petri nets copyright, 2003 © jerzy r. nawrocki [email protected] models and analysis of...

Petri NetsPetri Nets

Copyright, 2003 © Jerzy R. Nawrocki

[email protected]

www.cs.put.poznan.pl/jnawrocki/mse/mss/

Models and Analysis of Software Models and Analysis of Software

Lecture 7Lecture 7

Models and Analysis of Software Models and Analysis of Software

Lecture 7Lecture 7

J. Nawrocki, Models & Analysis of Soft.

From the previous lecture..From the previous lecture..From the previous lecture..From the previous lecture..

A Be(c) / a

e - event

c - conditiona - actionExamples:

• sending a signal,• assignment statement.

Action:

• an instantaneous occurrence

• ideally takes zero time.

whilewhile (a != b) (a != b)

ifif (a > b) (a > b)

a -= b;a -= b;

elseelse b -= a; b -= a;

whilewhile (a != b) (a != b)

ifif (a > b) (a > b)

a -= b;a -= b;

elseelse b -= a; b -= a;

(a>b) /a-=b

(b>a) /b-=a

OrthogonalityOrthogonalityOrthogonalityOrthogonality

(in G)

Activity:• similar to action but• takes a nonzero amount of time

Examples:• reading input• beeping• displaying• lengthy computations

Plan of the lecturePlan of the lecturePlan of the lecturePlan of the lecture

• From the previous lecture..

• Introduction• Simple examples• Concurrent transitions• Transitions in conflict• Another example• Readers and writers• Properties of Petri Nets

Carl Adam PetriCarl Adam PetriCarl Adam PetriCarl Adam Petri

Born on July, 12, 1926 in Leipzig

Since 1988:Honorary Prof. at the Univ. of Hamburg

Kommunikation mit Automaten. Petri, C.A., Bonn: Institut für Instrumentelle Mathematik, Schriften des IIM Nr. 2, 1962, Second Edition:, New York: Griffiss Air Force Base, Technical Report RADC-TR-65--377, Vol.1, 1966, English translation.

IntroductionIntroductionIntroductionIntroduction

Petri nets:• a graphical language• a good mathematical

background

Applications:• modelling concurrent and/or

distributed systems• communication protocols,

computer networks, manufacturing system, public transport systems etc.

TerminologyTerminologyTerminologyTerminology

A placeA place

A transitionA transition

A tokenA token

An inputAn input

arcarc

An outputAn output

arcarc

• From the previous lecture..• Introduction

• Simple examples• Concurrent transitions• Transitions in conflict• Another example• Readers and writers• Properties of Petri Nets

A very simple exampleA very simple exampleA very simple exampleA very simple example

A reader in a library:• waiting,• reading.

waiting

reading

strt_reading

stop_reading

waiting

reading

strt_reading

stop_reading

waiting

reading

strt_reading

stop_reading

waiting

reading

strt_reading

stop_reading

waiting

reading

strt_reading

stop_reading

waiting

reading

strt_reading

stop_reading

Another simple exampleAnother simple exampleAnother simple exampleAnother simple example

Two readers in a library:

• waiting,• reading.

waiting

reading

strt_reading

stop_reading

waiting

reading

strt_reading

stop_reading

waiting

reading

strt_reading

stop_reading

• From the previous lecture..• Introduction• Simple examples

• Concurrent transitions• Transitions in conflict• Another example• Readers and writers• Properties of Petri Nets

Concurrent transitionsConcurrent transitionsConcurrent transitionsConcurrent transitions

• From the previous lecture..• Introduction• Simple examples• Concurrent transitions

• Transitions in conflict• Another example• Readers and writers• Properties of Petri Nets

Transitions in conflictTransitions in conflictTransitions in conflictTransitions in conflict

A decision pointA decision point

• From the previous lecture..• Introduction• Simple examples• Concurrent transitions• Transitions in conflict

• Another example• Readers and writers• Properties of Petri Nets

Another exampleAnother exampleAnother exampleAnother example

while (true)

while (.. t4 ..)

... t5; ...

Client

Server

while (1)

while (..t4..)

... t5; ...

while (1)

while (..t4..)

... t5; ...

while (1)

while (..t4..)

... t5; ...

while (1)

while (..t4..)

... t5; ...

while (1)

while (..t4..)

... t5; ...

while (1)

while (..t4..)

... t5; ...

while (1)

while (..t4..)

... t5; ...

while (1)

while (..t4..)

... t5; ...

while (1)

while (..t4..)

... t5; ...

while (1)

while (..t4..)

... t5; ...

while (1)

while (..t4..)

... t5; ...

while (1)

while (..t4..)

... t5; ...

(t1 (t2 || t4 (t5 t4)*) t3)*

• From the previous lecture..• Introduction• Simple examples• Concurrent transitions• Transitions in conflict• Another example

• Readers and writers• Properties of Petri Nets

Readers and writersReaders and writersReaders and writersReaders and writers

while(1) { Read(d); Use(d); }

while(1) { Think(d); Write(d); }

LibraryLibrary

while(1) { Read(d); Use(d); }

while(1) { Think(d); Write(d); }

while(1) { StartRead(); Read(d); StopRead(); Use(d); }

while(1) { Think(d); StartWrite(); Write(d); StopWrite(); }

void StartRead() { ReadersIn++; }

void StartWrite() { WritersIn++; }

void StopRead() {ReadersIn--; }

void StopWrite() {WritersIn--; }

void StartRead() { wait(Readers); ReadersIn++; }

void StartWrite() { wait(Writers); WritersIn++; }

void StopRead() {ReadersIn--; signal(Writers); }

void StopWrite() {WritersIn--; signal(Writers); signal(Readers); }

void StartRead() {if(WritersIn > 0) wait(Readers); ReadersIn++; }

void StartWrite() {if(ReadersIn > 0 || WritersIn > 0) wait(Writers); WritersIn++; }

void StopRead() {ReadersIn--; if(ReadersIn == 0) signal(Writers); }

void StopWrite() {WritersIn--; if(!Empty(Writers)) signal(Writers); else if(!Empty(Readers)) signal(Readers); }

StartRead

StopReadStopWrite

StartWrite

StartRead

StopReadStopWrite

StartWrite

StartRead

StopReadStopWrite

StartWrite

StartRead

StopReadStopWrite

StartWrite

StartRead

StopReadStopWrite

StartWrite

StartRead

StopReadStopWrite

StartWrite

StartRead

StopReadStopWrite

StartWrite

StartRead

StopReadStopWrite

StartWrite

StartRead

StopReadStopWrite

StartWrite

StartRead

StopReadStopWrite

StartWrite

StartRead

StopReadStopWrite

StartWrite

StartRead

StopReadStopWrite

StartWrite

StartRead

StopReadStopWrite

StartWrite

• From the previous lecture..• Introduction• Simple examples• Concurrent transitions• Transitions in conflict• Another example• Readers and writers

• Properties of Petri Nets

Properties of Petri NetsProperties of Petri NetsProperties of Petri NetsProperties of Petri Nets

Reachability of a given marking

k-Boundedness:

A number of tokens in each place never exceeds k for any marking M reachable from the initial marking.

K-Boundedness

A PN is live if for every marking M reachable from the initial marking M0 it is possible eventually to fire every transition in the net in some marking M’ which is reachable from M.

Liveness

An infinite firing sequence s is said to be globally fair if every transition appears in s infinitely often.

Global fairness

SummarySummarySummarySummary

Petri nets: a simple modelling tool oriented towards concurrent and distributed systems.

Timed Petri Nets

Coloured Petri Nets

Stochastic Petri Nets

Further readingsFurther readingsFurther readingsFurther readings

James L. Peterson, Petri Nets,

Computing Surveys, vol. 9, No. 3, September 1977, 223 – 252.

N. Nissanke, Realtime Systems, Prentice Hall, London, 1997.

HomeworkHomeworkHomeworkHomework

Translate the statechart given below to Petri Nets.

(in G)

HomeworkHomeworkHomeworkHomework

• Prepare a model of a file access control protocol allowing up to four concurrent readers in read-only mode or a single writer in exclusive read and write mode.

• Draw statechart counterparts of the presented Petri Nets.

Quality assessmentQuality assessmentQuality assessmentQuality assessment

1. What is your general impression? (1 - 6)

2. Was it too slow or too fast?

3. What important did you learn during the lecture?

4. What to improve and how?

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