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
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 Model
Another (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 Outputs
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
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 P1
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 P1
Non-Determinism in Petri Nets
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).
t1
t2
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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
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• 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.
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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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
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.
black red
bb rr
br
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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.
black red
bb rr
br
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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..
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 Writers
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)?
rest
mail_box
receive_mail
type_mail
ready
rest
begin
send_mail
read_mail
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Exercise
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The Four Seasons
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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).
The Four Seasons
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0
Summer
Autumn
Winter
Spring
The Four Seasons
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0
Summer
Autumn
Winter
Spring
Hot
Cold
Dark
Bright
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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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
finished
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.
start
waiting
finish
busy
free name: Harry age: 28 experience: 2
name: Sally age: 28 hairtype: BL
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finished
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 Colour
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 := a
b sqrt a
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).
start
waiting
finish
busy
free
Tmin = 0 Tmax = 3
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Tmin = 5 Tmax = 10
finished
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.
start
waiting
finish
busy
free
Tmin = 0 Tmax = 3
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Tmin = 5 Tmax = 10
finished
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?
start
waiting
finish
busy
free
finished Tmin = 0 Tmax = 3
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Tmin = 5 Tmax = 10
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
waiting ready
h1
h2
h3
start finish busy
free
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Here we expand
subnet h3..
Exercise: Remove Hierarchy
waiting ready
h1
h2
h3
start finish busy
free
begin end pending
begin end pending
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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?
44 [1] - C. Ghezzi, M. Jazayeri, D. Mandrioli, “Fundamentals of Software Engineering”, Prentice Hall, Second Edition, page 196 - 198
Message Triplication
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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 = k3
Message Triplication (2)
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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 = k3
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.. 48
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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