1. Modeling based on Petri-nets.
Lecture 8

2. High-level Petri nets
The classical Petri net was invented by Carl Adam Petri in 1962.
A lot of research has been conducted (> 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)

3. 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.

4. 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.

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

6. 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.

7. Example

8. Non-determinism t1 t2
Two transitions fight for the same token: conflict.
Even if there are two tokens, there is still a conflict.

9. 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.

10. Example: traffic lightrg
red
yellow
green yr gy

11. Two traffic lights rg1 red1 yellow1 green1 yr1 gy1 rg2 red2 yellow2

12. Two safe traffic lights
rg1
red1
yellow1
green1
yr1
gy1
rg2
red2
yellow2
green2
yr2
gy2
safe

13. Two safe and fair traffic lights
red1
red2
safe2
yr1
yr2
yellow1
yellow2
rg1
rg2
gy1
gy2
safe1
green1
green2

14. Example: life-cycle of a person
child
puberty
marriage
bachelor
married
divorce
death
dead

15. br
red
black rr bb
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.

16. Some definitions black red bb rr br
current state The configuration of tokens over the places.
reachable state A state reachable form the current state by firing a sequence of enabled transitions.
dead state A state where no transition is enabled.
black
red bb rr br

17. 7 reachable states, 1 dead state.
black
red bb rr br
(3,2)
(1,3)
(3,1)
(1,2)
(3,0)
(1,1)
(1,0) rr br
bb\br
7 reachable states, 1 dead state.

18. Exercise: your life-cycle
sleeping
start
stop
active
die
dead
How many states are reachable?
Is there a dead state?

19. Exercise: readers and writers
begin
receive_mail
mail_box
rest
rest
type_mail
read_mail
send_mail
ready
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)?

20. 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

21. To explain the three extensions we use the following example of a hairdresser's saloon.
hairdresser ready to begin
client waiting
free
start
finish
waiting
busy
ready
Note how easy it is to model the situation with multiple hairdressers.

22. 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
start
waiting
finish
busy
free
ready

23. 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.

24. Examples Exercise: c := a+b b := -a a + a - b b c a >=0 | b := Ö a
sqrt b
select c
if a> 0
then b:= a
else c:=a fi
Exercise:
calculate Ö |a+b| using these buiding blocks

25. 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.
start
waiting
finish
busy
free
ready 1 3 9
D=3
D=0

26. 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
start
finish
busy
free

27. Exercise: remove hierarchy
waiting
ready h3
start
finish
busy
free
begin
pending
end
begin
pending
end