Petri Net1 :Abstract formal model of information flow Major use: Modeling of systems of events in which it is possible for some events to occur concurrently,

Petri Net 1

Petri Net:Abstract formal model of information flow

Major 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.

Petri Net 2

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.

Petri Net 3

Petri Net as a Graph (cont.)

(Figure 1) A simple graphrepresentation of a Petri net.

Petri Net 4

Petri Net as a Graph (cont.)

(Figure 2) A markedPetri net.

Petri Net 5

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

Petri Net 6

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

Petri Net 7

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

Petri Net 8

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

Petri Net 9

Petri Net as a Graph (cont.)

(Figure 5) A simple model of three conditions and an event

Petri Net 10

(Figure 6)Modeling of a simplecomputer system

Petri Net 11

Petri Net as a Graph (cont.)

(Figure 7) Modeling of a nonprimitive event

Petri Net 12

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.

Petri Net 15

(Figure 11) Hierarchicalmodeling in Petri nets byreplacing placesor transitionsby subnets(or vice versa).

Petri Net 16

(Figure 12) A portion of aPetri netmodeling acontrol unit fora computer withmultiple registersand multiplefunctional units

Petri Net 17

(Figure 13) Representation ofan asynchronouspipelined controlunit. The blockdiagram on the left is modeled bythe Petri neton the right

Petri Net 18

Petri Net as a Graph (cont.)

Petri Net 19

(Figure 15)A Petri net model of a P/V solutionto the mutualexclusionproblem

Petri Net 20

(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

Petri Net 22

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.

Petri Net 25

(Figure 19) A Petri netwith anonfirabletransition.Transition t3

is dead inthis marking

Petri Net 26

Petri Net as a Graph (cont.)

Petri Net 27

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) ......

• Complexity

reachability problem is exponential time-hard and exponential space-hard.