Modeling based on Petri-nets.

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Modeling based on Petri-nets.
Lecture 8
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High-level Petri nets

• The classical Petri net was invented by Carl Adam
  Petri in 1962.
• A lot of research has been conducted (gt10.000
  publications).
• Until 1985 it was mainly used by theoreticians.
• Since the 80-ties the practical use is increasing
  because of the introduction of high-level Petri
  nets and the availability of many tools.
• High-level Petri nets are Petri nets extended
  with
• color (for the modeling of attributes)
• time (for performance analysis)
• hierarchy (for the structuring of models, DFD's)

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The classical Petri net model

• A Petri net is a network composed of places ( )
  and transitions ( ).

t2
p2
t1
p1
p4
t3
p3
Connections are directed and between a place and
a transition. Tokens ( ) are the dynamic
objects. The state of a Petri net is determined
by the distribution of tokens over the places.
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p1
p4
t1
p2
p3

• Transition t1 has three input places (p1, p2 and
  p3) and two output places (p3 and p4).
• Place p3 is both an input and an output place of
  t1.

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Enabling condition

• Transitions are the active components and places
  and tokens are passive.
• A transition is enabled if each of the input
  places contains tokens.

Transition t1 is not enabled, transition t2 is
enabled.
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Firing

• An enabled transition may fire.
• Firing corresponds to consuming tokens from the
  input places and producing tokens for the output
  places.

Firing is atomic.
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Example
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Non-determinism
t1
t2

• Two transitions fight for the same token
  conflict.
• Even if there are two tokens, there is still a
  conflict.

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Modeling

• States of a process are modeled by tokens in
  places and state transitions leading from one
  state to another are modeled by transitions.
• Tokens represent objects (humans, goods,
  machines), information, conditions or states of
  objects.
• Places represent buffers, channels, geographical
  locations, conditions or states.
• Transitions represent events, transformations or
  transportations.

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Example traffic light
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Two traffic lights
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Two safe traffic lights
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Two safe and fair traffic lights
red1
red2
safe2
yr1
yr2
yellow1
yellow2
rg1
rg2
gy1
gy2
safe1
green1
green2
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Example life-cycle of a person
child
puberty
bachelor
marriage
married
divorce
death
dead
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br
black
red
bb
rr

• The number of arcs between two objects specifies
  the number of tokens to be produced/consumed.
• This can be used to model (dis)assembly processes.

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Some definitions

• current stateThe configuration of tokens over
  the places.
• reachable stateA state reachable form the
  current state by firing a sequence of enabled
  transitions.
• dead stateA state where no transition is
  enabled.

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• 7 reachable states, 1 dead state.

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Exercise your life-cycle
sleeping
start
stop
active
die
dead

• How many states are reachable?
• Is there a dead state?

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Exercise readers and writers
receive_mail
begin
mail_box
rest
rest
type_mail
read_mail
ready
send_mail

• How many states are reachable?
• Are there any dead states?
• How to model the situation with 2 writers and 3
  readers?
• How to model a "bounded mailbox" (buffer size 4)?

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High-level Petri nets

• In practice the classical Petri net is not very
  useful
• The Petri net becomes too large and too complex.
• It takes too much time to model a given
  situation.
• It is not possible to handle time and data.
• Therefore, we use high-level Petri nets, i.e.
  Petri nets extended with
• color
• time
• hierarchy

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• To explain the three extensions we use the
  following example of a hairdresser's saloon.

hairdresser ready to begin
free
client waiting
start
finish
waiting
busy
ready
Note how easy it is to model the situation with
multiple hairdressers.
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The extension with color

• A token often represents an object having all
  kinds of attributes.
• Therefore, each token has a value (color) with
  refers to specific features of the object modeled
  by the token.

name Sally age 28 hairtype BL
name Harry age 28 experience 2
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• Each transition has an (in)formal specification
  which specifies
• the number of tokens to be produced,
• the values of these tokens,
• and (optionally) a precondition.
• The complexity is divided over the network and
  the values of tokens.
• This results in a compact, manageable and natural
  process description.

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Examples
c ab
b -a
a

b
-
a
b
c
a gt0 b Ö a
a
b
b
sqrt
a
select
c
if agt 0 then b a else ca fi
Exercise calculate Ö ab using these buiding
blocks
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The extension with time

• For performance analysis we need to model
  durations, delays, etc.
• Therefore, each token has a timestamp and
  transitions determine the delay of a produced
  token.

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The extension with hierarchy

• A mechanism to structure complex Petri nets
  comparable to DFD's.
• A subnet is a net composed out of places,
  transitions and subnets.

h1
h2
waiting
ready
h3
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Exercise remove hierarchy
h1
h2
waiting
ready
h3
begin
end
pending
begin
end
pending