PETRINETS Nipun Devlekar Zauja Lahtau

PETRINETS DEFINITION : 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

 Prof. Carl Adam Petri, Germany

 Formal  Graphical  Executable  Specification  Analysis  Concurrent  Discreet

APPLICATIONS OF PETRINET  Software design  Workflow management  Data Analysis  Concurrent Programming  Readability Engineering  Diagnosis for finding the original error in the line of error -> error state -> visible error

 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 ->N initial marking

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.

Graphical Representation

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

FIRING

HOW TO USE PETRINET FOR MODELLEING

EG1:WAREHOUSE  Requirements  1>Truck arrives at a loading dock 2>Paper is processed and inventory chequed 3>Conveyor belts move from the loading dock to the warehouse transporting people 4>Conveyor belt moves back from the warehouse to loading dock with merchandise 5>Goods are loaded in the truck

EG2:Differential Equation from Petrinet.

 V1 = k1xy  V2 = k2z  dz /dt = V1 – V2 = k1xy – k2z  dy /dz = 2V2 – V1 = 2k2z – k1xy  dx /dt = -V1 = -k1xy

Non-Determinism Any of the enabled transitions may fire Model does not specify which fires, nor when it fires Starvation (i.e., a process never gets a resource) Starvation Deadlock Petri net reaches a state in which no transition is enabled i.e., a Deadlock State

How to avoid starvation

DeadLock

Why Other Types of Petri-Nets?  Token of PN’s do not carry information  no data concept nor manipulation  PN’s for simple systems get very complex  loss of usability  No hierarchical structures with PN’s  reusable modules

What is a Colored Petri-nets (CPN)?  Combination of Petri-Nets and a programming language  Control structures, synchronization, communication and resource sharing described by PN’s  Data and data manipulation described by functional programming language

Extensions in CPN’s Tokens (Each place contains a set of markers called tokens)  carry data value of different types  no longer shown as plain blank dots

Extensions in CPN’s 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

Extensions in CPN’s 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 of the CP-net changes when the transitions occur. when the transitions occur.

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

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.

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

Graphical representation of an SPN  Places are drawn as circles  Immediate transitions as thin bars as thin bars  Timed transitions as thick bars 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. 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

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, the instant it becomes enabled, whereas a timed transition fires whereas a timed transition fires after a positive after a positive (and usually random) (and usually random) amount of time amount of time

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

Disadvantage  The disappointing thing about SPN’s 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 SPN’s to specify the sharing of resources controlled by specific scheduling strategies is very cumbersome

Time Petri nets (TPN’s)  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 ∞)  Q is the set of rational numbers

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

Continue

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

Time Petri nets: transitions 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.

Time Petri nets: changes 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.

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

 References    ml ml ml  ?arnumber= ?arnumber= ?arnumber=  petrinet.pdf petrinet.pdf petrinet.pdf  

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