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Petri Net 1
Petri Net:Abstract formal model of information flowMajor use:
Modeling of systems of events in which it is possible for some events to occur concurrently, but there are constraints on the occurrences, precedence, or frequency of these occurrences.
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Petri Net as a Graph:Models static properties of a system• Graph contains 2 types of nodes
– Circles (Places)– Bars (Transitions)
• Petri net has dynamic properties that result from its execution– Markers (Tokens)– Tokens are moved by the firing of transitions of
the net.
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Petri Net as a Graph (cont.)
(Figure 1) A simple graphrepresentation of a Petri net.
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Petri Net as a Graph (cont.)
(Figure 2) A markedPetri net.
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Petri Net as a Graph (cont.)(Figure 3) The marking resulting fromfiring transitiont2 in Figure 2.Note that the token in p1 wasremoved andtokens wereadded to p2 and p3
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Petri Net as a Graph (cont.)(Figure 4) Markings resulting fromthe firing of different transitions in the net of Figure 3.
(a) Result offiring transition t1
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Petri Net as a Graph (cont.)(Figure 4) Markings resulting fromthe firing of different transitions in the net of Figure 3.
(b) Result offiring transition t3
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Petri Net as a Graph (cont.)(Figure 4) Markings resulting fromthe firing of different transitions in the net of Figure 3.
(c) Result offiring transition t5
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Petri Net as a Graph (cont.)
(Figure 5) A simple model of three conditions and an event
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(Figure 6)Modeling of a simplecomputer system
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Petri Net as a Graph (cont.)
(Figure 7) Modeling of a nonprimitive event
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Petri Net as a Graph (cont.)
(Figure 8) Modeling of “simultaneous”which mayoccur in eitherorder
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Petri Net as a Graph (cont.)(Figure 9) Illustration ofconflictingtransitions.Transitions tj
and tk conflictsince thefiring of onewill disablethe other
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Petri Net as a Graph (cont.)(Figure 10) An uninterpretedPetri net.
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(Figure 11) Hierarchicalmodeling in Petri nets byreplacing placesor transitionsby subnets(or vice versa).
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(Figure 12) A portion of aPetri netmodeling acontrol unit fora computer withmultiple registersand multiplefunctional units
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(Figure 13) Representation ofan asynchronouspipelined controlunit. The blockdiagram on the left is modeled bythe Petri neton the right
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Petri Net as a Graph (cont.)
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(Figure 15)A Petri net model of a P/V solutionto the mutualexclusionproblem
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(Figure 16)Example of a Petri netused to represent theflow of control in programs containingcertain kind of constructs
L: S0
Do while P0
if P2 then S1
else S2
endif parbegin S3,S4,S5, parend enddo goto L
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(Figure 17)A Petri netmodel forprotocol 3
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Other properties for analysis• Boundeness
– Safe net (bound = 1)– K-bounded net
• Conservation ==> conservative net• Live transition• Dead transition
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State of a Petri net• State - defined by its marking, • State space - set of all markings: (, , , ...)• Change in state - caused by firing a transition,
defined by partial Fn, (example) = ( , tj)
• Note: marking --For a marking , (Pi) = i
A marked Petri net: m = (P, T, I, O, )
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= (1, 0, 1, 0, 2)
(, t3)= (1, 0, 0, 1, 2) = (, t4)= (1, 1, 1, 0, 2) = etc.
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(Figure 19) A Petri netwith anonfirabletransition.Transition t3
is dead inthis marking
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Petri Net as a Graph (cont.)
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Petri Net as a Graph (cont.)
(Figure 21)The reachability tree of thePetri net ofFigure 19
(1, 0, 1, 0)
(1, 0, 0, 1)
(1, , 1, 0)
(1, , 0, 0) (1, , 0, 1)
(1, , 1, 0)
t3
t2
t1 t3
t2
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Unsolvable Problems• Subset problem - given 2 marked Petri nets, is the
reachability of one net a subset of the reachability of the other net undecidable (Hack) ......
• Complexityreachability problem is exponential time-hard and exponential space-hard.
