PETRINETS

1
PETRINETS

• Nipun Devlekar Zauja Lahtau

2
PETRINETS

• DEFINITION
• PETRINET (place/ transition net) a
  formal, graphical, executable technique for the
  specification and analysis of concurrent,
  discrete-event dynamic systems a technique
  undergoing standardization.
• Petri nets derive their name from the
  inventor of this tool Prof. Carl Adam Petri

3

• Prof. Carl Adam Petri, Germany

4

• Formal
• Graphical
• Executable
• Specification
• Analysis
• Concurrent
• Discreet

5
APPLICATIONS OF PETRINET

• Software design
• Workflow management
• Data Analysis
• Concurrent Programming
• Readability Engineering
• Diagnosis for finding the original error in the
  line of error -gt error state -gt visible error

6

• A simple PN is a 5-tuple (P,T,IN,OUT,Mo)
• P, finite number of Places p1,p2.,p
• T, finite set of transitions t1,t2.tm
• Input function Defines directed arcs from
  places to transitions.
• Output function Defines directed arcs from
  transition to places
• Mo P -gtN initial marking

7

• Various attributes of PN
• ? Safeness Number of tokens in each place cannot
  exceed one
• ? Boundedness The number of tokens in each place
  cannot exceed some threshold k
• ? Conservation Total number of tokens in the
  system is invariant.
• ? Dead transition A transition that can never be
  fired in future.
• ? Live Transitions transitions that can be
  enabled
• ? Deadlock No transition can fire
• ? Live PN Every transition is live
• ? Reachable markings A marking M obtained by
  firing a sequence of transitions from initial
  marking .

8
Graphical Representation
9
FIRING

• RULES
• A transition is enabled if each of its input
  places contains a number of tokens that is
  greater than or equal to the weight of the arrow
  connecting the input place to the transition
• An enabled transition may fire
• Firing of a transition T removes from each input
  place Pi the number of token equal to the weight
  of arrow from Pi to T and then inserts into each
  output place Pj the number of tokens equal to the
  weight of arrow from T to Pj

10
(No Transcript)
11
FIRING
12
HOW TO USE PETRINET FOR MODELLEING
13
EG1WAREHOUSE

• Requirements ?
• 1gtTruck arrives at a loading dock
• 2gtPaper is processed and inventory chequed
• 3gtConveyor belts move from the loading dock to
  the warehouse transporting people
• 4gtConveyor belt moves back from the warehouse to
  loading dock with merchandise
• 5gtGoods are loaded in the truck

14
(No Transcript)
15
(No Transcript)
16
EG2Differential Equation from Petrinet

• .

17

• V1 k1xy
• V2 k2z
• dz /dt V1 V2 k1xy k2z
• dy /dz 2V2 V1 2k2z k1xy
• dx /dt -V1 -k1xy

18
Non-Determinism

• Any of the enabled transitions may fire
• Model does not specify which fires, nor when it
  fires

Starvation
Starvation (i.e., a process never gets a
resource)
Deadlock
Petri net reaches a state in which no transition
is enabled i.e., a Deadlock State
19
(No Transcript)
20
How to avoid starvation
21
DeadLock
22
Why Other Types of Petri-Nets?

• Token of PNs do not carry information
• no data concept nor manipulation
• PNs for simple systems get very complex
• loss of usability
• No hierarchical structures with PNs
• reusable modules

23
What is a Colored Petri-nets (CPN)?

• Combination of Petri-Nets and a programming
  language
• Control structures, synchronization,
  communication and resource sharing described by
  PNs
• Data and data manipulation described by
  functional programming language

24
Extensions in CPNs

• Tokens (Each place contains a set of markers
  called tokens)
• carry data value of different types
• no longer shown as plain blank dots

25
Extensions in CPNs

• Places (The ellipses and the circles are called
  places)
• Color set specifies the type of tokens the place
  can hold
• Initial Marking multi-set of token colors

Multi-set for identical token colors
Token color describe different object
26
Extensions in CPNs

• Transitions (The rectangles are called
  transitions)
• Inspect token colors
• Guard-function decides if transition is enable or
  not
• Arcs (The arrows are called arcs)
• describe how the state
• of the CP-net changes
• when the transitions occur.

27
Why Colored Petri-Nets?

• CP-nets have a graphical representation
• CP-nets are very general and can be used to
  describe a large variety of different systems
• CP-nets have an explicit description of both
  states and actions
• CP-nets have a semantics which builds upon true
  concurrency, instead of interleaving
• CP-nets offer hierarchical descriptions
• CP-nets offer interactive simulations where the
  results are presented directly on the CPN diagram
• CP-nets have computer tools supporting their
  drawing, simulation and formal analysis

28
Stochastic Petri Nets

• Introduced by the computer science community in
  the early 1980s.
• SPNs are a probabilistic extension of the
  original nets introduced by Carl Adam Petri in
  his 1962 Ph.D. dissertation.

29
Why SPN?

• It has the same modelling power as GSMPs.
  (Generalized semi-Markov process)
• Graphically oriented (user friendly)
• Solid mathematical foundation
• Popular and enduring
• Useful to describe more or less complex systems

30
Graphical representation of an SPN

• Places are drawn as circles
• Immediate transitions
• as thin bars
• Timed transitions as
• thick bars
• Directed arcs connect
• transitions to output places and normal input
  places to transitions arcs terminating in open
  dots connect inhibitor input places to
  transitions.
• Tokens are drawn as black dots

31
Continue

• A transition is enabled whenever there is at
  least one token in each of its normal input
  places and no tokens in any of its inhibitor
  input places otherwise, it is disabled
• An enabled transition fires by removing one token
  per place from a random subset of its normal
  input places and depositing one token per place
  in a random subset of its output places.
• An immediate transition fires
• the instant it becomes enabled,
• whereas a timed transition fires
• after a positive
• (and usually random)
• amount of time

32
Advantage

• Graphical format for system design and
  specification
• The possibility and existing rich theory for
  functional analysis with Petri nets
• The facility to describe synchronization
• The natural way in which time can be added to
  determine quantitative properties of the
  specified system

33
Disadvantage

• The disappointing thing about SPNs is that the
  integration of time changes the behavior of the
  PN significantly
• So properties proven for the PN might not hold
  for the corresponding time-augmented PN
• E.g., a live PN might become deadlocked or a
  non-live PN might become live.
• Using SPNs to specify the sharing of resources
  controlled by specific scheduling strategies is
  very cumbersome

34
Time Petri nets (TPNs)

• First introduced by Merlin and Farber
• TPN is a 6-tuple (P, T, I, O, Mo, SI)
• (P, T, In, Out, Mo) is a Petri net
• SI is mapping called static interval
• SI T ? Q (Q U 8)
• Q is the set of rational numbers

35
Continue

• In a TPN, two time values are defined for each
  transition, a and b,
• where a is the minimum time the transition must
  wait for after it is enabled and before it is
  fired,
• b is the maximum time the transition can wait for
  before firing if it is still enabled
• Time a and b, for transition t are relative to
  the moment at which transition t is enabled
• Transition t enable at time , then t, cannot
  fire before time a and must fire before or at
  time b, otherwise disabled

36
Continue
37
Time Petri nets syntax

• Places logical part of the state
• Tokens current value of the logical part of the
  state
• Transitions events, actions,
• Labels observable behavior
• Arcs Pre and Post (logical) conditions of events
  occurrence
• Time (closed) intervals temporal conditions of
  events occurrence

38
Time Petri netstransitions occurrence

• Logical part
• The logical part of a state is a marking, i.e. a
  number of tokensper place.
• A transition is enabled if the tokens required by
  the pre conditions are present in the marking.
• Timed part
• There is an implicit clock per enabled transition
  and its value defines the timed part of the
  state.
• An enabled transition is fireable if its clock
  value lies in its interval.

39
Time Petri netschanges of state

• Time elapsing
• The marking is unchanged.
• Time may elapse (with updates of clocks) if every
  clock value does not go beyond the corresponding
  interval.
• Transition firing
• Tokens required by the pre condition are consumed
  and tokens specified by the post condition are
  produced.
• Clocks values of newly enabled transitions are
  reset.

40
Why Time Petri Nets?

• Basic Petri nets lack a temporal description and
  fail to represent any timing constraints for
  time-dependent systems.
• Time Petri nets (TPN) is a concise model for
  managing simultaneously concurrency and time
• TPNs are most widely used for real-time system
  specification and verification

41

• References
• http//en.wikipedia.org/wiki/Petri_net
• http//www.cs.arizona.edu/
• http//www.ento.vt.edu/sharov/PopEcol/lec1/petrin
  et.html
• http//ieeexplore.ieee.org/iel5/3477/18734/0086517
  3.pdf?arnumber865173
• http//www.mfn.unipmn.it/bobbio/BIBLIO/PAPERS/ANN
  O90/kluwerpetrinet.pdf
• http//www.informs-sim.org/wsc04papers/013.pdf
• http//www.daimi.au.dk/CPnets/intro

42

• Thank You..?