analysis Methods • Enumeration • Reachability Tree • Coverability Tree • Linear Algebraic Technique • State Matrix Equation • Invariant Analysis: P-Invariant and T-invariant • Simulation

  13. Analysis Methods • Reachability Tree: An exmple

  14. Analysis Methods • Reachability Tree: Another exmple

  15. Analysis Methods • Linear Algebraic Technique

  16. Analysis Methods • Linear Algebraic Technique

  17. Other Types of Petri Nets • Coloured Petri nets • Timed Petri nets • Object-Oriented Petri nets

  18. THE END

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