COMP 201 Lecturer Sebastian Coope Ashton Building Room G18 Email coopesliverpoolacuk COMP 201 webpage httpwwwcsclivacukcoopescomp201 Lecture 9 10 Modelling Based on Petri Nets ID: 313894
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Slide1
Software EngineeringCOMP 201
Lecturer:
Sebastian
Coope
Ashton Building, Room G.18
E-mail:
coopes@liverpool.ac.uk
COMP 201 web-page:
http://www.csc.liv.ac.uk/~coopes/comp201
Lecture 9, 10 – Modelling Based on Petri NetsSlide2
High-Level Petri NetsThe 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 withcolour (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.3COMP201 - Software EngineeringSlide4
The Classical Petri Net ModelA Petri net is a network composed of
places
( ) and
transitions ( ).
t2
p1
p2
p3
p4
t3
t1
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 ModelAnother (equivalent
) notation is to use a solid bar for the transitions:
t2
p1
p2
p3
p4
t3
t1
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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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.
p1
p2
p3
p4
t1
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Transitions with Multiple
Inputs and OutputsSlide7
Enabling ConditionTransitions 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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FiringAn 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
bachelor
child
married
puberty
marriage
divorce
death
dead
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Creating/Consuming Tokens
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A transition without any input can fire at any time and produces tokens in the connected places:
After firing 3 times..
T1
T1
T1
T1
P1
P1
P1
P1Slide12
Creating/Consuming Tokens
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A transition without any output must be enabled to fire and deletes (or consumes) the incoming token(s):
After firing 3 times..
T1
T1
T1
T1
P1
P1
P1
P1Slide13
Non-Determinism in Petri NetsTwo 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).
t1
t2
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ModellingStates 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.14COMP201 - Software EngineeringSlide15
Modelling a Traffic Light15COMP201 - Software EngineeringSlide16
Modelling Two Traffic Lights16COMP201 - Software Engineering
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:
A single traffic light turns from “Red” to “Green” to “Amber” and then back to “Red” (we’ll ignore “red and amber” for now). 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.Slide17
Example: Traffic Light
rg
red
amber
green
yr
gy
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Two Traffic Lights
rg1
red1
amber1
green1
yr1
gy1
rg2
red2
amber 2
green2
yr2
gy2
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Two Safe Traffic Lights
rg1
red1
amber1
green1
yr1
gy1
rg2
red2
amber 2
green2
yr2
gy2
safe
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Two Safe and Fair Traffic Lights
rg1
red1
yellow1
green1
yr1
gy1
rg2
red2
yellow2
green2
yr2
gy2
safe2
safe1
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Exercise1) 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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ExerciseCOMP201 - Software Engineering22Slide23
Arcs in Petri NetsThe 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.
black
red
bb
rr
br
23
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Some DefinitionsCurrent 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.
black
red
bb
rr
br
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Some DefinitionsIf 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.. black
red
bb
rr
br
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7 reachable states, 1 deadlock state.
black
red
bb
rr
br
(3,2)
(1,3)
(3,1)
(1,2)
(3,0)
(1,1)
(1,0)
rr
rr
rr
br
br
bb\br
bb\br
bb\br
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Exercise: Readers and WritersHow 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)?restmail_box
receive_mail
type_mail
ready
rest
begin
send_mail
read_mail
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ExerciseCOMP201 - Software Engineering28Slide29
The Four Seasons29COMP201 - Software Engineering
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).Slide30
The Four Seasons30COMP201 - Software Engineering
0
Summer
Autumn
Winter
SpringSlide31
The Four Seasons31COMP201 - Software Engineering
0
Summer
Autumn
Winter
Spring
Hot
Cold
Dark
BrightSlide32
High-Level Petri NetsIn 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:colourtimehierarchy32
COMP201 - Software EngineeringSlide33
To explain the three extensions we use the following example of a hairdresser's salon:
start
waiting
finish
busy
free
client waiting
hairdresser ready to begin
Note how easy it is to model the situation with multiple hairdressers
..
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Example - High-Level
Petri
Nets
finishedSlide34
The Extension with ColourA 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.
start
waiting
finish
busy
free
name: Harry
age: 28
experience: 2
name: Sally
age: 28
hairtype: BL
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finishedSlide35
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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The Extension with ColourSlide36
Examples
c := a+b
a
b
c
+
b := -a
b
neg
a
if a> 0
then b:= a
else c:=a
fi
a
b
c
select
a >=0 | b :=
Ö
a
b
sqrt
a
Exercise:
calculate
Ö
|a+b| using these buiding blocks
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The Extension with TimeTo 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).
start
waiting
finish
busy
free
Tmin
= 0
Tmax
= 3
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Tmin
= 5
Tmax
= 10
finishedSlide38
The Extension with TimeThe values t
min
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.
start
waiting
finish
busy
free
Tmin
= 0
Tmax
= 3
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Tmin
= 5
Tmax
= 10
finishedSlide39
The Extension with TimeQuestion: 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?
start
waiting
finish
busy
free
finished
Tmin
= 0
Tmax
= 3
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Tmin
= 5
Tmax
= 10Slide40
ExerciseCOMP201 - Software Engineering40Slide41
The Extension with HierarchyA 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..41COMP201 - Software EngineeringSlide42
The Extension with Hierarchy
waiting
ready
h1
h2
h3
start
finish
busy
free
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Here we expand subnet h3..Slide43
Exercise: Remove Hierarchy
waiting
ready
h1
h2
h3
start
finish
busy
free
begin
end
pending
begin
end
pending
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Another ExampleRecall 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?44[1] - C. Ghezzi, M. Jazayeri, D. Mandrioli, “Fundamentals of Software Engineering”, Prentice Hall, Second Edition, page 196 - 198Slide45
Message TriplicationCOMP201 - Software Engineering45
P1
P2
P3
Original Message
Tvoting1
Tvoting2
Tvoting3
Message Copies
Tmin
= c1
Tmax
= k1
Tmin
= c2
Tmax
= k2
Tvoting1: P1 = P2
Tvoting2: P1 = P3
Tvoting3: P2 = P3
Tmin
= c3
Tmax
= k3Slide46
Message Triplication (2)COMP201 - Software Engineering46
P1
P2
P3
Original Message
Tvoting
Message Copies
Tmin
= c1
Tmax
= k1
Tmin
= c2
Tmax
= k2
Tvoting
: (P1 = P2) or (P2 = P3) or (P1 = P3) else “ERROR”
Tmin
= c3
Tmax
= k3Slide47
A Final Note on Petri NetsWe 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
.47Slide48
A Final Note on Petri NetsImagine 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..48Slide49
Lecture Key PointsPetri 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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