1 Petri nets Classical Petri nets: The basic model.

1

Petri netsClassical Petri nets: The basic model

2

Process modeling

• Emphasis on dynamic behavior rather than structuring the state space

• Transition system is too low level

• We start with the classical Petri net

• Then we extend it with:– Color

– Time

– Hierarchy

3

Classical Petri net

• Simple process model– Just three elements: places, transitions and arcs.– Graphical and mathematical description.– Formal semantics and allows for analysis.

• History:– Carl Adam Petri (1962, PhD thesis)– In sixties and seventies focus mainly on theory.– Since eighties also focus on tools and applications

(cf. CPN work by Kurt Jensen).– “Hidden” in many diagramming techniques and

systems.

4

Elements

(name)

(name)

place

transition

arc (directed connection)

token

t34 t43

t23 t32

t12 t21

t01 t10

p4

p3

p2

p1

p0

place

transition

token

5

Rules

• Connections are directed.

• No connections between two places or two transitions.

• Places may hold zero or more tokens.

• First, we consider the case of at most one arc between two nodes.

wait enter before make_picture after leave gone

free

occupied

6

Enabled

• A transition is enabled if each of its input places contains at least one token.


free

occupied

enabled Not enabled

Not enabled

7

Firing

• An enabled transition can fire (i.e., it occurs).

• When it fires it consumes a token from each input place and produces a token for each output place.


free

occupied

fired

8

Play “Token Game”

• In the new state, make_picture is enabled. It will fire, etc.


free

occupied

9

Remarks

• Firing is atomic.

• Multiple transitions may be enabled, but only one fires at a time, i.e., we assume interleaving semantics (cf. diamond rule).

• The number of tokens may vary if there are transitions for which the number of input places is not equal to the number of output places.

• The network is static.

• The state is represented by the distribution of tokens over places (also referred to as marking).

10

Non-determinism

t34 t43

t23 t32

t12 t21

t01 t10

p4

p3

p2

p1

p0

transition t23fires

t34 t43

t23 t32

t12 t21

t01 t10

p4

p3

p2

p1

p0

Two transitions are enabled but only one can fire

11

Example: Single traffic light rg

go

or

red

green

orange

12

Two traffic lights

rg

go

or

red

green

orange

rg

go

or

red

green

orange

rg

go

or

red

green

orange

OR

13

Problem

14

Solution

rg1

go1

or1

r1

g1

o1

rg2

go2

or2

r2

g2

o2

x

How to make them alternate?

15

Playing the “Token Game” on the Internet

• Applet to build your own Petri nets and execute them: http://www.tm.tue.nl/it/staff/wvdaalst/Downloads/pn_applet/pn_applet.html

• FLASH animations: http://www.tm.tue.nl/it/staff/wvdaalst/courses/pm/flash/

16

Exercise: Train system (1)

• Consider a circular railroad system with 4 (one-way) tracks (1,2,3,4) and 2 trains (A,B). No two trains should be at the same track at the same time and we do not care about the identities of the two trains.

17


• Consider a railroad system with 4 tracks (1,2,3,4) and 2 trains (A,B). No two trains should be at the same track at the same time and we want to distinguish the two trains.

18


• Consider a railroad system with 4 tracks (1,2,3,4) and 2 trains (A,B). No two trains should be at the same track at the same time. Moreover the next track should also be free to allow for a safe distance. (We do not care about train identities.)

19


• Consider a railroad system with 4 tracks (1,2,3,4) and 2 trains. Tracks are free, busy or claimed. Trains need to claim the next track before entering.

20

WARNINGIt is not sufficient to understand the (process) models. You have to be

able to design them yourself !

21

Multiple arcs connecting two nodes

• The number of arcs between an input place and a transition determines the number of tokens required to be enabled.

• The number of arcs determines the number of tokens to be consumed/produced.


free

22

Example: Ball game

red

rr

rb

bb

black

23

Exercise: Manufacturing a chair

• Model the manufacturing of a chair from its components: 2 front legs, 2 back legs, 3 cross bars, 1 seat frame, and 1 seat cushion as a Petri net.

• Select some sensible assembly order.

• Reverse logistics?

24

Exercise: Burning alcohol.

• Model C2H5OH + 3 * O2 => 2 * CO2 + 3 * H2O

• Assume that there are two steps: first each molecule is disassembled into its atoms and then these atoms are assembled into other molecules.

25

Exercise: Manufacturing a car

• Model the production process shown in the Bill-Of-Materials.

car

engine

subassembly1

subassembly2

wheelchassis

chair2

4

26

Formal definition

A classical Petri net is a four-tuple (P,T,I,O) where:

• P is a finite set of places,

• T is a finite set of transitions,

• I : P x T -> N is the input function, and

• O : T x P -> N is the output function.

Any diagram can be mapped onto such a four tuple and vice versa.

27

Formal definition (2)

The state (marking) of a Petri net (P,T,I,O) is defined as follows:

• s: P-> N, i.e., a function mapping the set of places onto {0,1,2, … }.

28

Exercise: Map onto (P,T,I,O) and S

red

rr

rb

bb

black

29

Exercise: Draw diagram

Petri net (P,T,I,O):

• P = {a,b,c,d}

• T = {e,f}

• I(a,e)=1, I(b,e)=2, I(c,e)=0, I(d,e)=0, I(a,f)=0, I(b,f)=0, I(c,f)=1, I(d,f)=0.

• O(e,a)=0, O(e,b)=0, O(e,c)=1, O(e,d)=0, O(f,a)=0, O(f,b)=2, O(f,c)=0, O(f,d)=3.

State s:

• s(a)=1, s(b)=2, s(c)=0, s(d) = 0.

30

Enabling formalized

Transition t is enabled in state s1 if and only if:

31

Firing formalized

If transition t is enabled in state s1, it can fire and the resulting state is s2 :

32

Mapping Petri nets onto transition systems

A Petri net (P,T,I,O) defines the following transition system (S,TR):

33

Reachability graph

• The reachability graph of a Petri net is the part of the transition system reachable from the initial state in graph-like notation.

• The reachability graph can be calculated as follows:

1. Let X be the set containing just the initial state and let Y be the empty set.

2. Take an element x of X and add this to Y. Calculate all states reachable for x by firing some enabled transition. Each successor state that is not in Y is added to X.

3. If X is empty stop, otherwise goto 2.

34

Examplered

rr

rb

bb

black

(3,2)

(1,3) (1,2)

(3,1) (3,0)

(1,1)

(1,0)

Nodes in the reachability graph can be represented by a vector “(3,2)” or as “3 red + 2 black”. The latter is useful for “sparse states” (i.e., few places are marked).

35

Exercise: Give the reachability graph using both notations

rg1

go1

or1

r1

g1

o1

rg2

go2

or2

r2

g2

o2

x

36

Different types of states

• Initial state: Initial distribution of tokens.• Reachable state: Reachable from initial state.• Final state (also referred to as “dead states”):

No transition is enabled.• Home state (also referred to as home marking):

It is always possible to return (i.e., it is reachable from any reachable state).

How to recognize these states in the reachability graph?

37

Exercise: Producers and consumers

• Model a process with one producer and one consumer, both are either busy or free and alternate between these two states. After every production cycle the producer puts a product in a buffer. The consumer consumes one product from this buffer per cycle.

• Give the reachability graph and indicate the final states.

• How to model 4 producers and 3 consumers connected through a single buffer?

• How to limit the size of the buffer to 4?

38

Exercise: Two switches

• Consider a room with two switches and one light. The light is on or off. The switches are in state up or down. At any time any of the switches can be used to turn the light on or off.

• Model this as a Petri net.

• Give the reachability graph.

39

Modeling

• Place: passive element

• Transition: active element

• Arc: causal relation

• Token: elements subject to change

The state (space) of a process/system is modeled by places and tokens and state transitions are modeled by transitions (cf. transition systems).

40

Role of a token

Tokens can play the following roles:

• a physical object, for example a product, a part, a drug, a person;

• an information object, for example a message, a signal, a report;

• a collection of objects, for example a truck with products, a warehouse with parts, or an address file;

• an indicator of a state, for example the indicator of the state in which a process is, or the state of an object;

• an indicator of a condition: the presence of a token indicates whether a certain condition is fulfilled.

41

Role of a place

• a type of communication medium, like a telephone line, a middleman, or a communication network;

• a buffer: for example, a depot, a queue or a post bin;

• a geographical location, like a place in a warehouse, office or hospital;

• a possible state or state condition: for example, the floor where an elevator is, or the condition that a specialist is available.

42

Role of a transition

• an event: for example, starting an operation, the death of a patient, a change seasons or the switching of a traffic light from red to green;

• a transformation of an object, like adapting a product, updating a database, or updating a document;

• a transport of an object: for example, transporting goods, or sending a file.

43

Typical network structures

• Causality

• Parallelism (AND-split - AND-join)

• Choice (XOR-split – XOR-join)

• Iteration (XOR-join - XOR-split)

• Capacity constraints– Feedback loop

– Mutual exclusion

– Alternating

44

Causality

45

Parallelism

46

Parallelism: AND-split

47

Parallelism: AND-join

48

Choice: XOR-split

49

Choice: XOR-join

50

Iteration: 1 or more times

XOR-join before XOR-split

51

Iteration: 0 or more times

XOR-join before XOR-split

52

Capacity constraints: feedback loop

AND-join before AND-split

53

Capacity constraints: mutual exclusion


54

Capacity constraints: alternating


55

We have seen most patterns, e.g.:

rg1

go1

or1

r1

g1

o1

rg2

go2

or2

r2

g2

o2

x

How to make them alternate?

Example of mutual exclusion

56

Exercise: Manufacturing a car (2)

• Model the production process shown in the Bill-Of-Materials with resources.

• Each assembly step requires a dedicated machine and an operator.

• There are two operators and one machine of each type.

• Hint: model both the start and completion of an assembly step.

car

engine

subassembly1

subassembly2

wheelchassis

chair2

4

57

Modeling problem (1): Zero testing

• Transition t should fire if place p is empty.

t ?

p

58

Solution

• Only works if place is N-bounded

tN input and output arcs

Initially there are N tokens

p

p’

59

Modeling problem (2): Priority

• Transition t1 has priority over t2

t1

t2

?

Hint: similar to Zero testing!

60

A bit of theory

• Extensions have been proposed to tackle these problems, e.g., inhibitor arcs.

• These extensions extend the modeling power (Turing completeness*).

• Without such an extension not Turing complete.• Still certain questions are difficult/expensive to

answer or even undecidable (e.g., equivalence of two nets).

* Turing completeness corresponds to the ability to execute any computation.

61

Exercise: Witness statements

• As part of the process of handling insurance claims there is the handling of witness statements.

• There may be 0-10 witnesses per claim. After an initialization step (one per claim), each of the witnesses is registered, contacted, and informed (i.e., 0-10 per claim in parallel). Only after all witness statements have been processed a report is made (one per claim).

• Model this in terms of a Petri net.

62

Exercise: Dining philosophers

• 5 philosophers sharing 5 chopsticks: chopsticks are located in-between philosophers

• A philosopher is either in state eating or thinking and needs two chopsticks to eat.

• Model as a Petri net.

63

High level Petri netsExtending classical Petri nets with color, time and hierarchy (informal introduction)

Prof.dr.ir. Wil van der AalstEindhoven University of Technology, Faculty of Technology Management,

Department of Information and Technology, P.O.Box 513, NL-5600 MB,

Eindhoven, The Netherlands.

64

Limitations of classical Petri nets

• Inability to test for zero tokens in a place.

• Models tend to become large.

• Models cannot reflect temporal aspects

• No support for structuring large models, cf. top-down and bottom-up design

65

Inability to test for zero tokens in a place

t ?

p

“Tricks” only work if p is bounded

66

Models tend to become (too) larger1

incr1

bike

decr1

l1

r2

incr2

wheel

decr2

l2

r3

incr3

bell

decr3

l3

r4

incr4

steeringwheel

decr4

l4

r5

incr5

frame

decr5

l5

Size linear in the number of products.

67

Models tend to become (too) large (2)

Size linear in the number of tracks.

transfer

c1

b1

f1

c 2

b 2

f 2

transfer transfer

c 3

b 3

f 3

c 4

b 4

f 4

transfer transfer

c1

b1

f1

claim_track

c 2

b 2

f 2

transfer transfer

c 3

b 3

f 3

c 4

b 4

f 4

transfer

claim_track claim_track claim_track

68

Models cannot reflect temporal aspects

rg1

go1

or1

r1

g1

o1

rg2

go2

or2

r2

g2

o2

x

Duration of each phase is highly

relevant.

69

rg1

go1

or1

r1

g1

o1

rg2

go2

or2

r2

g2

o2

x

No support for structuring large models

70

High-level Petri nets

• To tackle the problems identified.• Petri nets extended with:

– Color (i.e., data)– Time– Hierarchy

• For the time being be do not choose a concrete language but focus on the main concepts.

• Later we focus on a concrete language: CPN.• These concepts are supported by many variants

of CPN including ExSpect, CPN AMI, etc.

71

Running example: Making punch cards

start

donewait

stop

busy

free

waiting patients

served patients

free desk employees

patient/ employees

72

Extension with color (1)

• Tokens have a color (i.e., a data value)

{Brand="BMW", RegistrationNo="GD-XW-11", Year=1993, Colour="blue", Owner= "Inge"}

{Brand="Lada", RegistrationNo="PH-14-PX", Year=1986, Color="grey", Owner="Inge"}

73


• Places are typed (also referred to as color set).

{Brand="BMW", RegistrationNo="GD-XW-11", Year=1993, Colour="blue", Owner= "Inge"}

{Brand="Lada", RegistrationNo="PH-14-PX", Year=1986, Color="grey", Owner="Inge"}

record Brand:string * RegistrationNo:string * Year:int * Color:string * Owner:string

74


• The relation between production and consumption needs to be specified, i.e., the value of a produced token needs to be related to the values of consumed tokens.

add

in sum

0

3

23

1 The value of the token produced for place sum is the sum of the values of

the consumed tokens.

75

Running example: Tokens are colored

start

donewait

stop

busy

free

{Name="Klaas", Address="Plein 10", DateOfBirth="13-Dec-1962", Gender="M"}

{EmpNo=641112, Experience=7}

76

Running example: Places are typed

start

donewait

stop

busy

free

record EmpNo:int * Experience:int

record Name:string *Address:string *DateOfBirth:str *Gender:string

record Name:string *Address:string *DateOfBirth:str *

Gender:stringrecord Name:string *Address:string *DateOfBirth:str *Gender:string *EmpNo:int *Experience:int

77

Running example: Initial state

start

donewait

stop

busy

free



start is enabled

78

Running example: Transition start fired

stop is enabled

start

donewait

stop

busy

free

{Name="Klaas", Address="Plein 10", DateOfBirth="13-Dec-1962", Gender="M", EmpNo=641112, Experience=7}

New value is created by simply merging the two records.

79

Running example: Transition stop fired

New values are created by simply spliting the record into two parts.

start

donewait

stop

busy

free



80

The number of tokens produced is no longer fixed (1)

test

samplenegative

positive

{sample_number=931101011, measurement_outcomes="XYV"}

{sample_number=931101023, measurement_outcomes="VXY"}

Note that the network structure is no longer a complete specification!

81

The number of tokens produced is no longer fixed (2)

test

samplenegative

positive

{sample_number=931101011, measurement_outcomes="XYV"}

The number of tokens produced for each output place is between 0 and 3 and the sum should be 3.

82

Example

Model as a colored Petri net.

r1

incr1

bike

decr1

l1

r2

incr2

wheel

decr2

l2

r3

incr3

bell

decr3

l3

r4

incr4

steeringwheel

decr4

l4

r5

incr5

frame

decr5

l5

83

in

increase

[{prod="bike", num=4},{prod="wheel", num=2},{prod="bell", number=3},{prod="steering wheel", num=3},{prod="frame", num=2}]

decrease

out

stock

{prod="bell", num=2}

The entire stock is

represented by the value of a single token, i.e., a list of

records.

Product and quantity are in the value of the

token

84

Types

in

increase

[{prod="bike", num=4},{prod="wheel", num=2},{prod="bell", number=3},{prod="steering wheel", num=3},{prod="frame", num=2}]

decrease

out

stock

{prod="bell", num=2}

color Product = string;

color Number = int;

color StockItem = record prod:Product * num:Number;

color Stock = list StockItem;

StockItem

StockItem

Stock

85

Extension with time (1)

• Each token has a timestamp.

• The timestamp specifies the earliest time when it can be consumed.

2 5

86

Extension with time (2)

• The enabling time of a transition is the maximum of the tokens to be consumed.

• If there are multiple tokens in a place, the earliest ones are consumed first.

• A transition with the smallest firing time will fire first.• Transitions are eager, i.e., they fire as soon as they

can.• Produced token may have a delay.• The timestamp of a produced token is the firing

time plus its delay.

87

Running example: Enabling time

• Transition start is enabled at time 2 = max{0,min{2,4,4}}.

start

donewait

stop

busy

free

4

2

4

0

88

Running example: Delays

• Tokens for place busy get a delay of 3

• @+3 = firing time plus 3 time units

start

donewait

stop

busy

free

4

2

4

0

@+3

@+0

@+0

89

Running example: Transition start fired

• Transition start fired a time 2.

start

donewait

stop

busy

free

4 5

4

@+0

@+0

@+3

Continue to play (timed) token game…

90

Exercise: Final state?

x

y

d

e

b

a

c

4

3

5

1

@+1

@+2

91

Exercise: Final state?

red

rr

rb

bb

black

3

1

2

4

2

@+1@+1

@+2

92

Extension with hierarchy

• Timed and colored Petri nets result in more compact models.

• However, for complex systems/processes the model does not fit on a single page.

• Moreover, putting things at the same level does not reflect the structure of the process/system.

• Many hierarchy concepts are possible. In this course we restrict ourselves to transition refinement.

93

rg1

go1

or1

r1

g1

o1

rg2

go2

or2

r2

g2

o2

x

Instead of

94

rg1

go1

or1

r1

g1

o1

rg2

go2

or2

r2

g2

o2

x

tl1 tl2

x

x

We can use hierarchy

95

rg

go

or

r

g

tl1 tl2

x

x

o

Reuse

• Reuse saves design efforts.

• Hierarchy can have any number of levels

• Transition refinement can be used for top-down and bottom-up design

96

Exercise: model three (parallel) punch card desks in a hierarchical manner

start

donewait

stop

busy

free

97

Analysis of Process Models:Reachability graphs, invariants, and simulation

Prof.dr.ir. Wil van der AalstEindhoven University of Technology, Faculty of Technology Management,

Department of Information and Technology, P.O.Box 513, NL-5600 MB,

Eindhoven, The Netherlands.

98

Questions raised when considering the handling of customer orders

• How many orders arrive on average?

• How many orders can be handled?

• Do orders get lost?

• Do back orders always have priority?

• What is the utilization of office workers?

• If the desired product is no longer available, does the order get stuck?

• Etc.

99

Questions raised when considering the handling of customers in the canteen

• What is the average waiting time from 12.30-13.00?

• What is the variance of waiting times?

• What is the effect of an additional cashier on the queue length?

• Etc.

100

Questions raised when considering the an intersection with multiple traffic lights

• How much traffic can be handled per hour?

• Give some volume of traffic, what is the probability to get a red light?

• Is the intersection safe, i.e., crossing flows have never a green light at the same time?

• Can a light go from yellow to green?

• Is the intersection fair (i.e., a trafficlight cannot turn green twice whilecars are waiting on the other side)?

101

Questions raised when considering a printer shared by multiple users

• Can two print jobs get mixed?

• Do small jobs always get priority?

• Can the settings of one job influence the next job?

• Do out-of-paper events cause jobsto get lost?

• How many jobs can be handled per day?

• What is the probability of a paper jam?

102

Questions raised when considering a teller machine

• What is the average response time?

• Is there a balance, i.e., the amount of money leaving the machine matches the amount taken from bank accounts?

• How often should the machine be filledto guarantee 90% availability?

• Is fraud possible?

• Etc.

103

Analysis

Operational process I nf ormation

System

Model

• Analysis is typically model-driven to allow e.g. what-if questions.

• Models of both operational processes and/or the information systems can be analyzed.

• Types of analysis:

– validation– verification– performance analysis

104

Three analysis techniques (Chapter 8)

• Reachability graph• Place & transition invariants• Simulation

• Each can be applied to both classical and high-level Petri nets. Nevertheless, for the first two we restrict ourselves to the classical Petri nets.

• Use:– reachability graph (validation, verification)– invariants (validation, verification)– simulation (validation, performance analysis)

105

Reachability graph

rg1

go1

or1

r1

g1

o1

rg2

go2

or2

r2

g2

o2

x

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

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

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

Five reachable states.Traffic lights are safe!

106

Alternative notation

rg1

go1

or1

r1

g1

o1

rg2

go2

or2

r2

g2

o2

x

r1+r2+x

g1+r2 r1+g2

r1+o2o1+r2

107

Reachability graph (2)

• Graph containing a node for each reachable state.

• Constructed by starting in the initial state, calculate all directly reachable states, etc.

• Expensive technique.

• Only feasible if finitely many states (otherwise use coverability graph).

• Difficult to generate diagnostic information.

108

Infinite reachability graphrg1

go1

or1

r1

g1

o1

rg2

go2

or2

r2

g2

o2

x

109

Exercise: Construct reachability graph


free

occupied

110

Exercise: Dining philosophers (1)

• 5 philosophers sharing 5 chopsticks: chopsticks are located in-between philosophers

• A philosopher is either in state eating or thinking and needs two chopsticks to eat.

• Model as a Petri net.

111


• Assume that philosophers take the chopsticks one by one such that first the right-hand one is taken and then the left-hand one.

• Model as a Petri net.• Is there a deadlock?

112


• Assume that philosopher take the chopsticks one by one in any order and with the ability to return a chopstick.

• Model as a Petri net.• Is there a deadlock?

113

Structural analysis techniques

• To avoid state-explosion problem and bad diagnostics.

• Properties independent of initial state.

• We only consider place and transition invariants.

• Invariants can be computed using linear algebraic techniques.

114

Place invariant

• Assigns a weight to each place.

• The weight of a token depends on the weight of the place.

• The weighted token sum is invariant, i.e., no transition can change it

couple

man

woman

marriage divorce

1 man + 1 woman + 2 couple

115

Other invariants

• 1 man + 0 woman + 1 couple

(Also denoted as: man + couple)

• 2 man + 3 woman + 5 couple

• -2 man + 3 woman + couple

• man – woman

• woman – man

(Any linear combination of invariants is an invariant.)

couple

man

woman

marriage divorce

116

Example: traffic light

• r1 + g1 + o1

• r2 + g2 + o2

• r1 + r2 + g1 + g2 + o1 + o2

• x + g1 + o1 + g2 + o2

• r1 + r2 - x

rg1

go1

or1

r1

g1

o1

rg2

go2

or2

r2

g2

o2

x

117

Exercise: Give place invariants

free producer

start_production

end_production

wait consumer

start_consumption

end_consumption

product

118

Transition invariant• Assigns a weight to

each transition.• If each transition

fires the number of times indicated, the system is back in the initial state.

• I.e. transition invariants indicate potential firing sets without any net effect.

couple

man

woman

marriage divorce

2 marriage + 2 divorce

119

Other invariants

• 1 marriage + 1 divorce

(Also denoted as: marriage + divorce)

• 20 marriage + 20 divorce

Any linear combination of invariants is an invariant, but transition invariants with negative weights have no obvious meaning.

Invariants may be not be realizable.

couple

man

woman

marriage divorce

120

Example: traffic light

• rg1 + go1 + or1

• rg2 + go2 + or2

• rg1 + rg2 + go1 + go2 + or1 + or2

• 4 rg1 + 3 rg2 + 4 go1 +3 go2 + 4 or1 + 3 or2

rg1

go1

or1

r1

g1

o1

rg2

go2

or2

r2

g2

o2

x

121

Exercise: Give transition invariants

free producer

start_production

end_production

wait consumer

start_consumption

end_consumption

product

122

Exercise: four philosophers

• Give place invariants.

• Give transition invariants

t1

t2

t3

t4

e1

e2

e3

e4

c4 c1

c2c3

st1

se1

st2

st3

st4 se4

se3

se2

123

Two ways of calculating invariants

• "Intuitive way": Formulate the property that you think holds and verify it.

• "Linear-algebraic way": Solve a system of linear equations.

Humans tend to do it the intuitive way and computers do it the linear-algebraic way.

124

Incidence matrix of a Petri net

• Each row corresponds to a place.

• Each column corresponds to a transition.

• Recall that a Petri net is described by (P,T,I,O).

• N(p,t)=O(t,p)-I(p,t) where p is a place and t a transition.

couple

man

woman

marriage divorce

N =−1 1−1 11 −1

125

Example

couple

man

woman

marriage divorce N =−1 1−1 11 −1

manwoman

couple

marriage

divorce

126

Place invariant

• Let N be the incidence matrix of a net with n places and m transitions

• Any solution of the equation X.N = 0 is a transition invariant – X is a row vector (i.e., 1 x n matrix)

– O is a row vector (i.e., 1 x m matrix)

• Note that (0,0,... 0) is always a place invariant.

• Basis can be calculated in polynomial time.

127

Example

Solutions:

• (0,0,0)

• (1,0,1)

• (0,1,1)

• (1,1,2)

• (1,-1,0)

couple

man

woman

marriage divorce

X −1 1−1 11 −1 =0,0

man , woman , couple −1 1−1 11 −1 =0,0

128

Transition invariant

• Let N be the incidence matrix of a net with n places and m transitions

• Any solution of the equation N.X = 0 is a place invariant – X is a column vector (i.e., m x 1 matrix)

– 0 is a column vector (i.e., n x 1 matrix)

• Note that (0,0,... 0)T is always a place invariant.

• Basis can be calculated in polynomial time.

129

Example

Solutions:• (0,0)T

• (1,1)T

• (32,32)T

couple

man

woman

marriage divorce−1 1−1 11 −1 X =0

00

−1 1−1 11 −1 marriage

divorce =000

130

Exercise

• Give incidence matrix.• Calculate/check place invariants.• Calculate/check transition invariants.

free producer

start_production

end_production

wait consumer

start_consumption

end_consumption

product

131

Simulation

• Most widely used analysis technique.• From a technical point of view just a "walk" in

the reachability graph.• By making many "walks" (in case of transient

behavior) or a very "long walk" (in case of steady-state) behavior, it is possible to make reliable statements about properties/ performance indicators.

• Used for validation and performance analysis.• Cannot be used to prove correctness!

132

Stochastic process

• Simulation of a deterministic system is not very interesting.

• Simulation of an untimed system is not interesting.• In a timed and non-deterministic system, durations

and probabilities are described by some probability distribution.

• In other words, we simulate a stochastic process!

• CPN allows for the use of distributions using some internal random generator.

133

Uniform distribution

pdf cumulative

134

Negative exponential distribution

135

Normal distribution

136

Distributions in CPN Tools

Assume some library with functions:

• uniform(x,y)

• nexp(x)

• erlang(n,x)

• Etc.

A nice function is also C.ran() which returns a randomly selected element of finite color set C, e.g.,color C = int with 1..5;fun select1to5() = C.ran()returns a number between 1 and 5

137

Example

throw_dice()trigger

color BT = unit;color Dice = int with 1..6;

Dice.ran()()

BT Diceoutcome

()

138

Example(2)

throw_dice

(x+1)@+(Delay.ran())

trigger

color INT = int;color TINT = int timed;color Dice = int with 1..6;color Delay = int with 0..99;

Dice.ran()0

TINT Diceoutcome

x

139

Subruns and confidence intervals

• A single run does not provide information about reliability of results.

• Therefore, multiple runs or one run cut into parts: subruns.

• If the subruns are assumed to be mutually independent, one can calculate a confidence interval, e.g., the flow time is with 95% confidence within the interval 5.5+/-0.5 (i.e. [5,6]).

140

Example of a simulation model

• Gas station with one pump and space for 4 cars (3 waiting and 1 being served).

• Service time: uniform distribution between 2 and 5 minutes.

• Poisson arrival process with mean time between arrivals of 4 minutes.

• If there are more than 3 cars waiting, the "sale" is lost.

• Questions: flow time, waiting time, utilization, lost sales, etc.

141

Top-level page: main

environment

HS HS

gas_station

arrive Car

Cardrive_on

depart Car

color Car = string

142

Subpage gas_station

color Car = string;color Pump = unit;color TCar = Car timed;color Queue = list Car;var c:Car;var q:Queue;fun len(q:Queue) = if q=[] then 0 else 1+len(tl(q));

arrive Car

depart Car

In

[len(q)<3] [len(q)>=3]

put_in_queue drive_on

queue Queue

[]

fill_up

pump_free

TCar Pump

()

Cardrive_on

Out Out

()

()

c c

c@+uniform(2,5)

c

c

c

qq^^[c]

c::qq

start

end

q

q

143

Assuming pages for the environment and measurements the last two pages allow for ...

• Calculation of flow time (average, variance, maximum, minimum, service level, etc.).

• Calculation of waiting times (average, variance, maximum, minimum, service level, etc.).

• Calculation of lost sales (average).• Probability of no space left.• Probability of no cars waiting.For each of these metrics, it is possible to formulate

a confidence interval given sufficient observations.

144

Alternatives

Model the following alternatives:

• 5 waiting spaces• 2 pumps• 1 faster pump

color Car = string;color Pump = unit;color TCar = Car timed;color Queue = list Car;var c:Car;var q:Queue;fun len(q:Queue) = if q=[] then 0 else 1+len(tl(q));

arrive Car

depart Car

In

[len(q)<3] [len(q)>=3]

put_in_queue drive_on

queue Queue

[]

fill_up

pump_free

TCar Pump

()

Cardrive_on

Out Out

()

()

c c

c@+uniform(2,5)

c

c

c

qq^^[c]

c::qq

start

end

q

q

145

Simulation of a Production system

X Y Z CSABC

146

Data X Y Z

A 2 5 8

B 3 6 9

C 4 7 1

Processing times

X 2

Y 3

Z 4

Resources per work center

A 2

B 1

C 2

Replenishment lead times

A 2

B 1

C 1

Kanbans in-between work

centers

A 7

B 9

C 8

Time in-between

subsequent orders

Use distributions

147

Top level page: maincolor INT = int;color Prod = string;color PT = product Prod * INT;color PTimed = Prod timed;color PTTimed = PT timed;var p:Prod;var t:INT;var i:INT;

kanban1

supplier

product1

HS

work_center_

X

kanban2

product2

work_center_

YHS HS

kanban3

product3

work_center_

ZHS

kanban4

product4

HS

customer

Prod Prod Prod Prod

ProdProdProdProd

2`"A"++1`"B"++1`"C"

io_lead_time PT

1`("A",2)++1`("B",1)++

1`("C",2)

processing_time_X

PT

1`("A",2)++1`("B",3)++

1`("C",4)

PT

1`("A",5)++1`("B",6)++

1`("C",7)

PT

1`("A",8)++1`("B",9)++1`("C",1)

c_ia_time PTTimed

1`("A",7)++1`("B",9)++

1`("C",8)

processing_time_Y

processing_time_Z

2`"A"++1`"B"++

1`"C"

2`"A"++1`"B"++1`"C"

resources_X INT

4

resources_Y INT

5

resources_Z INT

6

supplierio_lead_time = io_lead_timekanban_in = kanban1product_out = product1

work_centerprocessing_time = processing_time_Xresources = resources_Xkanban_in = kanban2product_in = product1kanban_out = kanban1product_out = product2

work_centerprocessing_time = processing_time_Yresources = resources_Ykanban_in = kanban3product_in = product2kanban_out = kanban2product_out = product3

work_centerprocessing_time = processing_time_Zresources = resources_Zkanban_in = kanban4product_in = product3kanban_out = kanban3product_out = product4

customerc_ia_time = c_ia_timekanban_out = kanban4product_in = product4

148

Sub page: supplier

kanban_in

p

product_out

In

Out

Prod

Prod

PTimedio_lead_time PT

(p,t)

(p,t)In/Out

oip

accept_order

deliver_order

p@+t

p

p

149

Sub page: customer

kanban_out

p

product_in

In

Out

Prod

Prod

PTTimedc_ia_time

(p,t)

In/Out

place_order

consume

p

(p,t)@+t

150

Sub page: work_center

product_out

p

kanban_in

In

Out

Prod

Prod

INTresources

In/Out

end_proc

start_proc

p

wip PTimed

i+1

i

i-1

i

p@+t

p

[i>=1]

kanban_out

Out

Prod p

product_in

In

Prod

p

PTprocessing_time

In/Out(p,t)

(p,t)

151

Overview

color INT = int;color Prod = string;color PT = product Prod * INT;color PTimed = Prod timed;color PTTimed = PT timed;var p:Prod;var t:INT;var i:INT;

kanban1

supplier

product1

HS

work_center_

X

kanban2

product2

work_center_

YHS HS

kanban3

product3

work_center_

ZHS

kanban4

product4

HS

customer

Prod Prod Prod Prod

ProdProdProdProd

2`"A"++1`"B"++1`"C"

io_lead_time PT

1`("A",2)++1`("B",1)++

1`("C",2)

processing_time_X

PT

1`("A",2)++1`("B",3)++

1`("C",4)

PT

1`("A",5)++1`("B",6)++

1`("C",7)

PT

1`("A",8)++1`("B",9)++1`("C",1)

c_ia_time PTTimed

1`("A",7)++1`("B",9)++

1`("C",8)

processing_time_Y

processing_time_Z

2`"A"++1`"B"++

1`"C"

2`"A"++1`"B"++1`"C"

resources_X INT

4

resources_Y INT

5

resources_Z INT

6

supplierio_lead_time = io_lead_timekanban_in = kanban1product_out = product1

work_centerprocessing_time = processing_time_Xresources = resources_Xkanban_in = kanban2product_in = product1kanban_out = kanban1product_out = product2

work_centerprocessing_time = processing_time_Yresources = resources_Ykanban_in = kanban3product_in = product2kanban_out = kanban2product_out = product3

work_centerprocessing_time = processing_time_Zresources = resources_Zkanban_in = kanban4product_in = product3kanban_out = kanban3product_out = product4

customerc_ia_time = c_ia_timekanban_out = kanban4product_in = product4

kanban_in

p

product_out

In

Out

Prod

Prod

PTimedio_lead_time PT

(p,t)

(p,t)In/Out

oip

accept_order

deliver_order

p@+t

p

p

kanban_out

p

product_in

In

Out

Prod

Prod

PTTimedc_ia_time

(p,t)

In/Out

place_order

consume

p

(p,t)@+tproduct_out

p

kanban_in

In

Out

Prod

Prod

INTresources

In/Out

end_proc

start_proc

p

wip PTimed

i+1

i

i-1

i

p@+t

p

[i>=1]

kanban_out

Out

Prod p

product_in

In

Prod

p

PTprocessing_time

In/Out(p,t)

(p,t)

Results:• response time• utilization• % backorders• average stock• etc.

152

Classical versus high-level Petri nets

• Simulation clearly works for all types of nets.

• Hierarchy is never a problem.

• Time allows for new types of analysis.

• Reachability graphs and invariants can also be extended to high-level nets.– More complex (both technique and computation)

• Sometimes abstraction from color is possible to derive invariants (consider previous example).

153

Exercise: Five Chinese philosophers• Recall hierarchical CPN model of five

Chinese philosophers alternating between states thinking and eating. – Give place invariants– Give transition invariants

• Change the model such that philosophers can take one chopstick at a time but avoid deadlocks and a fixed ordering of philosophers.– Give place invariants– Give transition invariants

154

Top-level pagecolor BlackToken = unit;var b:BackToken

BlackTokenPH1

CS1CS2

BlackToken

PH2PH5

PH3PH4

CS3

BlackTokenBlackToken

CS5

CS4

BlackToken

() ()

()

()

()

philosopherleft = CS5right = CS4

HS


HS

HS


HS


HS


155

Page philosopher

take_chopsticks

b

think

put_down_chopsticks

eat

BlackToken

b

()

bb

BlackToken

left right

BlackToken BlackTokenIn/Out In/Out

bb

b

b

156

Flat model is obtained by replacing substitution transitions by subpagescolor BlackToken = unit;var b:BackToken

BlackTokenPH1

CS1CS2

BlackToken

PH2PH5

PH3PH4

CS3

BlackTokenBlackToken

CS5

CS4

BlackToken

() ()

()

()

()


HS


HS

HS


HS


HS


take_chopsticks

b

think

put_down_chopsticks

eat

BlackToken

b

()

bb

BlackToken

left right


bb

b

b

Repeat 5 times...

Naming:•PH3.think•PH3.eat•PH3.take_chopsticks•PH3.put_down_chopsticks

157

Alternative page

start_eating

b

think

start_thinking

eat

BlackToken

b

()

bb

BlackToken

hold_left hold_ right

BlackToken BlackToken

bb

b

b

take_left

b

take_right

b

left right


bb

return_left return_right

b

b

b

b

158

You should be able to ...• Construct a reachability graph for a classical Petri

net.• Give meaningful place and transition invariants for

a classical Petri net.• Construct a reachability graph and give

meaningful place and transition invariants for a hierarchical CPN after abstracting from data and time and removing hierarchy.

• Build a simple simulation model using CPN.• Motivate the use of each of the analysis

techniques.

1 Petri nets Classical Petri nets: The basic model.

Documents

Transcript of 1 Petri nets Classical Petri nets: The basic model.