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

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Based on: Petri Nets and Industrial Applications: A Tutorial
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

Petri Nets Basics Bipartite directed graph Made of: Places Transitions
Directed Arcs Tokens Arcs connect transitions to places and places to transitions

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

Petri Nets Basics Transitions Tokens
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.

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

Petri Nets Formal Notation
Formalize this Petri Net PN = {P,T,I,O,M0}

Petri Nets Formal Notation
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

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

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.

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 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.

Petri Net Properties Boundedness and Conservativness

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: 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’

Petri Net Analysis Boundedness and Conservativness

Petri Net Uses Uses: Software Development Industrial Engineering
Chemistry