2Properties Reachability Boundedness Safeness “Can we reach one particular state from another?”Boundedness“Will a storage place overflow?”the number of tokens in a place is boundedSafeness- the number of tokens in a place never exceeds oneDeadlock-free- none of markings in R(PN, M0) is a deadlockLiveness“Will the system die in a particular state?”
3Properties Reachability Question: “Can we reach one particular state from another?”To answer: find R(PN, M0)
4Properties Reachability : Vending machine example p1 t8 t1 t2 p2 t3 p3 Initial marking:M0M0 = (1,0,0,0,0)M1 = (0,1,0,0,0)M2 = (0,0,1,0,0)M3 = (0,0,0,1,0)M4 = (0,0,0,0,1)M0M1M2M3M4t3t1t5t8t2t6“M2 is reachable from M1 and M4 is reachable from M0.”In fact, all markings are reachable from every marking.
5Properties Boundedness The PN for vending machine is 1-bounded. “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 k for any marking reachable from M0.The PN for vending machine is 1-bounded.A 1-bounded Petri net is also safe.
6Properties Safeness The number of tokens in a place never exceeds one i.e., a 1-bounded PN is a safe PNe.g., the PN for vending machine is safe
7PropertiesLiveness: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
8Properties Liveness: An Example A bounded but non-live Petri net t1 p3
9Properties Liveness: Another Example An unbounded but live Petri net
10PropertiesLiveness:a PN is live if, no matter what marking has been reached, it is possible to fire any transition with an appropriate firing sequenceequivalent to deadlock-free
11Properties Other properties: Reversibility Coverability Persistency Synchronic distanceFairness….
12Analysis Methods Enumeration Linear Algebraic Technique Simulation Reachability TreeCoverability TreeLinear Algebraic TechniqueState Matrix EquationInvariant Analysis: P-Invariant and T-invariantSimulation