Petri Nets

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Petri Nets

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Agenda

• Introduction

• Petri Net

• Modelling with Petri Net

• Analysis of Petri net

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Introduction

• Techniques for Evaluating SW / HW:

– Simulation – Testing

– Deductive Verification – Model Checking

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Introduction (2)

• Simulation & Testing:

– Needs experiments before system deployment

– Simulation is performed on an abstraction of a

model of the system

– Testing is performed on the real system. – It is very diffucult to evaluate all possible

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Introduction (3)

• Deductive Verification:

– Validate a system’s performance based on deductive rules

– Applicable to infinite-state systems – A time-consuming process

– Needs expert people with good experiments

– Usually is used for critical systems (e.g., security protocols)

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Introduction (4)

• Model Checking:

– An automated technique to evaluate finite-state systems

– Perform a thorough search in the state space of the system in order to determine some properties of the system.

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Introduction (5)

• The process of Model Checking:

• Modeling

: Transform a design to formalism (e.g., a

state-transition graph of the system)

• Verification: Perform a thorough search in the state

space of the system in order to determine whether some properties of the system are true or not.

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Petri Net

• A

‘Petri Net’

was introduced by a German

mathematician and used

to model a system

– Condition: a Boolean description of the state of the system

– _Event_{: an action that depends on the state of the}

system

• Petri net is a graphical and mathematical tools

for

the analysis of discrete event dynamic systems

.

– The system model is represented by a set of

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Petri Net: Building Blocks

place counter

transition

Place for user input

Basic Elements

• PN consists of three types of components: places (circles), transitions (rectangles) and arcs (arrows):

– Places represent possible states of the system;

– Transitions are events or actions which cause the change of

state; And

– Every arc simply connects a place with a transition or a transition

arcs

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Petri Net

• Building Blocks:

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Petri Net: Marks (Tokens)

• A marking M is an assignment of tokens

(dots) to places of a petri net.

• Dots are placed in the circle to represent

place in which its condition is satisfied

– e.g. a resource is available or operation in a process

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Petri Net: Marks (Tokens)

• A place can have more than one token and

therefore can represent a queue

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Petri Net : Firing a transition

• Executing a marked Petri net causes the number and positions of the token to change. The rules for the execution are:

– A transition is enabled if all its input places contain at least one token.

– Any enabled transition may fire

– Firing of a transition results in one token being removed from each of its input places and being deposited of its output places

– _{Execution halts when there are no enabled transitions.}

– Each time a transitions fires, the marking of the petri net will change.

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Petri Net: Formal Definition

A Petri net (PN) is a 5 tuple

PN (P,T,IN,OUT,M) where:

P = {p1,p2,....,p~} is a finite set of places, T = {t₁, t₂, …,t_n} is a finite set of transitions IN: (PxT)→S

OUT: (TxP)→S

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Petri Net: Formal Definition (cont’d)

• IN are input functions defining directed arcs

from

• places to transitions

• OUT are output functions defining directed arcs

• from transitions to places

• S is a set of all nonnegative integers k such

that:

• If k = 1 a directed arc is drawn without a label • If k > 1 a directed arc is drawn with label k. • If k = 0 no arc is drawn.

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Petri Net: Formal Definition (cont’d)

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Petri Net: Firing Rules for Transitions

• A specific transition t_i is said to be firable or enabled if

each input place p_i is marked with at least w(p_i,t_i) tokens where w(p_i,t_i) is the weight of the arc from p_i to t_i.

• An enabled transition may or may not fire depending on whether or not the event actually takes place .

• The firing of an enabled transition t_i removes w(p_i,t_i) tokens from each input place p_i of t_i, and adds w(p_j,t_i) tokens to

each output place p_j of t_i where w(p_i,t_i) is the weight of the arc from input place p to t , and w(p ,t) is the weight of the

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Petri Net: Firing Rules for Transitions

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PN: Firing Rules for Transitions

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PN: Firing Rules for Transitions

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PN: Firing Rules for Transitions

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PN: Firing Rules for Transitions

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PN: Firing Rules for Transitions

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PN: Change of States (1)

• is denoted by a movement of token(s)

(black dots) from place(s) to place(s); and is caused by the firing of a transition.

• The firing represents an occurrence of the event or an action taken.

• The firing is subject to the input

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PN: Change of States (2)

• A transition is firable or enabled when there are sufficient tokens in its input places.

• After firing, tokens will be transferred from the input places (old state) to the output places, denoting the new state.

• Note that the examples are Petri nets

representation of a finite state machine (FSM). PNs are much more powerful to model systems beyond FSMs.

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Modelling with petri net

• Petri net is mainly used for modelling.

– Many systems can be modelled by petri net.

• The system may be of many different kinds

like computer hardware, computer software,

physical system and so on.

– Petri net is used to model the occurrence of various events and activities in the system.

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Modelling with petri net

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Modelling with petri net

– Concurrency: two events

could take place in parallel, without

interfering with one another.

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Modelling with petri net

• Synchronisation: By having two places as input conditions to the same transition, these two conditions become synchronised.

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Modelling with petri net

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Modelling with petri net

• Confilincts:

• Two transitions t₁ and t₂ are said to be conflict if either t₁ or t₂ can occur but not both of them,

– i.e., when there is one place that is acting as input condition to a number of transitions.

– Therefore, only one transition can be fire.

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Modelling with petri net

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Modelling with petri net

• Example: Font Selection

Bold on User presses bold T1 T2 Italic on User presses italic T3 T4 User presses italic

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Modelling with petri net

• Another Example: a finite-state machine

– Consider a vending machine

• It accepts either nickels or dimes • Sells 15c or 20c candy bars

• The vending machine can hold up to 20c • Coin return transitions are omitted

the next slides are the state diagram of this

vending machine which represented by the Petri net Any finite-state machine (or its state diagram) can be

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Modelling with petri net

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Modelling with petri net

• Mutual exclusion

– Recall: Mutual exclusion is a technique of defining entry and exit code so that at most one process is allowed to access the critical region at the same time.

• The idea is that no process is allowed to enter the critical region, unless it checks that no other process is executing its own critical region.

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Mutual exclusion

• The place s represents the permission to enter the critical region and there must be token in p₁ and p₂.

– If both processes want to enter the critical region simultaneously, then transition t₁ and t₂ are in conflict

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Producer consumer problem

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Analysis of Petri net

• Petri nets are capable of modelling a large

variety of systems and properly representing

the interactions between the various actions

which can occur.

– The strength of Petri net is its capability to model the system.

• However, modelling by itself is of little use and its necessary to analyse the modelled system. This could lead to important insights into the behaviour of the

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Reachibility

• Given a Petri net, one would like to know

which marking M

_r

can be reached from an

initial marking M

_0.

• Example:

For the Petri net of the bounded buffer

consumer producer problem, M

₀

= (1, 0, 1, 0,

0, n) and M

₁

= (0, 1, 1, 0, 0, n)

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Analysis of Petri net : Safeness

• A place in a Petri net is

safe if the number of

tokens in that place

never exceeds one.

– A petri net is same if all

its places are safe.

what does this Petri net do?

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Analysis of Petri net : Boundedness

• A place is k-safe or k-bounded if the number of

tokens in that place cannot exceed an integer k.

– Therefore a place is 1-bounded is simply a safe place.

• Example:

– The Petri net for the producer/consumer problem

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Analysis of Petri net : Conservation

• A Petri net with an initial marking M

₀

, is strictly

conservative,

– if for all the reachable marking, the total number of

tokens in each marking is exactly the same as the initial marking.

• A Petri net can be used to model resource

allocation systems. In this systems some tokens

may represents the resources.

– Therefore, in such Petri nets conservation is an

important property such that resources can neither be created nor destroyed.

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Analysis of Petri net : Liveness

• Another problem that could occur in resource

allocation is deadlock.

– A deadlock in a Petri net is a transition (or a set of transition) which cannot fire.

– A transition is alive if it is not deadlock.

– A transition is live in a marking M if it is potentially fireable in every marking in the Petri net.

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Exercises

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Summary

• Petri net

– concepts

– Petri net representation – Firing a transition

– Marks (Tokens)