The document is a presentation by Mike Marin from IBM on Petri nets and their use in business process modeling. It introduces Petri nets as directed bipartite graphs that can model discrete systems and have been used as the theoretical foundation for workflow and business process management systems. It then provides an overview of Petri nets, including their history, applications, definitions, properties, analysis methods, and how they relate to business process modeling.

1. Mike Marin
29 April 2010
Petri-Nets an introduction
for Business Process Management (BPM)
© 2009 IBM Corporation

2. Petri Nets in Business Process Management
Abstract: Petri nets are directed bipartite graphs that can be used as a modeling
language for discrete distributed systems. Petri nets were introduced by Carl Adam
Petri in 1939, and have been used as the theoretical foundation for Workflow and
Business Process Management (BPM) systems. This presentation will be an introduction
to Petri nets, workflow nets, and their use in the modeling of business processes.
Bio: Mike Marin is an IBM Distinguished Engineer in the Software Group and the chief
architect of the IBM FileNet P8 Business Process Management technology. Marin is
also an ACM Distinguished Engineer with a MSCS in Artificial Intelligence. He has more
than twenty years of experience designing and developing system software. The last
Fifteen years, he has been developing business process management (BPM), workflow
products, and participating in standard organizations including WfMC, OMG, and
OASIS working on BPM and workflow standards. He has edited and contributed to the
definition several workflow and BPM standards; is a Fellow of the WfMC and has
received the WfMC Excellence Award for his technical contributions to the WfMC
standardization efforts.
2 © 2010 IBM Corporation

3. Petri Nets
3 © 2010 IBM Corporation

4. Petri Nets
■ Useful to model systems with concurrency and asynchronous behavior
– Able to model complex processes
■ Visual tool
– Based on bipartite graphs
■ Mathematical tool
– Useful to analyze the modeled system
■ Used by industry to analyze multiple type of systems
– Because they can be simulated, tested, and analyzed
■ Large variety of Petri Nets
– High level Petri Nets, colored Petri Nets, stochastic Petri Nets, workflow Nets,
etc.
4 © 2010 IBM Corporation

5. History
■ Invented by Carl Adam Petri
– Claims to have been invented “in August 1939 at the age of 13 for the purpose of
describing chemical processes” (Wikipedia & Scholarpedia)
– Ph.D. Dissertation: ”Kommunikation mit Automaten.”, Institut für Instrumentelle
Mathematik, Bonn, 1962
■ Nothing to do with Petri dishes
5 © 2010 IBM Corporation
Picture source: www.scholarpedia.org/article/Petri_net

6. Applications
■ Industrial control systems
■ Communication protocols
■ Performance evaluation
■ Distributed systems
■ Parallel and concurrent programs
■ Multiprocessor memory systems
■ Discreet event systems
■ Dataflow systems
■ Fault-tolerant systems
■ Workflow systems
■ Business Process Management (BPM) graphs
■ Etc.
6 © 2010 IBM Corporation
Source: Gabriel Eirea 2002 + mods

7. Informal definition
■ Directed, bipartite graph
2
■ Nodes
P1 P2
– Two types t1
• Places {p1, … , pn} (represented by circles)
• Transitions {t1, … , tn} (represented by boxes or bars)
■ Arcs (represented by arrows)
– Directed
– Connect nodes of different types
– Weighted (by default 1)
• w(p1,t1) = weight from place p1 to transition t1
■ Tokens (represented by dots inside a place)
– Places have zero or more tokens
■ Markings
– Describe the placement of tokens in the graph at a given moment on time
7 © 2010 IBM Corporation

8. Behavior (firing rule)
■ Transition t is enabled if each input place p has at least w(p,t) tokens
■ An enabled transition may or may not fire
■ A firing on an enabled transition t removes w(p i,t) from each input place pi, and adds
w(t,po) to each output place po
■ Firing is an atomic operation, resulting in a new marking
2 2
P1 P1
2 2
Fires to
P3 P3
t1 t1
P2 P2
8 © 2010 IBM Corporation
Graph source: Gabriel Eirea 2002 + mods

9. Non-determinism
■ Petri net execution is non-deterministic
■ Multiple transitions can be enabled at the same time
– Any one of which can fire
– None are required to fire
– They may fire at will (between time zero and infinite)
• They may not fire at all
9 © 2010 IBM Corporation

10. Interpretation of places and transitions
Input places Transitions Output places
input data computations output data
required resources task freed resources
input signal signal processing output signal
buffer/registers processor buffer/register
chemical substances chemical reactions new chemical substances
physical object transport geographical location
state event state
etc. etc. etc.
10 © 2010 IBM Corporation
Source: neo.dmcs.p.lodz.pl/oom/petri_nets.pdf + mods

11. Example – Elevator
Version 1 – token represent Version 2 – tokens represent
the elevator the number of movements the
(marking for ground floor) elevator can make up or down
in a certain state
3rd floor (marking for ground floor)
P4
t3 t4
P2
2nd floor Up Down
P3 t1 t2
t2 t5
P1
1 st floor
P2
t1 t6
Ground floor
P1
11 © 2010 IBM Corporation
Source: www.informatik.uni-hamburg.de/TGI/PetriNets/introductions/aalst/ + mods

12. Example - Dining Philosophers
 Five philosophers
alternatively think and
eating
 Chopsticks:
p0, p2, p4, p6, p8
 Philosophers eating:
p10, p11, p12, p13, p14
 Philosophers
thinking/meditating:
p1, p3, p5, p7, p9
12 © 2010 IBM Corporation
Source: Paul Fishwick, 2005

13. Formal definition
Definition 1 (Petri net): A Petri net is a 5-tuple, PN = (P, T, F, W, M 0)
where
P {p 1, p2, ⋯, pn }is a finite set ofplaces ,
T {t1, t 2, ⋯, t n }is a finite set oftransitions ,
F⊆P×T ∪T ×Pis a set of arcs ,
W :F  {1, 2,3, ⋯}is a weight function ,
M0 :P  {0, 1, 2,⋯}is an initialmarking ,
P∩T=∅∧P∪T ≠∅.
A Petri net structure N = (P, T, F, W) without initial marking is denoted by N
A Preti net with an initial marking is denoted as (N, M0)
13 © 2010 IBM Corporation

14. Behavioral properties (1)
 Properties that depend on the initial marking
 Reachability
 Mn is reachable from M0 if exists a sequence of firings that transform
M0 into Mn
 reachability is decidable, but exponential
 Boundedness
 a PN is bounded if the number of tokens in each place doesn’t exceed a
finite number k for any marking reachable from M0
 a PN is safe if it is 1-bounded
© 2010 IBM Corporation
Source: Gabriel Eirea 2002 + mods

15. Behavioral properties (2)
 Liveness
 a PN is live if, no matter what marking has been reached, it is possible
to fire any transition with an appropriate firing sequence
 equivalent to deadlock-free
 strong property, different levels of liveness are defined (L0=dead, L1,
L2, L3 and L4=live)
 Reversibility
 a PN is reversible if, for each marking M reachable from M , M is
0 0
reachable from M
 relaxed condition: a marking M’ is a home state if, for each marking M
reachable from M0, M’ is reachable from M
© 2010 IBM Corporation
Source: Gabriel Eirea 2002 + mods

16. Behavioral properties (3)
 Coverability
 a marking is coverable if exists M’ reachable from M s.t. M’(p)>=M(p)
0
for all places p
 Persistence
 a PN is persistent if, for any two enabled transitions, the firing of one
of them will not disable the other
 then, once a transition is enabled, it remains enabled until it’s fired
 all marked graphs are persistent
 a safe persistent PN can be transformed into a marked graph
© 2010 IBM Corporation
Source: Gabriel Eirea 2002 + mods

17. Behavioral properties (4)
 Synchronic distance
 maximum difference of times two transitions are fired for any firing
sequence
d12 = max σ (t1 ) − σ (t 2 )
σ
 well defined metric for condition/event nets and marked graphs
 Fairness
 bounded-fairness: the number of times one transition can fire while the
other is not firing is bounded
 unconditional(global)-fairness: every transition appears infinitely often
in a firing sequence
© 2010 IBM Corporation
Source: Gabriel Eirea 2002 + mods

18. Analysis methods (1)
 Coverability tree
 tree representation of all possible markings
• root = M0
• nodes = markings reachable from M0
• arcs = transition firings
 if net is unbounded, then tree is kept finite by introducing the symbol ω
 Properties
• a PN is bounded iff ω doesn’t appear in any node
• a PN is safe iff only 0’s and 1’s appear in nodes
• a transition is dead iff it doesn’t appear in any arc
• if M is reachable form M0, then exists a node M’ that covers M
© 2010 IBM Corporation
Source: Gabriel Eirea 2002 + mods

19. Analysis methods (2)
 Incidence matrix
 n transitions, m places, A is n x m
 aij = aij+ - aij-
 aij is the number of tokens changed in place j when transition i fires once
 State equation
 Mk = Mk-1 + ATuk
 uk=ei unit vector indicating transition i fires
© 2010 IBM Corporation
Source: Gabriel Eirea 2002 + mods

20. Business Process Management (BPM)
20 © 2010 IBM Corporation

21. Business Process Management (BPM) technology
■ Targeted to a team or line of business
– Examples
• Design review
• Loan origination
■ Designed to implement a business process
– Process improvement
■ Developed using a BPM system
– User Interface design
– Define how work flow between users and applications
– Define how to monitor key business metrics
– May still require some programming
© 2010 IBM Corporation

22. Evolution of Process Technology
“The ability to change is far
more prized than the ability to
create in the first place.”
Business Process Management — The Third Wave
Howard Smith & Peter Fingar
1st Wave: Taylorism 2nd Wave: Business Process 3rd Wave: Business Process
Reengineering Management (BPM)
Frederick Taylor’s “Scientific Processes manually re- Facilitating the ability to change
Management” theory engineered (typically a one time Manage and optimize business
Division of labour event)
processes
Managerial control of the Processes implemented via
ERP software Perceived as potential next “big
workplace thing” in consulting
Cost accounting based on Business & process logic hard-
systematic time-and-motion coded
study Led to EAI (application to
application focused)
Source: David Knight
© 2010 IBM Corporation

23. BPM or Workflow model
■ Defined using a directed graph using multiple icons as nodes
■ Executing the model means moving a package of data through the nodes
– similar to a token in a Petri net
■ Order processing example
– Accepted or rejected are alternatives
– Ship order and send invoice are done in parallel
– Send invoice, make payment, accept payment are done in series
23 © 2010 IBM Corporation

24. Modeling business processes requires
■ Sequential business activities
Activity1 Activity2
■ Parallel business activities
– Split (AND-split)
– Join (AND-join)
■ Branching (or alternative paths)
– Branch (XOR-split)
– Rendezvous (XOR-join)
■ Iteration
24 © 2010 IBM Corporation

25. Workflow nets definition
Definition 2 (Wf-net): A Petri net structure N = (P, T, F, W) is a Wf-net iff
(i) The weight function w(p,t) = w(t,p) = 1 for all p and t
(ii) N is a connected graph
(ii) N has two special places i and o,
where i is the start place, and o is the last place
25 © 2010 IBM Corporation

26. Modeling is just one
aspect of BPM
But, it is the base for all
the other aspects...
All based on graph and
Petri Net theory...
26 © 2010 IBM Corporation

27. © 2010 IBM Corporation

28. References
■ Murata, Tadao. Petri Nets: Properties, Analysis and Applications.
Proceedings of the IEEE, 77(4):541-580,. April 1989.
■ Van der Aalst, W.M.P. The Application of Petri Nets to Workflow
Management. . The Journal of Circuits, Systems and Computers, 8(1):21-66,
1998.
■ IBM Academic Initiative
– http://www.ibm.com/university
■ IBM Students Portal
– http://www.ibm.com/university/students
■ BPM Product Sample
– http://www.ibm.com/software/info/bpm
© 2010 IBM Corporation

29. 29 © 2010 IBM Corporation