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000 Petri Net 讲义(英文)


Petri nets
Classical 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.

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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

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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.
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p4 place

Elements
(name) place

t34 p3

t43

transition

t23 p2

t32

(name)

transition

token

arc (directed connection)
t12 t21 p1

token

t01 p0

t10

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free

Rules
wait enter before make_picture after leave gone

occupied

? ? ? ?

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.

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

wait

enter

before

make_picture

after

leave

gone

occupied

enabled
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Not enabled

Not enabled

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
fired
wait enter before make_picture after leave gone

occupied

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Play “Token Game”
? In the new state, make_picture is enabled. It will fire, etc.
free

wait

enter

before

make_picture

after

leave

gone

occupied

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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).
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Non-determinism
t34

p4

p4

t43 p3

t34 p3

t43

t23 p2

t32 transition t23 fires

t23 p2

t32

t12 p1

t21

t12 p1

t21

Two transitions are enabled but only one can fire
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t01 p0

t10

t01 p0

t10

Example: Single traffic light

rg

green

red

go

orange

or
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Two traffic lights
rg rg

rg

green
green green

red

go

red

go

OR
orange

red

go

orange

or

or

orange

or
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Problem

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Solution
rg1 rg2

g1

g2

r1

go1

x

go2

r2

How to make them alternate?
or1

o1

o2

or2

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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/

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Exercise: Train system (1)
? Consider a circular railroad system with 4 (oneway) 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.

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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.)

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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.

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WARNING
It is not sufficient to understand the (process) models. You have to be able to design them yourself !
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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

wait

enter

before

make_picture

after

leave

gone

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Example: Ball game
red rb black

rr

bb

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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?
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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.

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Exercise: Manufacturing a car
? Model the production process shown in the BillOf-Materials.
car subassembly2 engine 2 chair subassembly1 4 chassis
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wheel

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.
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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, … }.

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

rr

bb

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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:

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

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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:
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.
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?

Example
red rb black

(3,2)

(3,1)

(3,0)

(1,3)
rr bb

(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).
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Exercise: Give the reachability graph using both notations
rg1 rg2

g1

g2

r1

go1

x

go2

r2

o1

o2

or1

or2

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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?
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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?
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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

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Parallelism: AND-split

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Parallelism: AND-join

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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

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AND-join before AND-split

Capacity constraints: alternating

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AND-join before AND-split

We have seen most patterns, e.g.:
Example of mutual exclusion
rg1 rg2

g1

g2

r1

go1

x

go2

r2

How to make them alternate?
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or1

o1

o2

or2

Exercise: Manufacturing a car (2)
? Model the production process shown in the Bill-Of-Materials with car resources. ? Each assembly step engine requires a dedicated machine and an operator. ? There are two operators subassembly1 and one machine of each type. 4 wheel ? Hint: model both the start and completion of an assembly step.

subassembly2 2 chair

chassis
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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
N input and output arcs

t p’

Initially there are N tokens

p
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Modeling problem (2): Priority
? Transition t1 has priority over t2

t1

?
t2
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Hint: similar to Zero testing!

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.
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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.
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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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