Introduction to Petri Nets

1
Introduction to Petri Nets

• Hugo Andrés López
• lopez_at_dit.unitn.it

2
Plan for lectures

• 6th November 07
• Informal Introduction, Intuitions.
• Formal definition
• Properties for PNets.
• 8th November 07
• Examples on specifications.
• Analysis Techniques
• Applications.
• Petri Net Variants.

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The Scheduler
Deadlock-free

• A marked PNet is deadlock free iff for every
  marking mi there is a successor marking mi1.
• No deadlock-free marked PNet is terminating
• (but the converse does not necessarily hold)

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The Scheduler
Liveness

• Every live marked PNet is deadlock-free
• (this does not hold for nets without transitions)
  

A marked PNet is live iff there is no reachable
marking where a transition is dead (cannot become
enabled again)
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The Scheduler
Reversibility

• A marked PNet is reversible iff
• from any marking it is possible to reach the
  initial marking

1-safe non reversible PNet
Unbounded non reversible PNet
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Behavioural Properties of Marked Petri Nets

• A marked p/t-net is
• terminating if there is no infinite occurrence
  sequence
• deadlock-free if each reachable marking enables
  a transition
• live if each reachable marking enables an
  occurrence sequence containing all transitions
• bounded - if, for each place p, there is a bound
  b(p) s.t. m(p) lt b(p) for every reachable
  marking m
• 1-Safe - if b(s) 1 is a bound for each place s
• Reversible if m0 is reachable from each other
  reachable marking

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Analysis Techniques

• Petri Nets as powerful models for
• Simulation. ----(Execution of the token
  game)
• Verification.
• Techniques
• Marking Graphs
• Place/Transition Invariants
• Causal Semantics.
• ...

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Marking Graph

• A PNet can be represented as a graph connecting
  markings and transitions.
• Initial vertex m0
• Vertices reachable markings mi
• Labeled edges tuples ltmi,t,mjgt

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Representing Properties

• Instead of inspecting PNets, inspect the marking
  graph.

A marked PNet is deadlock-free if and only if
its marking graph has no vertex without successor
Deadlock-Free

A marked PNet is reversible iff its marking graph
is strongly connected
Reversibility
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Linear-Algebraic Representation of PNets

• Firing a transition can be represented as an
  algebraic operation between the marking and the
  transition.

M0 lt 4, 0, 0, 0, 1gt t2 lt-1, 1, 1, 0, -1gt
t
S
The composition of transitions can be seen as a
sum of vectors
t
S
See Van Der Aalst Tutorial
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Linear-Algebraic Representation of PNets

• Firing a transition can be represented as an
  algebraic operation between the marking and the
  transition.

Marking equation
A marking M is only reachable from M0 if
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Place Invariants

• Critical Places P2,P4.
• A place invariant will constraint the markings
  for a set of critical places.
• Place Invariant m(P2) m(P4) lt 1

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Place Invariants
for N, the sum of P2, P4 and P4 do not change
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Transition Invariants

• A transition invariant is a Parikh vector J such
  that any firing sequence s with J enabled in a
  marking M brings the system back to M.

j1 lt1,1,0,0gt (either allocation or deallocation
of a resource) j2 lt0,0,1,1gt j3 lt2,2,1,1gt
(multiple resources enable multiple firings)
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Applications

• Modelling
• Place passive element
• Transition active element
• Arc causal relation
• Token elements subject to change
• The state (space) of a process/system is modeled
  by places and tokens and state transitions are
  modeled by transitions (cf. transition systems).

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Modelling

• a physical object, for example a product, a part,
  a drug, a person
• an information object, for example a message, a
  signal, a report
• a collection of objects, for example a truck with
  products, a warehouse with parts, or an address
  file
• an indicator of a state, for example the
  indicator of the state in which a process is, or
  the state of an object
• an indicator of a condition the presence of a
  token indicates whether a certain condition is
  fulfilled.

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Modelling (II)

• a type of communication medium, like a telephone
  line, a middleman, or a communication network
• a buffer for example, a depot, a queue or a post
  bin
• a geographical location, like a place in a
  warehouse, office or hospital
• a possible state or state condition for example,
  the floor where an elevator is, or the condition
  that a specialist is available.

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Modelling (III)

• an event for example, starting an operation, the
  death of a patient, a change seasons or the
  switching of a traffic light from red to green
• a transformation of an object, like adapting a
  product, updating a database, or updating a
  document
• a transport of an object for example,
  transporting goods, or sending a file.

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PNets Variants
Extensions of Petri nets for

• zero-testing conditions.
• Compacting Models.
• Reflecting temporal aspects
• Structuring large models, cf. top-down and
  bottom-up design

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Zero-Testing

• The transition t will be fired whenever the event
  is out of tokens.

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Compacting Models

• More agents, more interactions HUGE models

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Timing aspects

• How to model event duration? How to model
  deadlines? delays?

??
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Modularization

• How to re-use a Petri-net? how to
  compact/hierarchize it?

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Color Petri Nets

• Extension for modelling different data structures
  in a single PNet.
• Coloured tokens (instances).
• Coloured Places (restrict tokens).
• Transition Includes a Coloring Function (Color
  the e according to the events of e).

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CPN (II)

• Example (assembly line)

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CPN (III)
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Modular PNets

• colored Petri nets result in more compact models.
• However, for complex systems/processes the model
  does not fit on a single page.
• Moreover, putting things at the same level does
  not reflect the structure of the process/system.
• Many hierarchy concepts are possible. In this
  course we restrict ourselves to transition
  refinement.

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Modular PNets
tl1
tl2
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Timed Petri Nets

• Each token has a timestamp.
• The timestamp specifies the earliest time when it
  can be consumed.
• The transition consumes x units of time (firing
  time) with a possible delay (_at_).

4
x_at_
2
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Timed Petri Nets

• The enabling time of a transition is the maximum
  of the tokens to be consumed.
• If there are multiple tokens in a place, the
  earliest ones are consumed first.
• A transition with the smallest firing time will
  fire first.
• Transitions are eager.
• The timestamp of a produced token is the firing
  time plus its delay.

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Timed Petri Nets (III)
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Bibliography

• J.L. Peterson. Petri Nets. Computing Surveys,
  Vol. 9 No. 3, 1977.
• Wil van Der aalst. Petri Nets Refresher.
  Eindhoven University of Technology, Faculty of
  Technology Management,
• Balbo et al. Lecture notes of the 21st. Int.
  Conference on Application and Theory of Petri
  Nets. 2000.
• The World of Petri netshttp//www.daimi.au.dk/Pe
  triNets/
• C. Ling. The Petri Net Method. http//www.utdallas
  .edu/gupta/courses/semath/petri.ppt