Petri Nets An Overview

1. Petri NetsAn Overview IE 680 Presentation April 30, 2007 Renata Kopach- Konrad

2. Overview • The building blocks of Petri Nets • An example • Analysis tools • Extensions • Elements of steady-state simulation theory for SPNs

3. What are Petri Nets? • Mathematical modeling tools that capture operational dynamics of discrete event systems • Graphical Representation • Modeling Language • State-transition mechanism • Event Scheduling mechanism Not this kind of Petri!

4. Applications • Software design • Workflow management • Data Analysis • Reliability Engineering

5. Suitable for Modeling • Concurrency • Synchronization • Precedence • Priority • Bottom up and top-down modeling

6. Modeling Tool: Petri Nets • Supports modularity and abstraction • Supports system simulation • Supports formal analysis for operational properties such as boundedness, liveness, reachabiltiy, etc. • Petri Nets have a rich research literature

7. p1 1 1 t1 p2 t2 p3 t3 2 1 1 1 1 p4 t4 2 1 What is a Petri Net?

8. p1 1 1 t1 p2 t2 p3 t3 2 1 1 1 1 p4 t4 2 1 System State: Tokens and Net Marking Mo=(M(p1), M(p2), M(p3), M(p4))=(1,1,2,1)

9. p1 1 1 t1 p2 t2 p3 t3 2 1 1 1 1 p4 t4 2 1 Initial State: (1,1,2,1) t2 is enabled

10. p1 1 1 t1 p2 t2 p3 t3 2 1 1 1 1 p4 t4 2 1 Fire t2, New State: (1,0,3,2)

11. p1 1 1 t1 p2 t2 p3 t3 2 1 1 1 1 p4 t4 2 1 Fire t3, New State: (2,0,2,2)

12. p1 p1 1 1 1 1 t1 t1 p2 p2 t2 t2 p3 p3 t3 t3 2 2 1 1 1 1 1 1 1 1 p4 p4 t4 t4 2 2 1 1 Fire t1, New State: (1,2,2,2)

13. Fire t4, New State: (1,3,2,0) p1 1 1 t1 p2 t2 p3 t3 2 1 1 1 1 p4 t4 2 1

14. In the context of DES • Marking of the SPN = state of the system • Firing of a transition = occurrence of an event

15. Notation… A Petri net is a 5-tuple , where • S is a set of places • T is a set of transitions • F is a set of arcs s.t. • M0is an initial marking • W is the set of arc weights/transition matrix/incidence matrix

16. …allows for many things The state of a net is an M vector so State equations are possible • Where is how many times each transition fires • WTstate transition matrix

17. p1 1 1 t1 p2 t2 p3 t3 2 1 1 1 1 p4 t4 2 1 • S={p1,p2,p3,p4} T={t1,t2,t3,t4} • F={(p1,t1) (p2,t2) (p3,t3) (p4,t4) (t1,p2)(t2,p3)(t2 p4) (t3,p1) (t4,p2)} W • M0 Initial state (1,1,2,1) • σFiring sequence (t2 t3 t1 t4) • Mn Final state (1,3,2,0)

18. Analysis Techniques • Reachability: used to find erroneous state (Karp and Miller, Hack) • Liveness: related to deadlock • Boundedness: if the # of tokens in any place cannot exceed k. (Covering) • Fairness: will there be an infinite sequence?

19. p1 1 1 t1 p2 t2 p3 t3 2 1 1 1 1 p4 t4 2 1 Reachability Starting State (Initial Marking) How do we get there? What sequence of event firings will get us to the final state? Final State

20. Reachability Example • Obtain the min sequence of events (transition firings) from an initial state to a final state • MIP Problem: solved multiple times • Constraints: State Equations • Objective Function: minimize the difference between the final and the intermediate state

21. An example

22. Part 1: Min number of events – Finding the Parikh Vector • Min S.t. = + X j>= 0 j Є J, ЄZ where, αi>= 0 initial marking (given) i Є I yi >= 0 final marking (given) i Є I, Z Фij- coefficient on the Incidence Matrix for place i, transition j; i Є I, j Є J, x is the number of times transition j fires j ЄJ

23. Part 2: Identifying the Sequence of Events using MIP Constraint (i) state equation Constraint (ii) limits the transitions per sequence to 1. Constraint (iii) enforces the total number of times a transition can fire across all sequences to be equal to the PV

24. Many extensions are possible • Coloured • Hierarchy • Prioritized • Timed Petri Nets • Stochastic

25. The Stochastic Element • Clocks • X(t) is the marking process of an SPN Ref: Haas, 2004 page 102

26. A Queuing Example 2 Service Centers N(>2) jobs p: rework at Center 1 The SPN of the model is Ref: Haas, 2004 page 103

27. Stability and Simulation • Interested in performance characteristics • Typically interested in terms of the marking process • Can apply • Regenerative Simulation Method

28. Expected value of data collected in regenerative interval Expected number of data • Concerns: • How do you find regenerative point in an SPN?

29. Key Assumptions There exists • a marking • set of transitions Such that the marking process probability restarts whenever the marking is and the transitions in fire simultaneously.

30. Regenerative method for the Marking Process • Select a sequence of regeneration points for the process • Simulate the process and observe a fixed number n of cycles defined by the random times • Compute the length of the kth cycle and the quantity • Form the strongly consistent point estimate (e.g. using jackknife method) • Haas (2004) provides the theorem for defining the sequence of regeneration points for a marking process

31. Challenges • Difficult to identify regeneration points in practice • Can be long intervals/ high utilization • Haas suggests using Standardized Time Series

32. Software • STNPlay • CPNTools (Colored PNs) • Petri Net Kernel (in Java) • YASPER (workflow analysis)

33. Summary • Petri Nets are a powerful modeling tool • Many extensions are possible • Allows for rigorous formulalism

34. References • J. Peterson. Petri Net Theory and the Modeling of Systems. Prentice Hall. 1981 • T. Murata, “Petri nets: Properties, Analysis And Applications”, Proc. Of The IEEE, Vol. 77 No.4, pp. 541-580, April 1989. • C. Cassandras. Discrete Event Systems: Modeling and Performance Analysis. Homewood, IL: Richard D. Irwin, Inc., and Aksen Associates, Inc., 1993. • P. Haas, “Stochastic Petri Nets for Modelling and Simulation,” Proc. Of the 2004 Winter Simulation Conference, Washington, D.C, pp. 101-112, December 2004.