Petri Nets

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Petri Nets

• Concurrent and distributed modelling
• Initiated by C.A.Petri in the early sixties
  (1962)
• Generic name for class of net-based models which
  can be devided in three main layers
• Elementary Net systems (most fundamental)
• Place/Transistion systems
• High-level nets (Predicate/Transition Nets,
  Coloured Petri Nets)
• Net-based model
• net - static structure
• rules of a token game - dynamic behaviour

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Three layers
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Static the net

• Build up from places and transitions,
• Arrows between them (never from place-gtplace or
  from transition-gttransition)

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Dynamic the rules of a token game

• Start configuration (Initial Configuration)
• A transition t can fire if all input places of t
  contain tokens and all output places of t are
  empty

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Local states and neighbourhood
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Definition

• A net is triple N ( P, T, F), where
• P and T are finite sets with P ? T ?,
• F ? (P ? T) ? (T ? P),
• for every t ? T there exists p, q ? P such that
  (p, t), (t, p) ? F, and
• for every t ? T and p, q ? P, if (p, t), (t, p) ?
  F, then p ? q.

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Example

• The net N (P, T, F) with
• P p1, p2,
• T t1,
• F (p1, t1), (t1, p2)

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Producer - Consumer Problem
c1
p1
producing
emptying
p2
c2
buffer
consuming
filling
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Mutual Exclusion Problem
A(1)
B(1)
in1
in2
A(2)
B(2)
out2
out1
B(3)
A(3)
d1
d2
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A(1) / B(1)
A(1) Component A is waiting to print
A(2) Component A is printing
A(3) Component A has finished printing
Sequential Configuration Graph
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Fundamental Situations

• 1. Causality
• 2. Concurrency
• 3. Conflict
• 4. Confusion

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1. Causality
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2. Concurrency
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3. Conflict
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4. Confusion
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Application areas of Petri Nets

• Modelling of concurrent distributed processes
• Analysis of concurrent distributed processes
• Analysis of performance
• Process Algebra
• Production Systems
• Computer Supported Cooperative Work (CSCW)
• Digital Hardware Design