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Petri Nets A Tutorial Based on: Petri Nets and Industrial Applications: A Tutorial

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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 2

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Petri Nets Basics Bipartite directed graph – Made of: Places Transitions Directed Arcs Tokens 3

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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 4

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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. 5

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Petri Nets Formal Notation PN = {P,T,I,O,M 0 } where – P={p 1,p 2,…,p n } 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 – M 0 : P→N is the initial marking 6

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Petri Nets Formal Notation Formalize this Petri Net PN = {P,T,I,O,M 0 } 7

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Petri Nets Formal Notation PN = {P,T,I,O,M 0 } P={p1,p2,p3} T={t1} I(p1,t1)=2 O(p2,t1)=2 O(p3,t1)=1 M 0 (p1)=2 8

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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 9

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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. 10

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Petri Net Properties Reachability – a marking M i is said to be reachable from M 0 if there exists a sequence of transitions firings which transforms a marking M 0 to M i. Boundedness and safeness – Petri net is k- bounded 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. 11

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Petri Net Properties 12 Boundedness and Conservativness

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Petri Net Analysis The Coverability Tree – Enumerate all possible markings reachable from the initial state. – Algorithm: Let M 0 be root of tree, tag as ‘new’ While ‘new’ markings exist: – 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’ 13

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Petri Net Analysis 14 Boundedness and Conservativness

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Petri Net Uses 15 Uses: Software Development Industrial Engineering Chemistry

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