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Mike Marin
29 April 2010




       Petri-Nets an introduction
 for Business Process Management (BPM)




                                 © 2009 IBM Corporation
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
Petri Nets




3                © 2010 IBM Corporation
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Business Process Management (BPM)




20                           © 2010 IBM Corporation
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
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
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
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
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
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
© 2010 IBM Corporation
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   © 2010 IBM Corporation

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2010 04-29 mm (carson, california - csu-dh) petri-nets introduction

  • 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