1. Petri-net based Analysis of workflows: Verification, validation, and performance analysis

2. Techniques to analyze workflows (design-time)
Validation is concerned with the relation between the model and reality.
Verification is typically used to answer qualitative questions
Is there a deadlock possible?
It is possible to successfully handle a specific case?
Will all cases terminate eventually?
It is possible to execute two tasks in any order?
Performance analysis is typically used to answer quantitative questions
How many cases can be handled in one hour?
What is the average flow time?
How many extra resources are required?
How many cases are handled within 2 days?

3. Three Analysis Techniques
Reachability graph
Place & transition invariants
Simulation

4. Traffic lights are safe!
Reachability Graph
(0,0,1,1,0,0,0)
(1,0,0,0,0,1,0)
(1,0,0,1,0,0,1)
(0,1,0,1,0,0,0)
(1,0,0,0,1,0,0)
Five reachable states.
Traffic lights are safe!

5. Alternative Notation
o1+r2
r1+o2
r1+r2+x
g1+r2
r1+g2

6. Reachability Graph Graph containing a node for each reachable state.
Constructed by starting in the initial state, calculate all directly reachable states, etc.
Expensive technique.
Only feasible if finitely many states
Difficult to generate diagnostic information.

7. Infinite Reachability Graph

8. Exercise: Construct Reachability Graph

9. Structural Analysis Techniques
To avoid state-explosion problem and bad diagnostics.
Properties independent of initial state.
We only consider place and transition invariants.
Proof the PN is live (transition invariants) and bounded (place invariants)

10. Place Invariant Assigns a weight to each place.
The weight of a token depends on the weight of the place.
The weighted token sum is invariant, i.e., no transition can change it
1 man + 1 woman + 2 couple

11. Other invariants 1 man + 0 woman + 1 couple
(Also denoted as: man + couple)
2 man + 3 woman + 5 couple
-2 man + 3 woman + couple
man – woman
woman – man
(Any linear combination of invariants is an invariant.)

12. Example: traffic light
r1 + g1 + o1
r2 + g2 + o2
r1 + r2 + g1 + g2 + o1 + o2
x + g1 + o1 + g2 + o2
r1 + r2 - x

13. Exercise: Give place invariants

14. Transition invariant Assigns a weight to each transition.
If each transition fires the number of times indicated, the system is back in the initial state.
i.e. transition invariants indicate potential firing sets without any net effect.
2 marriage + 2 divorce

15. Other invariants 1 marriage + 1 divorce
(Also denoted as: marriage + divorce)
20 marriage + 20 divorce
Any linear combination of invariants is an invariant, but transition invariants with negative weights have no obvious meaning.

16. Example: traffic light
rg1 + go1 + or1
rg2 + go2 + or2
rg1 + rg2 + go1 + go2 + or1 + or2
4 rg rg2 + 4 go1 + 3 go2 + 4 or1 + 3 or2

17. Incidence matrix of a Petri net
Each row corresponds to a place.
Each column corresponds to a transition.
Recall that a Petri net is described by (P,T,I,O).
N(p,t)=O(t,p)-I(p,t) where p is a place and t a transition.

18. man
Example
woman
marriage
couple
divorce

19. Place invariant
Let N be the incidence matrix of a net with n places and m transitions
Any solution of the equation X.N = 0 is a transition invariant
X is a row vector (i.e., 1 x n matrix)
O is a row vector (i.e., 1 x m matrix)
Note that (0,0,... 0) is always a place invariant.

20. Example
Solutions:
(0,0,0)
(1,0,1)
(0,1,1)
(1,1,2)
(1,-1,0)

21. Transition invariant
Let N be the incidence matrix of a net with n places and m transitions
Any solution of the equation N.X = 0 is a place invariant
X is a column vector (i.e., m x 1 matrix)
0 is a column vector (i.e., n x 1 matrix)
Note that (0,0,... 0)T is always a place invariant.
Basis can be calculated in polynomial time.

22. Example
Solutions:
(0,0)T
(1,1)T
(32,32)T

23. Performance Analysis Questions: Techniques:
throughput, waiting and service times
service levels
Techniques:
simulation