PN1

1
Petri nets refesher

• Prof.dr.ir. Wil van der Aalst
• Eindhoven University of Technology, Faculty of
  Technology Management,
• Department of Information and Technology, P.O.Box
  513, NL-5600 MB,
• Eindhoven, The Netherlands.

2
Petri netsClassical Petri nets The basic model

• Prof.dr.ir. Wil van der Aalst
• Eindhoven University of Technology, Faculty of
  Technology Management,
• Department of Information and Technology, P.O.Box
  513, NL-5600 MB,
• Eindhoven, The Netherlands.

3
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

4
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.

5
Elements
6
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.

7
Enabled

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

enabled
Not enabled
Not enabled
8
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.

fired
9
Play Token Game

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

10
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).

11
Non-determinism
Two transitions are enabled but only one can fire
12
Example Single traffic light
13
Two traffic lights
OR
14
Problem
15
Solution
How to make them alternate?
16
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/Downl
  oads/pn_applet/pn_applet.html
• FLASH animations http//www.tm.tue.nl/it/staff/wv
  daalst/courses/pm/flash/

17
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.

18
Exercise Train system (2)

• 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.

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Exercise Train system (3)

• 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.)

20
Exercise Train system (4)

• 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.

21
WARNINGIt is not sufficient to understand the
(process) models. You have to be able to design
them yourself !
22
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.

23
Example Ball game
24
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?

25
Exercise Burning alcohol.

• Model C2H5OH 3 O2 gt 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.

26
Exercise Manufacturing a car

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

car
subassembly2
engine
2
chair
subassembly1
4
chassis
wheel
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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 -gt N is the input function, and
• O T x P -gt N is the output function.
• Any diagram can be mapped onto such a four tuple
  and vice versa.

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Formal definition (2)

• The state (marking) of a Petri net (P,T,I,O) is
  defined as follows
• s P-gt N, i.e., a function mapping the set of
  places onto 0,1,2, .

29
Exercise Map onto (P,T,I,O) and S
30
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.

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Enabling formalized

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

32
Firing formalized

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

33
Mapping Petri nets onto transition systems

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

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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
• Let X be the set containing just the initial
  state and let Y be the empty set.
• 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.
• If X is empty stop, otherwise goto 2.

35
Example
(3,2)
(3,1)
(3,0)
(1,3)
(1,2)
(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).
36
Exercise Give the reachability graph using both
notations
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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?

38
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?

39
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.

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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).

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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.

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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.

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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.

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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

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Causality
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Parallelism
47
Parallelism AND-split
48
Parallelism AND-join
49
Choice XOR-split
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Choice XOR-join
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Iteration 1 or more times
XOR-join before XOR-split
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Iteration 0 or more times
XOR-join before XOR-split
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Capacity constraints feedback loop
AND-join before AND-split
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Capacity constraints mutual exclusion
AND-join before AND-split
55
Capacity constraints alternating
AND-join before AND-split
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We have seen most patterns, e.g.
Example of mutual exclusion
How to make them alternate?
57
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
subassembly2
engine
2
chair
subassembly1
4
chassis
wheel
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Modeling problem (1) Zero testing

• Transition t should fire if place p is empty.

?
t
p
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Solution

• Only works if place is N-bounded

t
Initially there are N tokens
N input and output arcs
p
p
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Modeling problem (2) Priority

• Transition t1 has priority over t2

t1
?
t2
Hint similar to Zero testing!
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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.

62
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.

63
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.

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

• Prof.dr.ir. Wil van der Aalst
• Eindhoven University of Technology, Faculty of
  Technology Management,
• Department of Information and Technology, P.O.Box
  513, NL-5600 MB,
• Eindhoven, The Netherlands.

65
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

66
Inability to test for zero tokens in a place
?
t
p
Tricks only work if p is bounded
67
Models tend to become (too) large
Size linear in the number of products.
68
Models tend to become (too) large (2)
Size linear in the number of tracks.
69
Models cannot reflect temporal aspects
Duration of each phase is highly relevant.
70
No support for structuring large models
71
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.

72
Running example Making punch cards
free desk employees
waiting patients
served patients
patient/ employees
73
Extension with color (1)

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

74
Extension with color (2)

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

record Brandstring RegistrationNostring
Yearint Colorstring Ownerstring
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Extension with color (3)

• 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.

The value of the token produced for place sum is
the sum of the values of the consumed tokens.
76
Running example Tokens are colored
77
Running example Places are typed
78
Running example Initial state
start is enabled
79
Running example Transition start fired
New value is created by simply merging the two
records.
stop is enabled
80
Running example Transition stop fired
New values are created by simply spliting the
record into two parts.
81
The number of tokens produced is no longer fixed
(1)
Note that the network structure is no longer a
complete specification!
82
The number of tokens produced is no longer fixed
(2)
The number of tokens produced for each output
place is between 0 and 3 and the sum should be 3.
83
Example

• Model as a colored Petri net.

84
Product and quantity are in the value of the token
The entire stock is represented by the value of a
single token, i.e., a list of records.
85
Types
StockItem
Stock
color Product string color Number int color
StockItem record prodProduct
numNumber color Stock list StockItem
StockItem
86
Extension with time (1)

• Each token has a timestamp.
• The timestamp specifies the earliest time when it
  can be consumed.

87
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.

88
Running example Enabling time

• Transition start is enabled at time 2
  max0,min2,4,4.

89
Running example Delays

• Tokens for place busy get a delay of 3
• _at_3 firing time plus 3 time units

90
Running example Transition start fired

• Transition start fired a time 2.

Continue to play (timed) token game
91
Exercise Final state?
92
Exercise Final state?
93
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.

94
Instead of
95
We can use hierarchy
96
Reuse

• Reuse saves design efforts.
• Hierarchy can have any number of levels
• Transition refinement can be used for top-down
  and bottom-up design

97
Exercise model three (parallel) punch card desks
in a hierarchical manner
98
Analysis of Process ModelsReachability graphs,
invariants, and simulation

• Prof.dr.ir. Wil van der Aalst
• Eindhoven University of Technology, Faculty of
  Technology Management,
• Department of Information and Technology, P.O.Box
  513, NL-5600 MB,
• Eindhoven, The Netherlands.

99
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.

100
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.

101
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)?

102
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?

103
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.

104
Analysis

• 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

105
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)

106
Reachability graph
(1,0,0,0,0,1,0)
(0,0,1,1,0,0,0)
(1,0,0,1,0,0,1)
(0,1,0,1,0,0,0)
(1,0,0,0,1,0,0)
Five reachable states. Traffic lights are safe!
107
Alternative notation
r1o2
o1r2
r1r2x
g1r2
r1g2
108
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.

109
Infinite reachability graph
110
Exercise Construct reachability graph
111
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.

112
Exercise Dining philosophers (2)

• 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?

113
Exercise Dining philosophers (3)

• 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?

114
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.

115
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

1 man 1 woman 2 couple
116
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.)

117
Example traffic light

• r1 g1 o1
• r2 g2 o2
• r1 r2 g1 g2 o1 o2
• x g1 o1 g2 o2
• r1 r2 - x

118
Exercise Give place invariants
119
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.

2 marriage 2 divorce
120
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.

121
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

122
Exercise Give transition invariants
123
Exercise four philosophers

• Give place invariants.
• Give transition invariants

124
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.

125
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.

126
Example
man
woman
marriage
couple
divorce
127
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.

128
Example

• Solutions
• (0,0,0)
• (1,0,1)
• (0,1,1)
• (1,1,2)
• (1,-1,0)

129
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.

130
Example

• Solutions
• (0,0)T
• (1,1)T
• (32,32)T

131
Exercise

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

132
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!

133
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.

134
Uniform distribution
pdf
cumulative
135
Negative exponential distribution
136
Normal distribution
137
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..5fun select1to5()
  C.ran()returns a number between 1 and 5

138
Example
139
Example(2)
140
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).

141
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.

142
Top-level page main
143
Subpage gas_station
144
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.

145
Alternatives

• Model the following alternatives
• 5 waiting spaces
• 2 pumps
• 1 faster pump

146
Simulation of a Production system
X
Y
Z
C
S
A
B
C
147
Data
Use distributions
148
Top level page main
149
Sub page supplier
150
Sub page customer
151
Sub page work_center
152
Overview

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

153
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).

154
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

155
Top-level page
156
Page philosopher
157
Flat model is obtained by replacing substitution
transitions by subpages

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

Repeat 5 times...
158
Alternative page
159
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.