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Petri Nets 2000

21st International Conference onApplication and Theory of Petri Nets

Aarhus, Denmark, June 26-30, 200

Introductory Tutorial

Petri Nets

Organised by

Gianfranco BalboJ org DeselKurt JensenWolfgang ReisigGrzegorz RozenbergManuel Silva

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Table of Contents

Wolfgang Reisig An Informal Introduction to Petri Nets . . . . . . . . . . . . . . . .

Grzegorz Rozenberg Elementary Net Systems . . . . . . . . . . . . . . . . . . . . . . .

J org Desel Place/Transition Nets I . . . . . . . . . . . . . . . . . . . . . . . .

Kurt Jensen Coloured Petri Nets . . . . . . . . . . . . . . . . . . . . . . . . .

Grzegorz Rozenberg Behaviour of Elementary Net Systems . . . . . . . . . . . . . . . .

Manuel Silva Place/Transition Nets II . . . . . . . . . . . . . . . . . . . . . . . .

Gianfranco BalboAn Introduction to Generalised Stochastic Petri Nets . . . . . . . . .

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Introductory Tutorial

Place/Transition-Nets I

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J org Desel, Catholic University in Eichst att

I. Introduction to place/transition-nets

II. Basic analysis techniques

I. Introduction to An exampleplace/transition nets Different features of place/tr

Place/transition-nets vs en-sy

Formal denitions

Place/transition-nets

Occurrence sequences and r

An example: a vending machine

Control structure of a vending machine

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eady for insertion

holdingcoin

dispenseitem

nsert coin

ccept coin

reject coin

ready to dispense

eady for insertion

ispenseitem

nser

ccep

rejec

eady to dispense

an en-system its behaviour

item storage ready for insertion

holding coin

refilldispense

item

insert coin

reject coin

Adding concurrency: a vending machine with capacity 1 ...

tem storage eady for insertion

holding coin

efillispense

item

nsert coin

reject coin

Adding bounded storages: a vending machine with capacity 4 ...

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equest for refill

efill item

accept coineady to dispense

ic

rc

ic ic ic ic

rc rc rc rc

refill refill refill

refillrefillrefillrefill

refill

ac ac ac acac

di di di di

refill refill refill refill

... and its behaviour

ready for insertion

holding coindispense

item

insert coin

reject coin

coins in machine

Adding unbounded counters: the control part with a counter ...


holding coin

efillispense

item

nsert coin

reject coin

Adding arc weights: the vending machine selling pairs ...

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equest for refill accept coineady to dispense

tem storage

equest for refill

eady for insertion

holding coin

efillispense

item

nsert coin

accept coin

reject coin

eady to dispense

... or storing pairs


holding coin

efillispense

item

nsert coin

reject coin

Adding limited capacities: replacing the place request for rell ...

Marked place/transition-nets generalize en-systems

Each contact-free en-system is a 1-safe marked place/transition-net

Terminology:

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Terminology:

en-system marked p/t-net

condition ; ! place

event ; ! transition

case / state ; ! marking

c conditions m : places ! f 0 1 g

sequential case graph ; ! marking graph

(reachability graph, state graph

Formal denition of marked place/transition-nets

A marked place/transition-net (p/t-net) is a tuple ( S T F

(S T F ) is a net with

S set of places (Stellen), nonempty, nite (often P

T set of transitions , nonempty, nite

The occurrence rule

A transition t is enabled at a marking m if

every place s 2 t satises m (s ) w (s t ) and

every place s 2 t satises m (s ) + w (t s ) k (s )

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2

2

32

2

3 k=4

The occurrence of t leads to the successor marking m 0 , den

m 0 (s ) =

8

>

>

>

>

>

<

>

>

>

>

>

:

m (s ) if s =2 t and s =2 t

m (s ) ; w (s t ) if s 2 t and s =2 t

m (s ) + w (t s ) if s =2 t and s 2 t

m (s ) ; w (s t ) + w (t s ) if s 2 t and s 2 t

Notation: m t

; ! m 0 (m [t i m 0 )

Occurrence sequences and reachability

A nite sequence = t 1 t 2 . . . t n of transitions is a

nite occurrence sequence leading from m 0 to m n if

m 0

t 1

; ! m 1

t 2

; ! t n

; ! m n

A m rking is re ch ble (from ) if

Marking graphs

The marking graph of a marked p/t-net is an edge-labeled graph wi

initial vertex initial marking m 0 (denoted ! )

vertices set of reachable markings [ m 0 i

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(1,1,0)t1 t2

s1

s2

s3 (0,2,0)

t2 (1,0,1)

t1

t2 (0,1,1)

t1

t2

t3

2t3

vertices set of reachable markings [labeled edges set of triples ( m t m 0 ) such that m ;

Example:

Lemma Each occurrence sequence corresponds to the

labels of a directed path of the marking graphstarting with the initial vertex,

and vice versa.

Behavioral properties of marked p/t-nets

A marked p/t-net is

terminating if there is no innite occurrence seque

deadlock-free if each reachable marking enables a tralive if each reachable marking enables

an occurrence sequence containing all

t1t2

t2

t1

t4

t3

A marked p/t-net which is not deadlock-free and its marking graph

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t3

t4

t4 t2

Proposition A marked p/t-net is deadlock-free if and only if

its marking graph has no vertex without successor

Proposition No deadlock-free marked p/t-net is terminating

(but the converse does not necessarily hold)

t2 t4

t1 t5

t3t2

t1

t3t4

t5

A deadlock-free marked p/t-net which is not live

Proposition Every live marked p/t-net is deadlock-free

Proposition A marked p/t-net is bounded if and only if

its set of reachable markings is nite

(its marking graph is nite)

Proof

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(( ) The maximal number of tokens on a place can be taken as its b

() ) If a place s is bounded by b (s ) then it can be in at most b (

m (s ) = 0 m (s ) = 1 . . . m (s ) = b (s ):

So the number of reachable markings does not exceed

(b (s 1 ) + 1) (b (s 2 ) + 1) (b (s n ) + 1)

where f s 1 s 2 . . . s n g is the (nite !) set of places

Corollary A 1safe marked p/t-net with n places has

at most 2 n reachable markings

t2 t4

t1 t5

t3

t2

t1

t3

Proposition A marked p/t-net is reversible if and only if

its marking graph is strongly connected

Example a 1safe non-live marked p/t-net which is not reversible

tem storage

eady for insertion

holding coin

efill ispenseitem

nsert coin

= 4

Substituting capacities ...

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accept coin

eject coin

eady to dispense

tem storage

equest for refill

eady for insertion

holding coin

efillispense

item

nsert coin

accept coin

reject coin

eady to dispense

... by complement places

Weak capacities. . . guarantee bounds of places

weak enabling condition:

a transitiont

is enabled at a markingm

ifevery place s 2 t satises m (s ) w (s t ) and

every place s 2 t n t satises m (s ) + w (t s) k (s ) an

Strong capacities

. . . generalize contact of en-systems

strong enabling condition:

a transition t is enabled at a marking m if

l ti ( ) ( ) d

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= 3

every place s 2 t satises m (s ) w (s t ) and

every place s 2 t satises m (s ) + w (t s ) k (s )

Proposition each en-system is equivalent to a

marked p/t-net without arc weights andwith the strong capacity restriction k (s ) = 1 for eve

Replacing a strong capacity restrictiction by a strong complem

Inhibitor arcs for null tests

inhibitor enabling condition: If (s t ) is an inhibitor arc then

t is only enabled at a marking

II. Basic Linear-algebraic techniquesanalysis The marking equationtechniques Place invariants

Transition invariants

Structural techniques

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Structural techniques

Siphons

Traps

The siphon/trap property

Restricted net classes

State machines

Marked graphs

Free-choice nets

Causal Semantics

Occurrence nets

Process nets

1

s2

s3

s4t1 t2

3

4

t5

5

Linear-algebraic representation of markings and transitions

1 s3

s4t1 t2

3

t5

Matrix representation of a net

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s2 45

incidence matrix of the net:

N

;

=

~ t 1

~ t 2

~ t 3

~ t 4

~ t 5

~ s 1 1 ; 1 0 0 0

~ s 2 ; 1 1 0 0 0

~ s 3 0 1 ; 1 0 1

~ s 4 0 0 1 ; 1 ; 1

~ s 5 0 ; 1 0 1 0

The marking equation

m 0

t 2 t 3 t 5 t 1 t 3

; ; ; ! m ) ~ m 0 + ~ t 2 + ~ t 3 + ~ t 5 + ~ t 1 + ~ t 3 = ~ m

~ m 0 + (1 ~ t 1 ) + (1 ~ t 2 ) + (2 ~ t 3 ) + (0 ~ t 4 ) +

~ m 0

+

N ;

(1 1 2 0 1)| { z } Parikh vector of t 2 t 3 t 5 t 1 t 3

=~ m

t1

2

3

t4

1

2

3

4

5

5

Example: a live and 1safe marked p/t-net

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2 t43 5

reachable markings solutions to the marking equation

(1 0 0 0 0) (0 0 0 0 0) (1 0 1 0 1)

(0 1 0 0 1) (1 0 0 0 0) . . .

(0 0 1 1 0) (0 1 0 0 0) . . .

(0 0 0 1 1) (1 0 1 0 0) (0 1 0 1 0) .

non-reachable marking solutions to the marking equation

(0 1 1 0 0) (1 1 0 0 1 ) . . .

Consequence: solubility of the marking equation is not sufcient for r

s1 s2s3

t1

s4

t3

s5

process 1 process 2

Place invariants

Example: mutual exclusion

Place invariants

Three equivalent denitions:

A place invariant of a net N is a vector ~ i satisfying

(1)X

~ i s =

X

~ i s for every transition t of N

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s 2

t s 2 t

(2) ~ i ~ t = 0 for every transition t of N

(3) ~ i

N ;

= (0 0 . . . 0)

The token conservation law for a place invariant ~ i :

If m is reachable from m 0 then ~ i ~ m 0 = ~ i ~ m

Proof: m 0

; ! m ) ~ m 0 +

N ;

P [ ] = ~ m

) ~ i ~ m

0 + ~ i

N ;

| { z }

= ( 0 . . . 0 )

P [ ] = ~ i ~ m

) ~ i ~ m

0 = ~ i ~ m

s1 s2s3

t1

t2

s4

t3

t4

s5

process 1 process 2

Proving stability

s1

s2

s3

t1

s4

t3

s5

process 1 process 2

Further place invariants

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t2 t4

(0 1 1 1 0) mutual exclusion

(0 1 1 0 ; 1) m (s 2 ) + m (s 3 ) = m (s 5 )

if s 2 is marked then s 5 is marked

(1 1 0 0 0) m (s 1 ) + m (s 2 ) = 1

m (s 1 ) m (s 2 ) 1, the places s 1 and s 2 are bou

A necessary condition for liveness

Proposition: In a live marked p/t-net without isolated places,

each place invariant ~ i without negative entries and

with some positive entry ~ i s satises ~ i ~ m 0 > 0.

Proof: otherwise transitions in s s are dead.

A sufcient condition for boundedness:

Proposition: Each marked p/t-net with a place invariant ~ i satis~ i

s > 0 for each place s is bounded.

Proof: m is reachable ) ~ i ~ m = ~ i ~ m 0 .~ ~ ~

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1 23

4

1

2

t3

t4

s5

) ~ i

s ~ m

s ~ i ~ m = ~ i ~ m 0 .

) m (s ) = ~ m s ~ i ~ m

0

~ i s

Example: place invariant (1 2 1 2 1)

3

s1

Place invariants and the marking equation

Proposition There is a place invariant ~ i satisfying ~ i ~ m 0 6= ~ i ~m

~ m 0 +

N ;

~x = ~ m has no rational-valued solution for

Example:

1 t3

Transition invariantsA transition invariant of a net N is a vector ~ j satisfying

N ;

~ j = (0 0 . . . 0)

Example:

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1 23

4

2 t4

s5

Transition invariants: (1 1 0 0), (0 0 1 1), (2 2 1 1)

Proposition Let m 0

; ! m be an occurrence sequence.

m 0 = m if and only if P [ ] is a transition invariant

Proof: follows immediately from ~ m 0 +

N ;

P [ ] = ~ m

A necessary condition for liveness and boundedness

Proposition: Each live and bounded marked p/t-net

has a transition invariant ~ j satisfying~ j (t ) > 0 for each transition t .

Proof: By liveness, there exist occurrence sequences

Structural Techniques

A siphon is a set of places which, once unmarked, never gains a token

S is a siphon if S S , i.e. if t \ S 6= implies t \ S 6= .

A trap is a set of places which, once marked, never looses all tokens

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a siphon{s1,s2}

a trap{s3, s4}

s1 s2 s3 s4

S is a trap if S S , i.e. if t \ S 6= implies t \ S 6= .

If a marking m satises m (s 1 ) = m (s 2 ) = 0 then so do all follower

If a marking m satises m (s 3 ) + m (s 4 ) > 0 then so do all follower

1

2

3

t4

1

2

3

s4

5

t5

Example for the use of a trap

Siphons and traps, liveness and deadlock-freedom

Proposition: In a live marked p/t-net without isolated places,

each nonempty siphon contains an initially marked pl

Proof: otherwise, for each place s of the siphon, all trans

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Proposition: Assume a marked p/t-net with some transition,

without capacity restrictions and arc weights.If each nonempty siphon includes an initially marked

then the marked p/t-net is deadlock-free

Proof: the set of unmarked places at a dead marking is a no

This siphon contains no marked trap.) It contains no initially marked trap.

Restricted net classes

State machines are marked p/t-nets without branched transitions,

i.e. j t j = j t j = 1 for each transition,

without arc weights andwithout capacity restrictions.

Marked graphs are marked p/t-nets without branched places,

i.e. j s j = j s j = 1 for each place,

without arc weights and

without capacity restrictions.

Example

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Proposition A marked graph is live if and only if

each cycle carries a token initially.

Proposition It is moreover 1safe if and only if

each place belongs to a cycle with exactly one token.

Free-choice nets are marked p/t-netswithout arc weights and capacity restrictions sati

(s t ) 2 F ) t s F for each place s

If then

produce send receive consume

produced

full

consumed

Causal semantics of marked p/t-nets

Example: a producer / consumer system

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sent received

ent

onsumed

end

roduce

roduced ent

ull ullroduce

roduced send sent

eceive

eceived

consume

onsumed

receive

eceived

a causal run of the producer /consumer system

Causal runsA causal run of a marked p/t-net is given by a labeled Petri net (

Interpretation of causal runs

net element name symbol interpretation

Occurrence netsAn occurrence net is a net ( B E K ) with the following properti

it has no cycles (i.e. K + is a partial order )

it has no branched places, i.e.

j

b j j b

j 1 for each condition b

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events have nite fan-in and fan-out, i.e.

e and e are nite sets for each event e

it has neither input nor output-events, i.e.j

e j j e

j 1 for each event e

no node has innitely many predecessors, i.e.

the set f x 2 (B E ) j x y g is nite for each node y

Process nets of marked p/t-nets represent causal runsAssume a marked p/t-net ( S T F k w m 0 ) without capacity restr

An occurrence net ( B E K ) together with

labels : (B E ) ! (S T ) is a process net of N if

sorts of nodes are respected by , i.e.

(B ) S and (E ) T

Occurrence sequences versus process nets

occurrence provide total orders of events that respect causality

sequences but add arbitrary interleavings of independent events.

Information about causal relationships can get lost.

process nets provide partial orders reecting causality.

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d a

e b

f

g

c h

d a

e b

f

g

c

f

d a

e b

f

g

cf

Example:

a b c

b a c

a c b

b c a

maximal occurrence sequences maximal process nets

s1 s3 s4

s2 s6

s5

t1 t2

t3

t5

t4

b4 b10s4 s4 b1s4

Two process nets corresponding to t 1 t 3 t 5 t 2 t 3 t 5 t 2

s1 s3 s4

s2 s6

s5

t1 t2

t3

t5

t4

b4 b104 s4 b14

Two process nets without a common occurrence sequence

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b8

b3

b1 e1

b4

b5 e2 b6

e3

b2

e4

b14

b7

b9

e5

b10

b11 e6 b12

e7

b13s3 t3

s4

s5 t5 s6 t2

s1

s3 t3

s4

s5 t5 s6 t2 s3

s2s2 t1

s1

s2

e8

b1

b16t3

s4

s5

b2

b8

b1

b3

e1

b4

b5 e2 b6

e3

b7

s3 t3

s4

s5 t5 s6 t2 s3

s2

s1

s2

e4

b9

b10 e6 b11

t3

s4

s5 t4 s3

b12e5t1 s1

e7

b13

b14 e8 b15

t3

s4

s5 t5 s6

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C o l o u r e d P e t r i N e t s 4

f e n a b l e d b i n d i n g

B P

2 2 ` ( p

, 0 )

T 2 C P

B P

2 2 ` ( p

, 0 )

T 2 C P

B P

1 1 ` ( p

, 0 )

T 2 C P

1 1 ` ( p

, 0 )

( x , i )

( x , i ) 1

1 ` ( p

, 0 )

( x , i ) 1

1 ` ( p

, 0 )

( x , i )

( x , i )

( x , i )

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C o l o u r e d P e t r i N e t s

3

R e s o u r c e a l l o c a t i o n e x a m p l e

A

P

3 ` ( q

, 0 )

B

P

2 ` ( p

, 0 )

C

P

D

P

E

P T 1

[ x = q

]

T 2 T 3 T 4 T 5

R E

1 ` e D e c

l a r a

t i o n s :

t y p e

U = w

i t h p

| q ;

t y p e

I = i n t ;

t y p e

P = p r o d u c

t U * I ;

t y p e

E = w

i t h e ;

v a r x : U ;

v a r

i : I ;

S E

3 ` e

T E

2 ` e

( x , i )

( x , i )

( x , i )

( x , i )

( x , i )

( x , i )

( x , i )

( x , i )

( x , i )

i f x = q

t h e n

1 ` ( x , i +

1 )

e l s e e m p

t y

e i f x = q

t h e n

1 ` e

e l s e e m p t y

c a s e x o

f

p = >

2 ` e

| q = >

1 ` e

2 ` e

e

i f x = p

t h e n

1 ` e

e l s e e m p t y e

c a s e x o f

p = >

2 ` e

| q = >

1 ` e

i f x = p

t h e n

1 ` ( x , i +

1 )

e l s e e m p

t y

O c c u r r e n c e o f

S

E

3 3 ` e

S

E

3 3 ` e

B i n d i n g :

< x = p , i

= 0 >

S

E

1 1 ` e

N e w

M a r

k i n g

c a s e x o f

p = >

2 ` e

| q = >

1 ` e

c a s e x o f

p = >

2 ` e

| q = >

1 ` e

2

2 ` e

c a s e x o f

p = >

2 ` e

| q = >

1 ` e

112


p l e x e x a m p l e

B

P

3

1 ` ( p

, 2 ) + 1 ` ( p

, 4 ) + 1 ` ( q

, 3 )

T 2 C

P

B

P

3

1 ` ( p

, 2 ) + 1 ` ( p

, 4 ) + 1 ` ( q

, 3 )

T 2 C

P

B

P

3

1 ` ( p

, 2 ) + 1 ` ( p

, 4 ) + 1 ` ( q

, 3 )

T 2 C

P ( x , i )

( x , i )

1 1 ` ( p

, 2 )

( x , i )

1 1 ` ( p

, 2 )

( x , i )

1 1 ` ( q

, 3 )

( x , i )

1 1 ` ( q

, 3 )

( x , i )

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5

B i n d i n g w h i c h i s n o t e n a b l e d

S

E

3 3 ` e

B

P

2 2 ` ( p

, 0 )

T 2 C

P

S

E

3 3 ` e

B

P

2 2 ` ( p

, 0 )

T 2 C

P

B i n d i n g :

< x = q , i

= 2 >

( x , i )

c a s e x o f

p = >

2 ` e

| q = >

1 ` e

( x , i )

1 1 ` ( q

, 2 )

( x , i )

c a s e x o f

p = >

2 ` e

| q = >

1 ` e

1

1 ` e

1 1 ` ( q

, 2 )

( x , i )

B i n d i n g c a n n o

t o c c u r

A m o r e c o m p

S

E

3 3 ` e

S

E

3 3 ` e

B i n d i n g :

< x = p ,

i = 2 >

S

E

3 3 ` e

B i n d i n g :

< x = q ,

i = 3 >

c a s e x o f

p = >

2 ` e

| q = >

1 ` e

c a s e x o f

p = >

2 ` e

| q = >

1 ` e

2

2 ` e

c a s e x o f

p = >

2 ` e

| q = >

1 ` e

1

1 ` e

113


B

P

3

1 ` ( p

, 2 ) + 1 ` ( p

, 4 ) + 1 ` ( q

, 3 )

T 2 C

P

B

P

3

1 ` ( p

, 2 ) + 1 ` ( p

, 4 ) + 1 ` ( q

, 3 )

T 2 C

P

1 1 ` ( p

, 2 )

( x , i )

1 1 ` ( p

, 2 )

( x , i )

1 1 ` ( p

, 4 )

( x , i )

1 1 ` ( p

, 4 )

( x , i )

g s c a n n o t o c c u r c o n c u r r e n t l y .

tt h e y n e e d

t h e s a m e t o k e n s .

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7

C o n c u r r e n c y

S

E

3 3 ` e

B P

3 1 ` ( p

, 2 ) + 1 ` ( p

, 4 ) + 1 ` ( q

, 3 )

T 2 C P

B i n d i n g :

< x = p , i

= 2 >

S

E

B P

1 1 ` ( p

, 4 )

T 2 C P

2 1 ` ( p

, 2 ) + 1 ` ( q

, 3 )

B i n d i n g :

< x = q , i

= 3 >

2 1 ` ( p

, 2 ) + 1 ` ( q

, 3 )

( x , i )

c a s e x o f

p = >

2 ` e

| q = >

1 ` e

3

3 ` e

2 1 ` ( p

, 2 ) + 1 ` ( q

, 3 )

( x , i )

( x , i )

c a s e x o f

p = >

2 ` e

| q = >

1 ` e

( x , i )

T h e t w o

b i n d

i n g s m a y o c c u r c o n c u r r e n t l y .

T h i s i s p o s s i

b l e

b e c a u s e

t h e y u s e d i f f e r e n t

t o k e n s .

C o n f l i c t

S

E

3 3 ` e

B i n d i n g :

< x = p ,

i = 2 >

S

E

3 3 ` e

B i n d i n g :

< x = p ,

i = 4 >

c a s e x o f

p = >

2 ` e

| q = >

1 ` e

2

2 ` e

c a s e x o f

p = >

2 ` e

| q = >

1 ` e

2

2 ` e

T h e s e

t w o

b i n d

i n g

T h e r e a s o n

i s t h a t

t

114


1 0

tr i N e t s

o p e r a t i o n s a n

d v a r i a b l e s .

f o l l o w

i n g i n s c r i p

t i o n s :

fic a t

i o n ) .

ify i n g

t h e t y p e o f

t o k e n s w

h i c h

p l a c e ) .

u l t i - s e

t o f t o k e n c o

l o u r s ) .

t h e

f o l l o w i n g

i n s c r i p

t i o n s :

fic a t

i o n ) .

x p r e s s

i o n c o n t a i n i n g s o m e o f

l l o w

i n g i n s c r i p

t i o n s :

o n t a i n i n g s o m e o f

t h e v a r i a

b l e s ) .

r e s s

i o n i s e v a l u a

t e d i t y i e l

d s a

c o l o u r s .

8/3/2019 Petri Nets 2000

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C o l o u r e d P e t r i N

e t s

9

R e s o u r c e a l l o c a t i o n s y s t e m

T w o

k i n d s o

f p r o c e s s e s :

T h r e e c y c l

i c q - p r o c e s s e s ( s t a t e s

A , B , C , D

a n d E )

.

T w o c y c l

i c p - p r o c e s s e s

( s t a t e s B , C , D

a n d E ) .

T h r e e

k i n d s o

f r e s o u r c e s :

R e p r e s e n t e d

b y t h e p l a c e s

R , S

a n d T .

D u r i n g a c y c l e a p r o c e s s r e s e r v e s s o m e r e s o u r c e s

a n d r e l e a s e s

t h e m a g a i n :

T o k e n s a r e r e m o v e d

f r o m a n

d a d d e d t o t h e

r e s o u r c e p l a c e s

R , S

a n d T .

A c y c l e c o u n t e r

i s i n c r e a s e d e a c h

t i m e a p r o c e s s

c o m p l e t e s a

f u l l c y c l e .

I t i s r a

t h e r s t r a

i g h t f o r w a r

d t o p r o v e

t h a t

t h e

r e s o u r c e a l

l o c a

t i o n s y s t e m

c a n n o t d e a d l o c k .

W h a t h a p p e n s

i f w e a d d

a n a d

d i t i o n a

l t o k e n t o

p l a c e

S

i . e . ,

i f w e s t a r

t w i t h f o u r

S - r e s o u r c e s

i n s t e a

d o f

t h r e e ?

C o l o u r e d P e t r

D e c l a r a t i o n s :

T y p e s , f u n c t i o n s , op

E a c h p l a c e

h a s t

h e f

N a m e

( f o r

i d e n

t i f i c

C o l o u r s e t

( s p e c i

f y

m a y r e s

i d e o n

t h ep

I n i t i a l m a r k i n g

( m

E a c h t r a n s i t i o n

h a s t

N a m e

( f o r

i d e n

t i f i c

G u a r d ( b o o

l e a n e x

t h e v a r i a

b l e s

) .

E a c h a r c

h a s t

h e f o l

A r c e x p r e s s i o n

( c o

W h e n t h e a r c e x p r

m u l t i - s e t o f

t o k e n

115


1 2

t e r i s t i c s o f C P - n e t s

t a n d g r a p h i c s .

e t i n s c r i p t i o n s a r e s p e c

i f i e d

b y

la n g u a g e , e . g . , a p r o g r a m m i n g

o p e r a t

i o n s , v a r i a

b l e s a n

d

is t s o f p l a c e s , t

r a n s

i t i o n s a n d

p a r t i

t e g r a p h

) .

t r e a d a b l e i t i s i m p o r t a n t

t o

h i c a

l l a y o u

t .

o u t h a s n o f o r m a l m e a n i n g .

am e

k i n d o

f c o n c u r r e n c y

/ T r a n s

i t i o n

N e t s .

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1 1

E n a b l i n g a n d o c c u r r e n c e

A b i n d i n g a s s i g n s a c o l o u r

( i . e . , a

v a l u e )

t o e a c

h

v a r i a b l e o f a

t r a n s

i t i o n .

A b i n d i n g e l e m e n t i s a p a

i r ( t , b ) w

h e r e

t i s a

t r a n s i t i o n w

h i l e b i s a b i n d i n g

f o r t

h e v a r i a

b l e s

o f t . E

x a m p l e :

( T 2 , < x = p ,

i = 2 > ) .

A b i n d i n g e l e m e n

t i s e n a b l e d

i f a n

d o n

l y i f :

T h e r e a r e e n o u g h t o k e n s

( o f t h e c o r r e c

t c o l o u r s o n

e a c

h i n p u

t - p l a c e

) .

T h e g u a r d e v a l u a

t e s t o

t r u e .

W h e n a

b i n d

i n g e l e m e n t

i s e n a b

l e d i t m a y o c c u r :

A m u l

t i - s e

t o f t o k e n s i s r e m o v e d

f r o m e a c h

i n p u

t - p l a c e .

A m u l

t i - s e

t o f t o k e n s i s a d d e d t o e a c h

o u t p u

t - p l a c e .

A b i n d i n g e l e m e n

t m a y o c c u r c o n c u r r e n t l y t o o t h e r

b i n d i n g e l e m e n

t s

i f f t h e r e a r e s o m a n y

t o k e n s

t h a t e a c

h b i n d

i n g e l e m e n

t c a n g e

t i t s " o w n s h a r e " .

M a i n c h a r a c t

C o m

b i n a t

i o n o f t e x t

D e c l a r a t i o n s a n

d n e

m e a n s o f a

f o r m a l

l a

l a n g u a g e .

T y p e s ,

f u n c

t i o n s ,

e x p r e s s

i o n s .

N e t s t r u c t u r e c o n s

i s

a r c s

( f o r m

i n g a

b i - p

T o m a k e a

C P - n e

t

m a k e a n i c e g r a p

h

T h e g r a p h

i c a l

l a y o

C P - n e t s h a v e

t h e s a m

p r o p e r t i e s a s

P l a c e /

116


1 4

n i t i o n o f b e h a v i o u r

m u l t i - s e t o f b i n d i n g e l e m e n t s

.

a m a r

k i n g

M i f f t h e

f o l l o w

i n g p r o p e r

t y i s

b > M ( p ) .

d i n a m a r

k i n g

M 1

i t m a y o c c u r , c h a n g i n g

e r m a r

k i n g

M 2 ,

d e f i n e d

b y :

( p )

( t , b )

Y E (

p , t ) < b > ) +

( t , b )

Y E

( t , p )

< b > .

t h e r e m o v e d

t o k e n s w

h i l e t h e s e c o n d

i s

s . M o r e o v e r w e s a y

t h a t

M 2

i s d i r e c

t l y

h e o c c u r r e n c e o f

t h e s t e p

Y , w

h i c h w e a l s o

e i s a s e q u e n c e o f m a r

k i n g s a n d s t e p s :

M

n [ Y

n M

n + 1

f o r a l

l i 1 . . n

. W e

t h e n s a y

t h a t

M n + 1 i s

e u s e [ M t o d e n o t e t h e s e t o f m a r k i n g s

M .

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1 3

F o r m a l d e f i n i t i o n

o f C P - n e t s

D e f i n i

t i o n : A C o l o u r e d P e t r i N e t i s a t u p l e C P N = (

, P , T ,

A , N ,

C , G

, E , I

) s a

t i s f y i n g

t h e

f o l l o w i n g r e q u

i r e m e n

t s :

( i )

i s a

f i n i t e s e

t o f n o n - e m p

t y t y p e s , c a

l l e d c o

l o u r s e

t s .

( i i )

P i s a

f i n i t e s e

t o f p l a c e s .

( i i i )

T i s a

f i n i t e s e

t o f t r a n s i t i o n s .

( i v )

A i s a f i n i t e s e t o f a r c s s u c h t h a t :

P T =

P A =

T A =

.

( v )

N i s a n o

d e f u n c

t i o n .

I t i s d e f i n e d

f r o m

A i n t o P T

T

P .

( v i )

C i s a c o

l o u r

f u n c

t i o n .


f r o m

P i n t o

.

( v i i ) G i s a g u a r

d f u n c t i o n . I t i s d e f i n e d f r o m T i n t o e x p r e s s i o n s

s u c h

t h a t :

t T : [ T y p e (

G ( t ) ) = B o o

l

T y p e

( V a r

( G ( t ) ) )

] .

( v i i i ) E i s a n a r c e x p r e s s i o n

f u n c

t i o n .


f r o m

A

i n t o

e x p r e s s i o n s s u c

h t h a t :

a

A : [

T y p e

( E ( a ) ) = C

( p ( a ) )

M S

T y p e

( V a r

( E ( a ) ) )

]

w h e r e p ( a )

i s t h e p l a c e o f N

( a ) .

( i x )

I i s a n

i n i t i a l i z a

t i o n

f u n c t

i o n .

I t i s d e f i n e

d f r o m

P

i n t o

c l o s e d e x p r e s s i o n s s u c

h t h a t :

p

P : [ T y p e (

I ( p )

) = C ( p ) M

S ] .

F o r m a l d e f i n

D e f

i n i t i o n : A s t e p i s a m

A s t e p

Y i s e n a b

l e d i n a

s a t i s f i e d : p

P :

( t , b )

Y E

( p , t )

< b

W h e n a s t e p

Y i s e n a b

l e d

t h e m a r k

i n g

M 1

t o a n o t

h e

p

P : M

2 ( p ) =

( M 1 (

T h e

f i r s t

s u m

i s c a

l l e d t

c a l l e d t h e a d

d e d

t o k e n s

r e a c

h a b l e f r o m

M 1

b y t h

d e n o

t e : M

1 [ Y M

2 .

A n o c c u r r e n c e s e q u e n c e

M 1 [

Y 1

M 2

[ Y 2

M 3

s u c h

t h a t M i [

Y i M i + 1

f

r e a c

h a b l e f r o m M

1 . W e

w h i c h a r e r e a c

h a b l e

f r o m

117


1 6

o n t r a l o w - l e v e l n e t s

e t w e e n

C P - n e t s a n d

e t s (

P T - n e

t s ) i s a n a l o g o u s

t o

t w e e n

h i g h - l e v e

l p r o g r a m m

i n g

e m b l y c o d e .

o l e v e

l s h a v e e x a c

t l y t h e s a m e

o w e r .

l e v e

l l a n g u a g e s h a v e m u c

h

p o w e r b e c a u s e

t h e y

h a v e

g f a c i

l i t i e s , e . g . , t

y p e s a n

d

n e q u i v a l e n t P T - n e

t a n

d v i c e

i s u s e d

t o d e r i v e

t h e

d e f i n i t i o n

ie s a n

d t o e s t a b

l i s h t h e

ho d s

.e v

e r t r a n s l a t e a

C P - n e

t i n t o a

v e r s a .

u l a t

i o n a n d v e r i f

i c a t

i o n a r e

d o n e

o f C P - n e t s .

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1 5

F o r m a l d e f i n i t i o n

T h e e x

i s t e n c e o f a f o r m a l d e f i n i t i o n

i s v e r y

i m p o r

t a n t :

I t i s t h e

b a s i s f o r s i m u l a t i o n ,

i . e . ,

e x e c u t

i o n o

f t h e

C P - n e

t .

I t i s a l s o

t h e

b a s i s f o r t h e f o r m a l v e r i f i c a t i o n

m e t

h o d s ( e

. g . ,

s t a t e s p a c e s a n d p l a c e

i n v a r i a n

t s ) .

W i t h o u

t t h e f o r m a l

d e f i n i t i o n , i

t w o u

l d h a v e

b e e n

i m p o s s

i b l e t o o b t a

i n a s o u n d n e

t c l a s s .

I t i s n o t n e c e s s a r y

f o r a u s e r

t o k n o w

t h e

f o r m a

l

d e f i n i t i o n o f

C P - n e

t s :

T h e c o r r e c

t s y n t a x

i s c h e c

k e d b y t h e

C P N e d i

t o r ,

i . e . , t

h e c o m p u

t e r t o o l b y w

h i c h

C P - n e

t s a r e

c o n s t r u c t e d .

T h e c o r r e c

t u s e o f

t h e s e m a n t i c s

( i . e . , t

h e

e n a

b l i n g r u

l e a n

d t h e o c c u r r e n c e r u

l e ) i s

g u a r a n t e e

d b y t h e

C P N s i m u l a t o r a n

d t h e

C P N t o o l s f o r

f o r m a l v e r

i f i c a

t i o n .

H i g h - l e v e l c o

T h e r e l a t

i o n s

h i p

b e

P l a c e / T r a n s

i t i o n

N e

t h e r e l a

t i o n s

h i p

b e t

l a n g u a g e s a n

d a s s e

I n t h e o r y ,

t h e

t w o

c o m p u t a t i o n a l p o

I n p r a c t

i c e ,

h i g h -

m o r e m o d e l l i n g p

b e t t e r s t r u c t u r

i n g

m o d u

l e s .

E a c h C

P - n e

t h a s a n

v e r s a .

T h i s e q u i v a

l e n c e

o f b a s i c p r o p e r t i e

v e r i f i c a t i o n m e t h o

I n p r a c t

i c e , w e n e

P T - n e t o r v i c e v

D e s c r i p

t i o n , s i m u

d i r e c t l y i n t e r m s o

118

C o l o u r e d P e t r i N e t s 1 8

to c o l

T r a n s m

i t

P a c

k e t

R e c e i v e

P a c

k e t

T r a n s m

i t

A c k n o w .

R e c e i v e

d

D A T A

" "

N e x

t R e c

I N T

1

B

I N T x D A T A

C I N T

R P

8

I n t_ 0_ 1 0

R A

I n t_ 0_ 1 0 8

N e

t w o r k

R e c e

i v e r

T =

i n t ;

O O L =

b o o l ;

A T A = s t r i n g ;

T x D A T A = p r o d u c t

I N T * D A T A ;

: I N T ;

tr : D

A T A ;

= " # # # # # # # # " ;

t 0_ 1 0 =

i n t w i t h 0 .

. 1 0 ;

t 1_ 1 0 =

i n t w i t h 1 .

. 1 0 ;

In t_ 0_ 1 0 ;

In t_ 1_ 1 0 ;

(s : I n t

_ 0_ 1

0 , r :

I n t_ 1_ 1

0 ) =

( r s ) ;

p )

i f O k ( s , r )

t h e n

1 ` ( n

, p )

e l s e e m p t y

( n , p

)

i f n =

k

a n d a l s o

p < > s

t o p

t h e n s t r ^ p

e l s e s t r

i f n =

k

t h e n

k + 1

e l s e

k

n

k i f n =

k

t h e n

k + 1

e l s e

k

( s , r

)1 `

ne m

p t y

s t r

s s

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1 7

O t h e r k i n d s o f h i g h - l e v e l n e t s

C P - n e t s h a v e

b e e n

d e v e l o p e

d f r o m

P r e d i c a

t e / T r a n s

i t i o n

N e t s .

H a r t m a n n G e n r i c h & K u r t L a u t e n b a c h .

W i t h r e s p e c

t t o d e s c r i p t i o n a n

d s i m u l a t i o n t h e

t w

o m o d e l s a r e n e a r

l y i d e n t

i c a l .

W i t h r e s p e c

t t o f o r m a l v e r i f i c a t i o n

t h e r e a r e

s o m e

d i f f e r e n c e s .

S e v e r a l o t

h e r k

i n d s o f

h i g h - l e v e

l P e t r i

N e t s e x i s t .

T h e y a l

l b u i

l d u p o n

t h e s a m e b a s i c i d e a s ,

b u t u s e

d i f f e r e n t l a n g u a g e s

f o r d e c l a r a t

i o n s a n

d

i n s c r i p

t i o n s .

A

d e t a i l e d c o m p a r i s o n

i s o u

t s i d e

t h e s c o p e o f

t h i s t a l k

.

S i m p l e p r o t o

S e n

d

P a c

k e t

R e c e i v e

A c k n o w .

S e n

d

I N T x D A T A

1 ` ( 1

, " M o d e l

l i n " ) +

1 ` ( 2

, " g a n

d A n "

) +

1 ` ( 3

, " a l y s

i s b " ) +

1 ` ( 4

, " y

M e a n s

" ) +

1 ` ( 5

, " o f

C o l o u

" ) +

1 ` ( 6

, " r e

d P e t r "

) +

1 ` ( 7

, " i N e t s #

# " ) +

1 ` ( 8

, " # # # # # # # # " )

N e x

t S e n

d

I N T 1

D I N T A

I N T x D A T A

S e n

d e r

t y p e

I N T

t y p e

B O

t y p e

D A

t y p e

I N T

v a r n , k

v a r p , s

t r

v a l s t o p

t y p e

I n t_

t y p e

I n t_

v a r s : I n

v a r r : I n

f u n

O k ( s

( n , p

)

( n , p)

( n , p

)

i f O k (

t h e n

1

e l s e e

n

k

n n

119


c k e t

r a n s m

i t

P a c

k e t

R P

1 1 ` 8

B

I N T x D A T A

i f O k ( s

, r )

t h e n

1 ` ( n , p )

e l s e e m p t y

s

e n a b l e d b i n d i n g s :

th e

f o r m

"M o d e l

l i n " , s =

8 , r =

> .

a n t a k e 1 0 d i f f e r e n t v a l u e s ,

e o f r i s d e f

i n e d

t o c o n t a i n

t h e

s, r ) c h e c k s w

h e t h e r r s .

s, r ) = t r u e .

T h i s m e a n s

t h a t

t h e

f r o m

A t o B

, i . e . ,

t h a t

t h e p a c k e t

i s

n s m i t t e d o v e r

t h e n e

t w o r

k .

k( s , r ) = f a l s e . T

h i s m e a n s

t h a t n o

o B

, i . e . , t h

a t t h e p a c k e t

i s l o s t .

la t o r m a k e r a n d o m c h o i c e s

d b i n d

i n g s .

H e n c e

t h e r e

i s 8 0 %

e s s f u l t r a n s f e r .

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1 9

S e n d p a c k e t

S e n

d

I N T x D A T A

8 1 ` ( 1

, " M o d e l

l i n " )

+ 1 ` ( 2

, " g a n

d A n "

)

+ 1 ` ( 3

, " a l y s

i s b " )

+ 1 ` ( 4

, " y

M e a n s

" )

+ 1 ` ( 5

, " o f

C o l o u "

)

+ 1 ` ( 6

, " r e

d P e t r " )

+ 1 ` ( 7

, " i N e t s #

# " )

+ 1 ` ( 8

, " # # # # # # # # " )

S e n

d

P a c

k e t

A I N T x D A T A

N e x

t S e n

d

I N T

1

1 1 ` 1 ( n

, p )

( n , p

) n

O n l y t h e

b i n d

i n g

< n = 1 , p

= " M o d e l

l i n " > i s e n a b l e d .

W

h e n

t h e

b i n d

i n g o c c u r s

i t a d

d s a

t o k e n

t o

p l a c e

A . T

h e t o k e n r e p r e s e n t s t

h a t t h e p a c k e t

( 1 , " M o d e l

l i n " ) i s s e n t t o

t h e n e

t w o r

k .

T h e p a c k e t

i s n o t r e m o v e

d f r o m p l a c e S e n d a n d

t h e N e x t S e n d c o u n

t e r i s n o t c h a n g e d .

T r a n s m i t p a c

A

I N T x D A T A

1 1 ` ( 1

, " M o d e l

l i n " )

T r P 8

I n t_ 0_ 1 0

( n , p

)

T h e r e a r e n o w 1 0 e

T h e y a r e a l

l o f t h

< n = 1 , p

= " M

T h e v a r

i a b l e r c a

b e c a u s e

t h e

t y p e

i n t e g e r s 1 . . 1

0 .

T h e f u n c t i o n

O k ( s ,

F o r r

1 . . 8 , O

k ( s ,

t o k e n i s m o v e d

f

s u c c e s s f u l l y t r a n

F o r r

9 . . 1

0 , O k (

t o k e n i s a d

d e d t o

T h e C

P N s i m u l a

b e t w e e n e n a b

l e d

c h a n c e

f o r s u c c e

120


k e t n u m b e r

) R e c e i v e

P a c

k e t

R e c e i v e

d

1 1 ` " M o d e l

l i n g a n

d A n "

p )

i f n =

k

a n d a l s o

p < > s t o p

t h e n s

t r ^ p

e l s e s

t r

1 1 ` " M o d e l

l i n g a n

d A n a

l y s i s

b "

i f n =

k

t h e n

k + 1

e l s e

k

1 1 ` 4

s t r 1

n d A n "

p a c k e t

i s c o n c a t e n a t e d

t o t h e

d a t a

d . u n t e r

i s i n c r e a s e d

b y o n e .

e m e n t m e s s a g e

i s s e n t . I

t c o n

t a i n s

h e n e x t p a c

k e t w h i c h

t h e r e c e

i v e r

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2 1

R e c e i v e p a c k e t

B

I N T x D A T A

1

1 ` ( 1

, " M o

d e

l l i n

" ) R e c e

i v e

P a c

k e t

N e x t

R e c

I N T

1

1

1 ` 1

R e c e

i v e

d

D A T A

" "

1

1 ` " "

C I N T

( n , p

)

i f n =

k

a n

d a

l s o

p < > s t o p

t h e n s t r ^ p

e l s e s

t r

i f n =

k

t h e n

k + 1

e l s e

k

k i f n =

k

t h e n

k + 1

e l s e

k

s t r

I t i s c

h e c k e d w

h e t h e r

t h e n u m

b e r o

f t h e i n c o m i n g

p a c k e t n m a t c h e s

t h e n u m

b e r o

f t h e e x p e c t e d

p a c k e t k

.

C o r r e c t p a c k

B

1 1 ` ( 3

, " a l y s

i s b " )

N e x

t R e c

1

1 ` 3

C

( n , p)

1

1 ` ( 3

, " a l y s

i s b " )

k

1 1 ` 3

i f n =

k

t h e n

k + 1

e l s e

k 1

1 ` 4

1 ` " M o d e l

l i n g a n

T h e d a t a

i n t h e p

a l r e a

d y r e c e

i v e d

T h e N e x t R e c c o u

A n a c k n o w l e d g e

t h e n u m

b e r o

f t h

w a n t s t o g e

t .

121


c k n o w l e d g e m e n t

C I N T

1

1 ` 2

r a n s m

i t

c k n o w .

R A

1

1 ` 8

n

s

o r k s i n a s i m

i l a r w a y a s

la c e R A d e t e r m

i n e s

t h e s u c c e s s

is s i o n o f a c k n o w

l e d g e m e n

t s .

i n s a

t o k e n w

i t h v a

l u e

8 , t h e

8 0 % .

i n s a

t o k e n w

i t h v a

l u e

1 0 , n o

e n t s a r e

l o s t .

i n s a

t o k e n w

i t h v a

l u e

0 , a l l

e n t s a r e

l o s t .

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2 3

W r o n g p a c k e t n u m b e r

B

1

1 ` ( 2

, " g a n

d A n

" ) R e c e

i v e

P a c k e

t

N e x t

R e c

1

1

1 ` 3

R e c e

i v e

d

1

1 ` " M o

d e

l l i n g a n

d A n

"

C

( n , p

)

1

1 ` ( 2

, " g a n

d A n

" )

i f n =

k

a n

d a

l s o

p < > s

t o p

t h e n s t r ^ p

e l s e s t r

1 1 ` " M o

d e

l l i n g a n

d A n

"

i f n =

k

t h e n

k +

1

e l s e

k

1 1 ` 3

k

1

1 ` 3

i f n =

k

t h e n

k + 1

e l s e

k

1

1 ` 3

s t r 1

1 ` " M o

d e

l l i n g a n

d A n

"

T h e d a t a i n t h e p a c k e t

i s i g n o r e d .

T h e N e x t R e c c o u n

t e r i s u n c h a n g e

d .

A n a c k n o w l e d g e m e n t m e s s a g e

i s s e n t . I

t c o n

t a i n s

t h e n u m

b e r o

f t h e n e x t p a c k e t w

h i c h

t h e r e c e

i v e r

w a n

t s t o g e

t .

T r a n s m i t a c T

r A c

D I N T

I n t_ 0_ 1 0

8

i f O k ( s , r )

t h e n

1 ` n

e l s e e m p

t y

T h i s t r a n s i

t i o n w o

T r a n s m i t P a c k e t .

T h e t o k e n o n p l a

r a t e f o r

t r a n s m

i s

W h e n R A c o n t a i

s u c c e s s r a

t e i s 80


a c k n o w

l e d g e m e


a c k n o w

l e d g e m e

122


6

te s t a t e T r a n

s m i t

P a c

k e t

R e c e i v e

P a c

k e t

T r a n s m

i t

A c k n o w .

) n " )

" ) s " ) " ) r"

)# "

)# #

" )

R e c e i v e

d

D A T A

" "

1

1 ` " M o d e l

l i n g a n

d

A n a

l y s i s

b y M e a n s

o f C o l o u

" N e x

t R e c

I N T

1

1 1 ` 6

B

I N T x D A T A

1 1 ` ( 5

, " o f C o l o u

" )

C I N T

1 1 ` 6

N e

t w o r

k

R e c e

i v e r

R P

8

I n t_ 0_ 1 0

1 1 ` 8

R A

I n t_ 0_ 1 0

8

1 1 ` 8

(n , p

)

i f O k ( s , r )

t h e n

1 ` ( n

, p )

e l s e e m p t y

( n , p

)

i f n =

k

a n d a l s o

p < > s

t o p

t h e n s t r ^ p

e l s e s t r

i f n =

k

t h e n

k + 1

e l s e

k

n

k i f n =

k

t h e n

k + 1

e l s e

k

O k ( s , r )

he n

1 ` n

ls e e m p t y

s t r

s s

s e x p e c

t i n g p a c

k a g e n o . 6 . T

h i s

a s s u c c e s s f u

l l y r e c e

i v e d

t h e

f i r s t 5

st i l l s e n d

i n g p a c k e t n o . 5 . I

n a

lr e c e i v e a n a c k n o w

l e d g e m e n

t

q u e s

t f o r p a c k e t n o . 6 .

o w l e d g e m e n

t i s r e c e i v e

d t h e

t e r i s u p d a t e

d a n

d t h e S e n d e r w i l l

a c k e t n o .

6 .

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2 5

R e c e i v e a c k n o w l e d g e m e n t

R e c e i v e

A c k n o w .

N e x

t S e n

d

I N T 1

1 1 ` 1

D I N T 1 1 ` 2

n

k

n

W h e n a n a c

k n o w

l e d g e m e n

t a r r i v e s

t o t h e S e n d e r

i t

i s u s e

d t o u p

d a t e t h e N e x t S e n d c o u n

t e r .

I n

t h i s c a s e

t h e c o u n

t e r v a

l u e

b e c o m e s

2 , a n d

h e n c e

t h e S e n d e r w

i l l b e g

i n t o s e n d p a c k e t

n u m

b e r 2 .

I n t e r m e d i a t e

S e n

d

P a c

k e t

R e c e i v e

A c k n o w .

S e n

d

I N T x D A T A

8 1 ` ( 1

, " M o d e l

l i n " )

+ 1 ` ( 2

, " g a n

d A n

+ 1 ` ( 3

, " a l y s

i s b

)

+ 1 ` ( 4

, " y

M e a n s

+ 1 ` ( 5

, " o f

C o l o u

+ 1 ` ( 6

, " r e

d P e t r

+ 1 ` ( 7

, " i N e t s #

#

+ 1 ` ( 8

, " # # # # # # #

N e x

t S e n

d

I N T 1

1 1 ` 5

D I N T

1 1 ` 6

A

I N T x D A T A

S e n

d e r

( n , p

)

( n

( n , p

)

i f O

t h e

e l s

n

k

n

n

T h e R e c e i v e r

i s

m e a n s t

h a t i t h a

p a c k e t s .

T h e S e n d e r

i s s t

m o m e n

t i t w i l l r

c o n t a

i n i n g a r e q

W h e n

t h e a c

k n o

N e x t S e n d c o u n

t

s t a r t s e n d

i n g p a

123

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0

b e u s e d m

o r e t h a n o n c e

R e c e

i v e

d

D A T A

" "

B 1

I N T x D A T A

C 1

I N T

e t w o r k

t w o r k

R e c N o

1

H S

R e c e

i v e r

B 1 - >

B

C 1 - >

C

B 2

I N T x D A T A

C 2

I N T

R e c N o

2

H S R

e c e

i v e r

B 2 - >

B

C 2 - >

C

R e c e

i v e

d

D A T A

" "

R e c e

i v e r s

T r a n s m

i t

P a c k e

t

R P

8

I n t_ 0_ 1 0

R A 1

I n t_ 0_ 1 0

8 T r a n s m

i t

A c k n o w .

T

C 1

I N T B

2

I N T x D A T A

A T A

B 1

I N T x D A T A

R A 2

I n t_ 0_ 1 0 8

T r a n s m

i t

A c k n o w .

C 2

I N T

N e x t

R e c

I N T

1

R e c e

i v e

P a c k e

t

C I N T

R e c e

i v e

d

D A T A

" "

B

I N T x D A T A

R e c e i v e r

tw o r k

O u

t

O u

t

O u

t I n

t

I n

I n

I / O

s s

( n , p

)

i f O k ( s

, r 2 )

t h e n

1 ` ( n , p

)

e l s e e m p

t y n

i f O k ( s , r )

t h e n

1 ` ( n , 1

)

e l s e e m p

t y

i f O k ( s

, r 1 )

t h e n

1 ` ( n , p

)

e l s e e m p

t y

s

n

i f O k ( s

, r )

t h e n

1 ` ( n , 2

)

e l s e e m p

t y

k i f n =

k

t h e n

k + 1

e l s e

k

( n , p

)

i f n =

k

a n

d a

l s o

p < > s t o p

t h e n s

t r ^ p

e l s e s

t r

i f n =

k

t h e n

k + 1

e l s e

k

s t r

if f e r e n

t i n s t a n c e s o f

t h e R e c e i v e r

n c e

h a s i

t s o w n m a r k i n g .

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2 9

S u b s t i t u t i o n t r a n s i t i o n s

A p a g e m a y c o n t a i n o n e o r e m o r e s u b s t i t u t i o n

t r a n s i t i o n s .

E a c

h s u

b s t i t u t

i o n

t r a n s

i t i o n

i s r e

l a t e d t o a p a g e ,

i . e . ,

a s u b n e t p r o v i

d i n g a m o r e d e t a i l e d

d e s c r i p t i o n

t h a n

t h e t r a n s

i t i o n

i t s e l

f .

T h e p a g e

i s a s u b p a g e o f

t h e s u

b s t i t u t

i o n

t r a n s

i t i o n .

T h e r e

i s a w e l l - d e f i n e d i n t e r f a c e

b e t w e e n a

s u b s t

i t u t i o n

t r a n s

i t i o n a n

d i t s s u

b p a g e :

T h e p l a c e s s u r r o u n d

i n g

t h e s u

b s t i t u t

i o n

t r a n s

i t i o n

a r e s o c k e t p l a c e s .

T h e s u

b p a g e c o n t a i n s a n u m

b e r o

f p o r t p l a c e s .

S o c

k e t p l a c e s a r e r e l a t e d

t o p o r t p l a c e s

i n a

s i m i l a r w a y a s a c

t u a l p a r a m e t e r s a r e r e

l a t e d t o

f o r m a l p a r a m e t e r s i n a p r o c e d u r e c a

l l .

A

s o c k e t p l a c e

h a s a

l w a y s t

h e s a m e m a r k i n g a s

t h e r e

l a t e d p o r t p l a c e . T

h e t w o p l a c e s a r e

j u s t

d i f f e r e n t v i e w s o f

t h e s a m e p l a c e .

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