PN1

1
Petri netsrefresher

• Prof.dr.ir. Wil van der Aalst
• Eindhoven University of Technology, Faculty of
  Technology Management,
• Department of Information and Technology, P.O.Box
  513, NL-5600 MB,
• Eindhoven, The Netherlands.

2
Process modeling

• Emphasis on dynamic behavior rather than
  structuring the state space
• Transition system is too low level
• We start with the classical Petri net
• Then we extend it with
• Color
• Time
• Hierarchy

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Classical Petri net

• Simple process model
• Just three elements places, transitions and
  arcs.
• Graphical and mathematical description.
• Formal semantics and allows for analysis.
• History
• Carl Adam Petri (1962, PhD thesis)
• In sixties and seventies focus mainly on theory.
• Since eighties also focus on tools and
  applications (cf. CPN work by Kurt Jensen).
• Hidden in many diagramming techniques and
  systems.

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Elements
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Rules

• Connections are directed.
• No connections between two places or two
  transitions.
• Places may hold zero or more tokens.
• First, we consider the case of at most one arc
  between two nodes.

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Enabled

• A transition is enabled if each of its input
  places contains at least one token.

enabled
Not enabled
Not enabled
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Firing

• An enabled transition can fire (i.e., it occurs).
• When it fires it consumes a token from each input
  place and produces a token for each output place.

fired
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Play Token Game

• In the new state, make_picture is enabled. It
  will fire, etc.

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Remarks

• Firing is atomic.
• Multiple transitions may be enabled, but only one
  fires at a time, i.e., we assume interleaving
  semantics (cf. diamond rule).
• The number of tokens may vary if there are
  transitions for which the number of input places
  is not equal to the number of output places.
• The network is static.
• The state is represented by the distribution of
  tokens over places (also referred to as marking).

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Non-determinism
Two transitions are enabled but only one can fire
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Example Single traffic light
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Two traffic lights
OR
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Problem
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Solution
How to make them alternate?
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Playing the Token Game on the Internet

• Applet to build your own Petri nets and execute
  them http//is.tm.tue.nl/staff/wvdaalst/workflowc
  ourse/pn_applet/pn_applet.htm
• FLASH animations www.workflowcourse.com

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Exercise Train system (1)

• Consider a circular railroad system with 4
  (one-way) tracks (1,2,3,4) and 2 trains (A,B). No
  two trains should be at the same track at the
  same time and we do not care about the identities
  of the two trains.

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Exercise Train system (2)

• Consider a railroad system with 4 tracks
  (1,2,3,4) and 2 trains (A,B). No two trains
  should be at the same track at the same time and
  we want to distinguish the two trains.

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Exercise Train system (3)

• Consider a railroad system with 4 tracks
  (1,2,3,4) and 2 trains (A,B). No two trains
  should be at the same track at the same time.
  Moreover the next track should also be free to
  allow for a safe distance. (We do not care about
  train identities.)

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Exercise Train system (4)

• Consider a railroad system with 4 tracks
  (1,2,3,4) and 2 trains. Tracks are free, busy or
  claimed. Trains need to claim the next track
  before entering.

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WARNINGIt is not sufficient to understand the
(process) models. You have to be able to design
them yourself !
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Multiple arcs connecting two nodes

• The number of arcs between an input place and a
  transition determines the number of tokens
  required to be enabled.
• The number of arcs determines the number of
  tokens to be consumed/produced.

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Example Ball game
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Exercise Manufacturing a chair

• Model the manufacturing of a chair from its
  components 2 front legs, 2 back legs, 3 cross
  bars, 1 seat frame, and 1 seat cushion as a Petri
  net.
• Select some sensible assembly order.
• Reverse logistics?

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Exercise Burning alcohol.

• Model C2H5OH 3 O2 gt 2 CO2 3 H2O
• Assume that there are two steps first each
  molecule is disassembled into its atoms and then
  these atoms are assembled into other molecules.

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Exercise Manufacturing a car

• Model the production process shown in the
  Bill-Of-Materials.

car
subassembly2
engine
2
chair
subassembly1
4
chassis
wheel
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Formal definition

• A classical Petri net is a four-tuple (P,T,I,O)
  where
• P is a finite set of places,
• T is a finite set of transitions,
• I P x T -gt N is the input function, and
• O T x P -gt N is the output function.
• Any diagram can be mapped onto such a four tuple
  and vice versa.

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Formal definition (2)

• The state (marking) of a Petri net (P,T,I,O) is
  defined as follows
• s P-gt N, i.e., a function mapping the set of
  places onto 0,1,2, .

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Exercise Map onto (P,T,I,O) and s
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Exercise Draw diagram

• Petri net (P,T,I,O)
• P a,b,c,d
• T e,f
• I(a,e)1, I(b,e)2, I(c,e)0, I(d,e)0, I(a,f)0,
  I(b,f)0, I(c,f)1, I(d,f)0.
• O(e,a)0, O(e,b)0, O(e,c)1, O(e,d)0, O(f,a)0,
  O(f,b)2, O(f,c)0, O(f,d)3.
• State s
• s(a)1, s(b)2, s(c)0, s(d) 0.

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Enabling formalized

• Transition t is enabled in state s1 if and only
  if

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Firing formalized

• If transition t is enabled in state s1, it can
  fire and the resulting state is s2

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Mapping Petri nets onto transition systems

• A Petri net (P,T,I,O) defines the following
  transition system (S,TR)

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Reachability graph

• The reachability graph of a Petri net is the part
  of the transition system reachable from the
  initial state in graph-like notation.
• The reachability graph can be calculated as
  follows
• Let X be the set containing just the initial
  state and let Y be the empty set.
• Take an element x of X and add this to Y.
  Calculate all states reachable for x by firing
  some enabled transition. Each successor state
  that is not in Y is added to X.
• If X is empty stop, otherwise goto 2.

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Example
(3,2)
(3,1)
(3,0)
(1,3)
(1,2)
(1,1)
(1,0)
Nodes in the reachability graph can be
represented by a vector (3,2) or as 3 red 2
black. The latter is useful for sparse states
(i.e., few places are marked).
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Exercise Give the reachability graph using both
notations
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Different types of states

• Initial state Initial distribution of tokens.
• Reachable state Reachable from initial state.
• Final state (also referred to as dead states)
  No transition is enabled.
• Home state (also referred to as home marking) It
  is always possible to return (i.e., it is
  reachable from any reachable state).
• How to recognize these states in the reachability
  graph?

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Exercise Producers and consumers

• Model a process with one producer and one
  consumer, both are either busy or free and
  alternate between these two states. After every
  production cycle the producer puts a product in a
  buffer. The consumer consumes one product from
  this buffer per cycle.
• Give the reachability graph and indicate the
  final states.
• How to model 4 producers and 3 consumers
  connected through a single buffer?
• How to limit the size of the buffer to 4?

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Exercise Two switches

• Consider a room with two switches and one light.
  The light is on or off. The switches are in state
  up or down. At any time any of the switches can
  be used to turn the light on or off.
• Model this as a Petri net.
• Give the reachability graph.

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Modeling

• 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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Role of a token

• Tokens can play the following roles
• 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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Role of a place

• 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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Role of a transition

• 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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Typical network structures

• Causality
• Parallelism (AND-split - AND-join)
• Choice (XOR-split XOR-join)
• Iteration (XOR-join - XOR-split)
• Capacity constraints
• Feedback loop
• Mutual exclusion
• Alternating

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Causality
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Parallelism
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Parallelism AND-split
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Parallelism AND-join
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Choice XOR-split
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Choice XOR-join
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Iteration 1 or more times
XOR-join before XOR-split
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Iteration 0 or more times
XOR-join before XOR-split
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Capacity constraints feedback loop
AND-join before AND-split
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Capacity constraints mutual exclusion
AND-join before AND-split
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Capacity constraints alternating
AND-join before AND-split
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We have seen most patterns, e.g.
Example of mutual exclusion
How to make them alternate?
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Exercise Manufacturing a car (2)

• Model the production process shown in the
  Bill-Of-Materials with resources.
• Each assembly step requires a dedicated machine
  and an operator.
• There are two operators and one machine of each
  type.
• Hint model both the start and completion of an
  assembly step.

car
subassembly2
engine
2
chair
subassembly1
4
chassis
wheel
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Modeling problem (1) Zero testing

• Transition t should fire if place p is empty.

?
t
p
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Solution

• Only works if place is N-bounded

t
Initially there are N tokens
N input and output arcs
p
p
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Modeling problem (2) Priority

• Transition t1 has priority over t2

t1
?
t2
Hint similar to Zero testing!
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A bit of theory

• Extensions have been proposed to tackle these
  problems, e.g., inhibitor arcs.
• These extensions extend the modeling power
  (Turing completeness).
• Without such an extension not Turing complete.
• Still certain questions are difficult/expensive
  to answer or even undecidable (e.g., equivalence
  of two nets).
• Turing completeness corresponds to the ability
  to execute any computation.

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Exercise Witness statements

• As part of the process of handling insurance
  claims there is the handling of witness
  statements.
• There may be 0-10 witnesses per claim. After an
  initialization step (one per claim), each of the
  witnesses is registered, contacted, and informed
  (i.e., 0-10 per claim in parallel). Only after
  all witness statements have been processed a
  report is made (one per claim).
• Model this in terms of a Petri net.

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Exercise Dining philosophers

• 5 philosophers sharing 5 chopsticks chopsticks
  are located in-between philosophers
• A philosopher is either in state eating or
  thinking and needs two chopsticks to eat.
• Model as a Petri net.

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Preview Analysis

• Various types of analysis techniques
• Simulation (repeatedly playing the token game)
• Reachability analysis (constructing the
  reachability graph)
• Markovian analysis (reachability graph with
  transition probabilities)
• Invariants place invariants and transition
  invariants (conservation of tokens and sequences
  without effect)
• Role of models (1) insight, (2) analysis, and
  (3) specification.

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Place invariant Example
waitbeforeaftergone freeoccupied
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Transition invariant Example
entermake_pictureleaveaccident
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High level Petri netsExtending classical Petri
nets with color, time and hierarchy (informal
introduction)
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Limitations of classical Petri nets

• Inability to test for zero tokens in a place.
• Models tend to become large.
• Models cannot reflect temporal aspects
• No support for structuring large models, cf.
  top-down and bottom-up design

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Inability to test for zero tokens in a place
?
t
p
Tricks only work if p is bounded
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Models tend to become (too) large
Size linear in the number of products.
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Models tend to become (too) large (2)
Size linear in the number of tracks.
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Models cannot reflect temporal aspects
Duration of each phase is highly relevant.
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No support for structuring large models
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High-level Petri nets

• To tackle the problems identified.
• Petri nets extended with
• Color (i.e., data)
• Time
• Hierarchy
• For the time being be do not choose a concrete
  language but focus on the main concepts.
• Later we focus on a concrete language CPN.
• These concepts are supported by many variants of
  CPN including ExSpect, CPN AMI, etc.

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Running example Making punch cards
free desk employees
waiting patients
served patients
patient/ employees
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Extension with color (1)

• Tokens have a color (i.e., a data value)

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Extension with color (2)

• Places are typed (also referred to as color set).

record Brandstring RegistrationNostring
Yearint Colorstring Ownerstring
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Extension with color (3)

• The relation between production and consumption
  needs to be specified, i.e., the value of a
  produced token needs to be related to the values
  of consumed tokens.

The value of the token produced for place sum is
the sum of the values of the consumed tokens.
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Running example Tokens are colored
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Running example Places are typed
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Running example Initial state
start is enabled
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Running example Transition start fired
New value is created by simply merging the two
records.
stop is enabled
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Running example Transition stop fired
New values are created by simply spliting the
record into two parts.
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The number of tokens produced is no longer fixed
(1)
Note that the network structure is no longer a
complete specification!
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The number of tokens produced is no longer fixed
(2)
The number of tokens produced for each output
place is between 0 and 3 and the sum should be 3.
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Example

• Model as a colored Petri net.

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Product and quantity are in the value of the token
The entire stock is represented by the value of a
single token, i.e., a list of records.
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Types
StockItem
Stock
color Product string color Number int color
StockItem record prodProduct
numNumber color Stock list StockItem
StockItem
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Extension with time (1)

• Each token has a timestamp.
• The timestamp specifies the earliest time when it
  can be consumed.

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Extension with time (2)

• 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, i.e., they fire as soon as
  they can.
• Produced token may have a delay.
• The timestamp of a produced token is the firing
  time plus its delay.

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Running example Enabling time

• Transition start is enabled at time 2
  max0,min2,4,4.

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Running example Delays

• Tokens for place busy get a delay of 3
• _at_3 firing time plus 3 time units

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Running example Transition start fired

• Transition start fired a time 2.

Continue to play (timed) token game
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Exercise Final state?
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Exercise Final state?
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Extension with hierarchy

• Timed and 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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Instead of
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We can use hierarchy
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Reuse

• Reuse saves design efforts.
• Hierarchy can have any number of levels
• Transition refinement can be used for top-down
  and bottom-up design

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Exercise model three (parallel) punch card desks
in a hierarchical manner