1. Ch8. Analysis
Joosung, Ko

2. outline Introduction What do we want to analyze? Reachability analysis
Structural analysis
Place invariants
Transition invariants
Automated calculation of invariants
Simulation
Example the taxi company
simulation using CPN Tools
Classical Petri nets Vs High level Petri nets

3. 1. Introduction It is reasonable to make such a process model.
Making a process model can help us to understand it better.
A Petri-net-based process model can be used as a specification or a design of a system to be developed.
A Petri-net-based process model can also be used for the analysis of a system.
In this chapter, we study how to use Petri nets for the purpose of analysis.

4. 2. What do we want to analyze?
We can ask some questions to know about some systems.
Two question types : qualitative and quantitative questions.
If you answer, ‘yes’ or ‘no’ -> qualitative
If you answer, ‘number, frequency’ -> quantitative
There are two reasons to distinct these questions

5. 3. Reachability Analysis
In reachability analysis, the reachability graph of a Petri net is analyzed.
With this graph of the Petri net, we can answer a number of qualitative Questions.
By inspecting the reachability graph Of a high-level Petri net, we can gain insight into the occupation rates, efficiency, performance and costs of the modeled situation.

6. A model for two traffic lights.
rg1
rg2 r1 r2
go1
go2 g1 g2
or1
or2 o1 o2 x
A model for two traffic lights.

7. 4. Structural Analysis
The number of nodes in the reachability graph on the initial state.
It is difficult to use reachability analysis when you have unlimited reachable states.
start_production
start_consumption
producer
free
consume
wait
product
end_production
end_consumption
Producers and consumers

8. a. Place Invariants
A place invariants assigns to each place a weight, in such a way that no transition can change the weighted token sum.
A place invariant represents a kind of preservation law.
man
woman
couple
marriage
divorce
A classical Petri net

9. A business process involving a specialist
free
docu
end
wait
done
start
change
busy
inside
A business process involving a specialist

10. rg1
rg2 g2 g1 r2 r1
go1 x
go2 o1 o2
or1
or2
Two traffic lights

11. b. Transition invariants
We use transition invariants to prove that a Petri net returns to the initial state after a certain series of firings
A transition invariant assigns each transition t a weight wt in such a way that firing each transition t wt times does not change the state. The weight of a transition is not allowed to be a negative number.
A transition invariant does not indicate in which order the transition should fire. Only the number of firings per transition is fixed.
Transition invariants are the counterparts of place invariants.
We can use invariants in two ways
We can verify whether previously determined properties indeed apply.
We can derive invariants and subsequently check which properties they represent.

12. A classical Petri net The four seasons winter p1 p4 p2 p3 man couple
divorce
marriage
woman
winter p1 p4
spring
autumn
The four seasons p2 p3
summer

13. c. Automated calculation of invariants
Place and transition invariants are structural techniques, and the invariants can be checked and calculated efficiently.
We can simply describe how invariants can be computed.
It is important to note that a Petri net can be represented by a matrix having a row for each place and a column for each T.
The incidence matrix of a Petri net can be used for all kinds of analysis.
It is not difficult to calculate a basis and there are many techniques to obtain the most interesting invariants.

14. 5. Simulation
The purpose of simulation is to avoid too costly or dangerous experiments.
We study the simulation in the sense of experimenting with a high-level Petri net representing processes of reality.
We can run the simulation through the reachability graph.
To clarify things we need to discuss the concept “probability distribution” since these are typically used to specify delays in a simulation model.
During the simulation all kinds of things can be measured.
To determine the reliability of the measurements, the simulation run is often divided into a number of subruns.

15. A uniform distribution of probability for service durations
0.3
0.2
0.1 2 4 6 8 10
duration (min) 2 4 6 8 10
duration (min)
0.1
0.2
0.3
probability
A negative exponential probability distribution for the time in between two arrivals

16. a. Example: The Taxi Company
color Car=string
arrive
Car
environment
gas_station
drive_on
depart HS HS
Page main modeling the gas station and its environment

17. fun len(q:Queue) = if q=[] then 0ln
arrive
Car
drive_on c
[len(q)<3]
put_in_queue
queue
Queue
q^^[c] q []
start ()
Pump
pump_free
TCar
end
out
depart
c::q
fill_up
color Car = string;
color Pump = until;
color TCar = Car timed;
color Queue = list Car;
var c:Car;
var q:Queue;
fun len(q:Queue) = if q=[] then 0
else 1+len(tl(q));
Page gas_station

18. b. simulation using cpn tools

19. Page gas_station in CPN Toolsln
arrive c
[len(q)<3]
[len(q)>=3] c
Car
put_in_queue q
drive_on
q^^[c] q []
queue q 1
1’0
Queue q
c::q
start () () c
fill_up
Car
pump_free
Pump c () 1
end c
depart
drive_on
out
out
Car
Car
Page gas_station in CPN Tools

20. Page environment in CPN Tools. 1 1
(“car_”^Int.toString(i),IntInf.toInt(time())) i
counter
create
arrive
out
Car
percentage 1
1’0
(i*100)div(i+1)
INT 1
1’0
i+1 k i c
count_do
drive_on
record_depart_on in
INT
Car 1
1’0
count_d
INT 1
1’0 i
average_d
i+1 k
INT 1
1’0 i
I div(i+1)
(cid,t) j
sum_do
depart
record_drive_do in
INT
Car
Page environment in CPN Tools.

21. The top level of the simulation model
arrive
Car
environment
gas_station
drive_on
depart
gas_station
environment
The top level of the simulation model

22. The effect of adding three extra places to the waiting space.

23. 6. classical petri nets Vs high-level petri nets
The classical Petri net allows for many types of analysis and many questions are tractable
To gain insight into applicability of analysis methods to high level nets we discuss each f the three extensions and their effect on analysis
The extension with color increases the expressive power of Petri nets considerably.
The extension with time adds a dimension to the Petri net
The extension with hierarchy allows for the construction of large and complex models but does not complicate analysis.

24. Definition of top-level page five _philosopher
left = CS2
right = CS1
color BlackToken = unit;
var b:BlackToken HS ()
CS2 ()
CS1
BlackToken
BlackToken
PH1 HS
PH2
PH5
philosopher
left = CS1
right = CS5 HS
philosopher
left = CS3
right = CS2
CS5
CS3 () ()
BlackToken
BlackToken
CS4 ()
PH4
PH3 HS HS
philosopher
left = CS5
right = CS4
philosopher
left = CS4
right = CS3
BlackToken
Definition of top-level page five _philosopher

25. Thank you