Transcript petrinet.pptx

Petri Nets
A Tutorial
Based on:
Petri Nets and Industrial
Applications: A Tutorial
Petri Net Intro.
• Often used for description of distributed
systems
• Provide a graphical notation for stepwise
processes
– Choice
– Iteration
– Concurrent execution
• Has exact mathematical definition of their
execution semantics
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Petri Nets Basics
• Bipartite directed graph
– Made of:
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Places
Transitions
Directed Arcs
Tokens
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Petri Nets Basics
• Places
– Circles
– Input place to a transition
if there is a directed arc
connecting the place to a
transition
– Output place to a
transition if there is an arc
connecting a transition to
the place
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Petri Nets Basics
• Transitions
– Bars (box)
– Represent events
• Tokens
– Dot in places
– Indicate if a condition is true or
false
• Petri Net Marking – defines
current state of modeled
system, distribution of tokens.
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Petri Nets Formal Notation
• PN = {P,T,I,O,M0} where
– P={p1,p2,…,pn} is a finite set of places
– T={12,t1,…,tn} is a finite set of transitions,
• P ᴜ T ≠ {} and P ∩ T = {}
– I: (P X T) → N is an input function that defines
directed arcs from places to transitions, N is set of
nonnegative integers
– O: (P X T) → N is an output function that defines
directed arcs from transitions to places
– M0: P→N is the initial marking
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Petri Nets Formal Notation
Formalize this Petri Net
• PN = {P,T,I,O,M0}
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Petri Nets Formal Notation
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PN = {P,T,I,O,M0}
P={p1,p2,p3}
T={t1}
I(p1,t1)=2
O(p2,t1)=2
O(p3,t1)=1
M0(p1)=2
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Petri Nets Token Flow Rules
• Enabling Rule
– t:T, t is enabled if each input place p of t has at
least the same number of tokens as weight of
directed arc
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Petri Nets Token Flow Rules
• Firing Rule
– A) Enabled transition may or may not fire depending
on additional interpretation
– B) Firing of an enabled transition t, removes an equal
number of tokens from each input place as the weight
of the transition, and puts tokens in each output
place.
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Petri Net Properties
• Reachability – a marking Mi is said to be
reachable from M0 if there exists a sequence of
transitions firings which transforms a marking M0
to Mi.
• Boundedness and safeness – Petri net is kbounded if the number of tokens in a p is always
less than or equal to k. A Petri Net is safe it k =1.
• Conservativeness – Petri Net is conservative if
number of tokens is conserved.
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Petri Net Properties
Boundedness and Conservativness
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Petri Net Analysis
• The Coverability Tree
– Enumerate all possible markings reachable from the
initial state.
– Algorithm:
• Let M0 be root of tree, tag as ‘new’
• While ‘new’ markings exist:
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Select a ‘new’ marking
If M is identical to another marking, tag as ‘old’
If no transitions are enabled in M, then M is ‘terminal’
For every transition in t enabled in M:
» Obtain the marking M’ , result from firing
» If a token value can increase indefinitely place ‘ω’
» Introduce M’ as a node, tag as ‘new’ draw arc from M to M’
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Petri Net Analysis
Boundedness and Conservativness
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Petri Net Uses
Uses:
Software Development
Industrial Engineering
Chemistry
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