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 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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 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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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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Another (equivalent) notation is to use a solid bar for the transitions: p2 t2 p1 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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p1 t1 p4 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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 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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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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child puberty marriage married death
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A transition without any input can fire at any time and produces tokens in the connected places:
T1 T1
P1
P1
T1
After firing 3 times..
P1
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T1
P1
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A transition without any output must be enabled to fire and deletes (or consumes) the incoming token(s):
T1 T1
P1 P1
T1
P1
After firing 3 times..
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P1
T1
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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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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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• 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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red yr amber rg gy green
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rg1
red1 red2 yr1 yr2 amber 2 amber1 rg2 gy1 gy2 green2 green1
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rg1
red1 red2 safe yr1 yr2 amber1 amber 2 rg2 gy1 gy2 green2 green1
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red1 red2 safe2 yr1 yr2 rg1 yellow1 gy1 green1 safe1 yellow2 gy2 rg2 green2
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 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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red
br 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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 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 bb
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 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 bb
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br rr
(3,2) bb\br red rr bb black
(1,3) bb\br
(1,2) rr
(3,1) br rr
(3,0) bb\br
(1,1) br
(1,0)
 7 reachable states, 1 deadlock state.
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Deposit 10p
Deposit 10p Deposit 10p Deposit 10p Deposit 10p
10p 20p 30p 40p 50p
Deposit 20p Deposit 20p Deposit 20p Deposit 20p Deposit 20p
 Is there a deadlock state?
 How could a “cancel” button be simulated?
(i.e. To return the person’s money)
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rest
begin receive_mail mail_box rest type_mail read_mail send_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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 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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Autumn
Summer
0 Spring
Winter
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Autumn
Summer
Bright
Hot
Cold
0 Spring
Dark
Winter
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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
: client waiting hairdresser ready to begin free start finish waiting busy finished
Note how easy it is to model the situation with multiple hairdressers..
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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 waiting start free busy name: Harry age: 28 experience: 2 finish finished
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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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a b
+ c := a+b
a c neg b := -a b a a >=0 | b :=
 a b a sqrt b select c if a> 0 then b:= a else c:=a fi
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Exercise: calculate

|a+b| using these buiding blocks
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To analyse performance, we must model durations, delays, etc.
A timed Petri net associates a pair t min and t max with each transition (there are other possible definitions for timed
Petri net, but we shall only consider this one).
free waiting start
Tmin = 0
Tmax = 3 busy finish
Tmin = 5
Tmax = 10 finished
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The values t min and t max
, 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 waiting start
Tmin = 0
Tmax = 3 busy finish
Tmin = 5
Tmax = 10 finished
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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 waiting start
Tmin = 0
Tmax = 3 busy finish
Tmin = 5
Tmax = 10 finished
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Exercise
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 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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waiting h1 h2 h3 free start busy finish
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Here we expand subnet h3..
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waiting
h1 h2 ready h3 free begin pending end start busy finish
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 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 45

Original Message
Tmin = c1
Tmax = k1
Message Copies
Tmin = c2
Tmax = k2
Tmin = c3
Tmax = k3
Tvoting1
P1 P2
Tvoting2
P3
Tvoting3
Tvoting1: P1 = P2
Tvoting2: P1 = P3
Tvoting3: P2 = P3
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Original Message
Tmin = c1
Tmax = k1
Message Copies
Tmin = c2
Tmax = k2
P1
Tmin = c3
Tmax = k3
Tvoting
P2 P3
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Tvoting: (P1 = P2) or (P2 = P3) or (P1 = P3) else “ERROR”
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 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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 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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 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 places.
of a net is an assignment of tokens to
 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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