PN modelling

Report
Lecturer: Sebastian Coope
Ashton Building, Room G.18
E-mail: [email protected]
COMP 201 web-page:
http://www.csc.liv.ac.uk/~coopes/comp201
Lecture 9, 10 – Modelling Based on Petri Nets
High-Level Petri Nets
 The classical Petri net was invented by Carl Adam Petri in 1962.
 A lot of research has been conducted (>10,000 publications).
 Until 1985 it was mainly used by theoreticians.
 Since the 80’s their practical use has increased because of the
introduction of high-level Petri nets and the availability of many
tools.
 High-level Petri nets are Petri nets extended with
 colour (for the modelling of attributes)
 time (for performance analysis)
 hierarchy (for the structuring of models, DFD's)
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Why do we need Petri Nets?
 Petri Nets can be used to rigorously define a system
(reducing ambiguity, making the operations of a system
clear, allowing us to prove properties of a system etc.)
 They are often used for distributed systems (with several
subsystems acting independently) and for systems with
resource sharing.
 Since there may be more than one transition in the Petri
Net active at the same time (and we do not know which
will ‘fire’ first), they are non-deterministic.
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The Classical Petri Net Model
A Petri net is a network composed of places ( ) and transitions
( ).
t2
t1
p2
p1
t3
p4
p3
Connections are directed and between a place and a transition, or a
transition and a place (e.g. Between “p1 and t1” or “t1 and p2” above).
Tokens ( ) are the dynamic objects.
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The Classical Petri Net Model
Another (equivalent) notation is to use a solid bar for the transitions:
p2
p1
t2
p4
t1
p3
t3
We may use either notation since they are equivalent, sometimes one
makes the diagram easier to read than the other..
The state of a Petri net is determined by the distribution of tokens
over the places (we could represent the above state as (1,2,1,1) for
(p1,p2,p3,p4))
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Transitions with Multiple
Inputs and Outputs
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 components.
 A transition is enabled if each of the input places contains tokens.
t1
t2
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.
t2
t2
Firing is atomic (only one transition fires at a time,
even if more than one is enabled)
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An Example Petri Net
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Example: Life-Cycle of a Person
child
puberty
marriage
bachelor
married
divorce
death
dead
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Creating/Consuming Tokens
A transition without any input can fire at any time and
produces tokens in the connected places:
T1
T1
P1
T1
T1
P1
After firing 3
times..
P1
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P1
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Creating/Consuming Tokens
A transition without any output must be enabled to fire
and deletes (or consumes) the incoming token(s):
T1
P1
T1
P1
T1
P1
After firing 3
times..
T1
P1
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Non-Determinism in Petri Nets
t1
t2
Two transitions fight for the same token: conflict.
Even if there are two tokens, there is still a conflict.
The next transition to fire (t1 or t2) is arbitrary (non-deterministic).
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Modelling
States of a process can be modelled by tokens in places and state
transitions leading from one state to another are modelled by
transitions.
 Tokens can represent resources (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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Modelling a Traffic Light
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Modelling Two Traffic Lights
• Imagine that we are designing a traffic light system for a crossroads
junction (i.e. with two sets of (simplified) lights).
• An informal specification of a traffic light junction:
o A single traffic light turns from “Red” to “Green” to “Amber”
and then back to “Red” (we’ll ignore “red and amber” for now).
o There are two sets of lights. When one of the traffic lights is
“Amber” or “Green”, the other must be “Red”.
• As a first step, we may decide to model the system as a Petri net.
This allows us to make sure the specification is rigorously defined
and reduces potential ambiguities later.
• We can also prove properties about the model if we wish.
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Example: Traffic Light
red
yr
amber
rg
gy
green
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Two Traffic Lights
red1
red2
yr1
rg1
yr2
amber1
gy1
amber 2
rg2
gy2
green1
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green2
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Two Safe Traffic Lights
red1
red2
safe
yr1
rg1
yr2
amber1
gy1
amber 2
rg2
gy2
green1
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green2
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Two Safe and Fair Traffic Lights
red1
red2
safe2
yr1
rg1
yr2
yellow1
rg2
yellow2
gy1
gy2
safe1
green1
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green2
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Exercise
 1) Can you prove that the Petri net from the previous slide
will never allow two red lights to be shown simultaneously?
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Exercise
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Arcs in Petri Nets
br
red
black
rr
bb
 The number of arcs between two objects specifies the number of
tokens to be produced/consumed (we can alternatively represent
this by writing a number next to a single arc).
 This can be used to model (dis)assembly processes.
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Some Definitions
 Current state (also called current marking) - The configuration of
tokens over the places.
 Reachable state - A state reachable form the current state by
firing a sequence of enabled transitions.
 Deadlock state - A state where no transition is enabled.
br
red
black
rr
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bb
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Some Definitions
 If we write the places in some fixed order (red, black say), then
we can use a tuple: (n,m) to denote the number of tokens in each
corresponding place (n tokens in “red” and m tokens in “black”).
 The example below is thus in state (3,2). After firing transition
“rr”, it will move to state (1,3) etc..
br
red
black
rr
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bb
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(3,2)
br
rr
red
black
bb\br
(1,3)
(3,1)
rr
br
bb\br
rr
bb
(1,2)
(3,0)
rr
bb\br
(1,1)
br
(1,0)
 7 reachable states, 1 deadlock state.
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Example: Simple Vending Machine
Deposit 10p
Deposit 10p Deposit 10p Deposit 10p
20p
10p
Deposit 20p
30p
Deposit 10p
40p
50p
Deposit 20p Deposit 20p Deposit 20p Deposit 20p
 Is there a deadlock state?
eat
 How could a “cancel” button be simulated?
(i.e. To return the person’s money)
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Exercise: Readers and Writers
begin
receive_mail
mail_box
rest
rest
type_mail
send_mail




read_mail
ready
How many states are reachable?
Are there any deadlock 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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Exercise
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The Four Seasons
 Let us try to model the four seasons of the year together with
their properties by a Petri net.
 We would like to denote the current season {spring, summer,
autumn, winter}, the temperature {hot, cold} and the light
level {bright, dark}.
 As a first step, let us model the seasons (with a token to
represent that it is currently autumn).
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The Four Seasons
Summer
Autumn
0
Spring
Winter
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The Four Seasons
Summer
Bright
Hot
Autumn
0
Spring
Cold
Dark
Winter
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High-Level Petri Nets
In practice, classical Petri nets have some modelling problems:
 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:
 colour
 time
 hierarchy
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Example - High-Level Petri Nets
To explain the three extensions we use the following example
of a hairdresser's salon:
hairdresser ready to begin
free
client waiting
start
waiting
finish
busy
finished
Note how easy it is to model the situation with multiple hairdressers..
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The Extension with Colour
A token often represents an object having all kinds of attributes.
Therefore, each token has a value (colour) with refers to specific
features of the object modelled by the token.
name: Sally
age: 28
hairtype: BL
free
start
waiting
name: Harry
age: 28
experience: 2
finish
busy
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finished
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The Extension with Colour
 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 := a+b
a
b := -a
+
b
neg
a
b
c
a >=0 | b :=  a
a
b
a
select
if a> 0
then b:= a
else c:=a
fi
sqrt
b
c
Exercise:
calculate |a+b| using these buiding blocks
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The Extension with Time
To analyse performance, we must model durations, delays, etc.
A timed Petri net associates a pair tmin and tmax with each
transition (there are other possible definitions for timed
Petri net, but we shall only consider this one).
free
start
waiting
Tmin = 0
Tmax = 3
finish
busy
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Tmin = 5
Tmax = 10
finished
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The Extension with Time
The values tmin and tmax, tell us the minimum and maximum
time that a transition will take to fire once enabled.
This allows us to model performance properties of the system,
although the analysis of such systems may be more difficult.
free
start
waiting
Tmin = 0
Tmax = 3
finish
busy
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Tmin = 5
Tmax = 10
finished
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The Extension with Time
Question: What is the minimum/maximum time for all three
people to have their hair cut in this system?
(Harder) Question: What about with n clients and m
hairdressers? Is there a general formula for the required time?
free
start
waiting
Tmin = 0
Tmax = 3
finish
busy
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Tmin = 5
Tmax = 10
finished
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Exercise
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The Extension with Hierarchy
 A hierarchy is a mechanism to structure complex Petri nets
comparable to Data Flow Diagrams.
 A subnet is a net composed out of places, transitions and
other subnets.
 This allows us to model a system at different levels of
abstraction and can reduce the complexity of the model.
 We shall see an example of this on the next slide..
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The Extension with Hierarchy
h1
h2
waiting
ready
h3
Here we
expand
subnet h3..
free
start
busy
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finish
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Exercise: Remove Hierarchy
h1
h2
waiting
ready
h3
free
begin
start
busy
pending end
finish
begin
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pending end
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Another Example
 Recall the following example of an informal specification from
a critical system [1] :
 The message must be triplicated. The three copies must be
forwarded through three different physical channels. The
receiver accepts the message on the basis of a two-out-of-three
voting policy.
 Questions: Can you identify any ambiguities in this
specification?
 How could we model this system with a Petri net?
[1] - C. Ghezzi, M. Jazayeri, D. Mandrioli, “Fundamentals of Software
Engineering”, Prentice Hall, Second Edition, page 196 - 198
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Message Triplication
Original Message
Tmin = c1
Tmax = k1
Message Copies
Tmin = c2
Tmax = k2
P1
Tmin = c3
Tmax = k3
Tvoting1
P2
Tvoting2
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P3
Tvoting3
Tvoting1: P1 = P2
Tvoting2: P1 = P3
Tvoting3: P2 = P3
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Message Triplication (2)
Original Message
Tmin = c1
Tmax = k1
Message Copies
Tmin = c2
Tmax = k2
P1
P2
P3
Tmin = c3
Tmax = k3
Tvoting
Tvoting: (P1 = P2) or (P2 = P3)
or (P1 = P3) else “ERROR”
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A Final Note on Petri Nets
 We can see from the previous example that the ambiguity (or
impreciseness) in the informal specification for the message
triplication protocol is clearly highlighted by the more formal
Petri net model.
 We can also perform some analysis on the model itself, for
example to see if certain “bad” states ever occur or if
deadlock/livelock is possible in the model.
 Finally we can represent timing constraints (to encode even
more constraints on the system) and use hierarchical models
to show different levels of abstration.
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A Final Note on Petri Nets
 Imagine modelling the elevator system of a skyscraper which
contains three elevators and twenty floors.
 What would be some of the advantages of using a Petri net
model for this?
 We can ensure if someone at a floor pushes the lift button (up or




down), the elevator will eventually come.
We can attempt to model the timing constraints of the system
(Timed Petri net).
We can also use hierarchies to simplify the system.
Finally we could try to optimize the model in some way if its
performance is not optimal.
Etc..
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Lecture Key Points
 Petri nets have Arcs, Places and Transitions.
 Petri nets are non-deterministic and thus may be used to
model discrete distributed systems.
 They have a well defined semantics and many variations and
extensions of Petri nets exist.
 The state or marking of a net is an assignment of tokens to
places.
 For those interested, the book “Fundamentals of Software
Engineering” (Prentice Hall) by C. Ghezzi, M. Jazayeri and D.
Mandrioli has an extensive example of using Petri nets for an
elevator system.
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