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.
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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.
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3. Petri Net as a Graph (cont.)
(Figure 1)
A simple graph
representation
of a Petri net.
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4. Petri Net as a Graph (cont.)
(Figure 2)
A marked
Petri net.
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5. Petri Net as a Graph (cont.)
(Figure 3)
The marking
resulting from
firing transition
t2 in Figure 2.
Note that the
token in p1 was
removed and
tokens were
added to p2
and p3
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6. Petri Net as a Graph (cont.)
(Figure 4)
Markings
resulting from
the firing of
different
transitions in the
net of Figure 3.
(a) Result of
firing transition t1
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7. Petri Net as a Graph (cont.)
(Figure 4)
Markings
resulting from
the firing of
different
transitions in the
net of Figure 3.
(b) Result of
firing transition t3
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8. Petri Net as a Graph (cont.)
(Figure 4)
Markings
resulting from
the firing of
different
transitions in the
net of Figure 3.
(c) Result of
firing transition t5
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9. Petri Net as a Graph (cont.)
(Figure 5) A simple model of three conditions and an event
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10. (Figure 6)
Modeling of
a simple
computer system
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11. Petri Net as a Graph (cont.)
(Figure 7) Modeling of a nonprimitive event
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12. Petri Net as a Graph (cont.)
(Figure 8)
Modeling of
“simultaneous”
which may
occur in either
order
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13. Petri Net as a Graph (cont.)
(Figure 9)
Illustration of
conflicting
transitions.
Transitions tj
and tk conflict
since the
firing of one
will disable
the other
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14. Petri Net as a Graph (cont.)
(Figure 10) An
uninterpreted
Petri net.
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15. (Figure 11) Hierarchical modeling in Petri nets by replacing places
or transitions
by subnets
(or vice versa).
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16. (Figure 12) A portion of a Petri net modeling a control unit for
a computer with
multiple registers
and multiple
functional units
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17. (Figure 13) Representation of an asynchronous pipelined control
unit. The block
diagram on the
left is modeled by
the Petri net
on the right
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18. Petri Net as a Graph (cont.)30

19. (Figure 15) A Petri net model of a P/V solution to the mutual
exclusion
problem
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20. (Figure 16) L: S0 Do while P0 if P2 then S1 else S2 endif parbegin
Example of a Petri net
used to represent the
flow of control in
programs containing
certain 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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21. (Figure 17)
A Petri net
model for
protocol 3
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22. Other properties for analysis
Boundeness
Safe net (bound = 1)
K-bounded net
Conservation ==> conservative net
Live transition
Dead transition
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23. State of a Petri net State - defined by its marking, m
State space - set of all markings: (m0, m1, m2, ...)
Change in state - caused by firing a transition, defined by partial Fn, d
(example) m1 = d (m0 , tj)
Note: marking --
For a marking m , m(Pi) = mi
A marked Petri net: m = (P, T, I, O, m)
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24. m0 = (1, 0, 1, 0, 2)
d(m0, t3) = (1, 0, 0, 1, 2) = m1
d(m1, t4) = (1, 1, 1, 0, 2) = m2 etc.
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25. (Figure 19) A Petri net with a nonfirable transition. Transition t3
is dead in
this marking
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26. Petri Net as a Graph (cont.)30

27. Petri Net as a Graph (cont.)
(1, 0, 1, 0)
(1, 0, 0, 1)
(1, w, 1, 0)
(1, w, 0, 0)
(1, w, 0, 1) t3 t2 t1
(Figure 21)
The reachability
tree of the
Petri net of
Figure 19
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28. 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.
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