Abstract Petri Net Notation
FalkoBause Peter Kemp er Pieter Kritzinger
Forschungsb ericht
der Universit at Dortmund
Lehrstuhl Informatik IV
Universit at Dortmund
D Dortmund
Germany
e mail fbause kemp erg ls informatik uni dortmund de
Pieter Kritzinger is from
Data Network Architecture Lab oratory
Department of Computer Science
UniversityofCapeTown
Private Bag RONDEBOSCH
South Africa psk cs uct ac za
Abstract
The need to interchange the description of Petri Net mo dels amongst re
searchers and users was recognised as long ago as This do cument
prop oses an Abstract Petri Net Notation APNN in whichvarious nets
can b e describ ed using a common notation The use of the notation in the
context of Petri Net software to ols is shown and the general requirements
for such a notation to b e generally acceptable are suggested Keywords in
a
the notation are similar to L T X commands in order to supp ort readabil
E
ity By employing an appropriate style le a net description can thus b e
a
included directly into a L T X source do cument The notation is given for
E
untimed Place Transition Nets and Coloured Petri Nets as well as Gener
alised Sto chastic Petri Nets and Hierarchical Petri Nets
So eine Arbeit wird eigentlich nie fertig
man muss sie fur fertig erkl aren
wenn man nach Zeit und Umst anden
das M oglichste getan hat
J W Go ethe Italienische Reise
Intro duction
Petri Nets have b ecome widely accepted as a formalism for describing
concurrent systems and an active community of researchers exist in the eld
As the theory of Petri Nets develop ed so were software to ols develop ed to
assist the analysis of Petri Nets
It was thus recognised as long ago as by Berthelot et al that
it would b e an advantage if Petri Net researchers could share their mo del
descriptions in some common notation or interface and thereby exploit the
existing variety of to ols and user interfaces amongst themselves Suchan
interface would have the functionality illustrated in Fig where the P
i
indicate translations from one to the other of the functions illustrated in that gure
PRINTED GRAPHICAL HUMAN ABLE TEXT
REPRESENTATION READ
P
P
PETRI NET DESCRIPTION IN ABSTRACT NOTATION
INDIVIDUAL TOOL
P P
LOCAL INTERNAL
REPRESENTATION
P
P P P
N
GRAPHICAL INTERFACE
ANALYZER ANALYZER N
DISPLAY
RESULTS RESULTS
Figure The context of the APNN
The biggest advantage of such a common notation is the ability to share
analysis algorithms and the implementations of these In this way one can
compare the correctness and e ciency of such algorithms and avoid their
rep eated implementation The authors have for instance b een able to use
the Markovchain analyser USENUM develop ed at one institution for
the analysis of Petri Net mo dels created with a Petri Net user interface
develop ed at the other
Once one accepts the idea of a common interface description various other
advantages other than sharing co de b ecome apparent One could translate
the description of a net directly into a printed graphical representation or
textual description for do cumentation
Requirements of the Notation
In general the net notation should satisfy the following criteria for it to b e
useful to a wide community
Exchangeable A net description should b e in such a format that it can
b e easily exchanged in electronic form via e g email or ftp b etween
di erent sites Exchangeability is the ma jor reason for having sucha
textual notation at all
Extensible More p owerful i e high level Petri Nets should have a nota
tion such that simpler net classes like Condition Event Nets app ear
as sp ecial cases of such a formalism in a natural way i e intuitive
and concise Missing information should default to appropriate val
ues A notation should extend from simple nets to high level nets
in a straightforward way and b e exible enough to allow for future
extensions
Mo dular and hierarchical These two requirements are strongly related
to each other The notation should b e suchthatPetri Net descriptions
in a le can b e re used as part s of the description of another larger
net or higher level Petri Net
Readable Although the graphical description of Petri Nets is one of its
ma jor advantages in terms of human readability it is a drawback for
inter to ol communication Softw are to ols always have a textual de
scription for nets in order to store them in les Such a le format
is in most cases very concise but not human readable A useful no
tation should compromise b etween conciseness and human readability
at least in a way that it is easily transformed into a human readable
format Ideally this transformation should b e p erformed by public
domain software which is generally available
The general idea is then
other p ersons should b e able to read the description even with
out access to the corresp onding to ols or graphical user interfaces
GUIs
it should b e easy to include the description into printed do cu
ments
Portability of the graphical representation of the net derived directly from
an abstract description is another matter Graphical layout where and
whether arcs cross and so on is very much a matter of p ersonal preference
One could sp ecify the p osition of elements of the net with resp ect to some
prede ned origin in the notation as T X do es in fact This is not simple to
E
do without drawing the net in some form or other on a co ordinate system
Although not imp ossible to do the authors have decided to avoid the issue
for now
For the notation the authors decided to cho ose key words in the textual
a
description of a net which are similar to L T X commands in order to supp ort
E
readability Byemploying an appropriate style le a net description can b e
a a
directly included into a L T X do cument Since L T X is public domain
E E
and widely accepted within the scienti c community we decided to use
a
the descriptivepower of L T X as a formal language to de ne a textual
E
description of Petri Nets
The remainder of this rep ort is structured as follows We distinguish b e
tween timed Petri Nets and untimed Petri Nets In the former category we
de ne Place Transition Nets and give the grammar describing their abstract
notation in Sec This grammar is then extended to Coloured Petri Nets
in Sec
Where timed Petri Nets are concerned we start with Generalised Sto chastic
Petri Nets GSPN de ned in Sec and Coloured Generalised Sto chastic
Petri Nets in Sec Hierarchies are considered in Sec In Sec we
de ne Hierarc hical Queueing Petri Nets which are new in that they include
b oth places in the ordinary sense of Petri Nets and a new typ e of place called
queueing places
The abstract notation of everyclassofPetri Net is describ ed in Backus Naur
Form BNF throughout the remainder of this rep ort Common features of
the extended BNF such as rep etition are not used since it clashes with the
terminals used for the description of the notation itself Non terminals are
written in upp er case terminals in lower case
UntimedPetri Nets
Petri Nets were inventedin by Carl Adam Petri and are a formalism
for the description of concurrency and synchronisation in systems Since rst
describ ed byPetri manyvariations of his original nets have b een de ned
Since the simplest of such nets known as Condition Event Nets are easily
describ ed by Place Transition Nets with a place capacityofone we shall
regard Place Transition Nets as the basic kind of net to b egin with In this
section we de ne an APNN for Place Transition Nets and then we extend
this notation for Coloured Petri Nets
In this do cumentwehave used simple style le macros but these can b e changed
and or extended easily to improve readabili ty
In general Petri Nets comprise the following comp onents
places usually drawn by circles whichtypically mo del conditions or ob
jects
tokens usually drawn by black dots which represent the sp eci c value of
the condition or ob ject
transitions usually drawn by rectangles These mo del activities which
change the values of conditions and ob jects
arcs sp ecifying the interconnection of places and transitions
Petri Nets are clearly bipartite graphs i e one may only connect a place to
a transition or vice versa but may not connect places to places or transitions
to transitions
Place Transition Nets
De nition A Place Transition Net is a tuple PN P T I I M
where
P fp p g is a nite and non empty set of places
n
T ft t g is a nite and non empty set of transitions
m
P T
e the backward and forward incidence functions I I P T N ar
respectively
M P N is the initial marking
APN Notation for Place Transition Nets
In the notation which follows the authors have adhered to the requirements
describ ed in Sec Nevertheless such a notation is obviously not unique and
frequently a matter of p ersonal preference For instance in the notation ID
is a non terminal which denotes a unique place transition or arc identi er
Place and transition identi ers are necessary to sp ecify arcs uniquely The
existence of a unique identi er is useful in that it allows one no de description
to refer to another one byanentity like ID
The notation also allows for an additional description whichmaybeempty
and need not b e unique asso ciated with an entity name at eachnode
description to allow the mo del builder to include a name explanation or
comment ab out the function of the no de in the mo del
A net description using the notation will obviously b e more concise if cer
tain attributes are assumed to have generally accepted default values We
thus de ne the default value for an arc weight to b e and for the initial
marking of a place A place has a capacity of in nity unless explicitly stated
otherwise
In the notation we assume the usual pro duction rules for integers and strings
The keyword empty denotes the empty string
Terminal symb ols
beginnet endnet place transition like arc
name init from to capacity weight
Non Terminal symbols
NET ELEMENT PLACE TRANSITION ARC ID NAME INIT
WEIGHT CAP STRING INTEGER
Pro ductions
NET beginnet ID ELEMENT endnet
ELEMENT empty
j PLACE ELEMENT
j TRANSITION ELEMENT
j ARC ELEMENT
ID STRING
PLACE place ID NAME INIT CAP
j place ID like ID
NAME empty j name STRING
INIT empty j init INTEGER
CAP empty j capacity INTEGER
TRANSITION transition ID NAME
ARC arc ID from ID to ID WEIGHT
WEIGHT empty j weight INTEGER
Start symbol NET
We omit the explicit de nition of non terminal and terminal symb ols in the
remainder of the descriptions of the APNN since these are distinguished
anywayby upp er and lower case letters resp ectively p1 p2 p3
2
t1t2 t3
2
p4 p5 p6
Figure Example Place Transition Net
Example Consider the Place Transition Net il lustratedinFig The
fol lowing is the corresponding APNN description of that net
beginnet examplenet
place p init place p
place p like p place p
place p init place p
transition t transition t transition t
arc a from p to t arc a from t to p
arc a from p to t arc a from t to p
arc a from t to p arc a from p to t
arc a from p to t arc a from t to p
arc a from p to t weight
arc a from t to p weight
endnet
As previously explained the authors decided to cho ose key words in the
a
textual description of a net which are similar to L T X commands In this
E
way the net de nition in the previous example can b e translated by use of
a
appropriate L T X macros to the following description intended to b e read
E
rather than b e pro cessed bymachine
Example examplenet
Place p marking f g
Place p
Place p is defined as for p
Place p
Place p marking f g
Place p
Transition s t t t
Arc a from p to t
Arc a from t to p
Arc a from p to t
Arc a from t to p
Arc a from t to p
Arc a from p to t
Arc a from p to t
Arc a from t to p
Arc a from p to t weight
Arc a from t to p weight
Coloured Petri Nets
In Coloured Petri Nets as de ned by K Jensen eachtoken has an
asso ciated data value or colour which b elongs to a sp eci ed typ e of arbitrary
complexity A place can contain various colours which form a set the so
called colour set A colour function assigns a colour set to each place of the
net
entions Since it is widely in use we follow Jensen and use the conv
The typ e of a variable v is denoted byTyp e v
The typ e of an expression expr is denoted byTyp e expr
Thesetofvariables in an expression expr is denoted byVar expr
S denotes the set of all multisets over S
MS
De nition A Coloured Petri Net is a tuple P T A C G E I where
is a set of non empty colour sets
P is a nite and non empty set of places
T is a nite and non empty set of transitions
A isa nitesetofarcs such that
P T P A T A
C is a colour function P
G is the guard function from T into boolean expressions such that
ype G t B and Type Var G t t T T
E is the arc function from A into expressions such that
a A Type E a C p a and Type Var E a
MS
where p a denotes the placeconnectedbyarc a
I is the initialisation function de nedfrom P into closedexpressions
such that
p P Type I p C p
MS
Programming languages can b e employed for the precise de nition of ex
pressions and colour sets In a derivative of Standard ML so called
CPN ML is used In this rep ort we do not distinguish b etween these two
languages explicitly
APN Notation for Coloured Petri Nets
In order to avoid tedious rep etition only the di erences b etween the nota
tion for Place Transition Nets and that needed to sp ecify Coloured Petri
Nets are given next
We assume that the non terminal ML EXPRESSION is an ML expression
according to whichevaluates to the appropriate typ e i e either integer
b o olean or multiset In fact since ML expressions can contain names of
arbitrary complex user de ned ML functions such ML functions can b e
sp eci ed in a separate ML sp eci cation le which can b e then referred to
in the net description by the APNN seeML FILENAME statement The
same holds for ML de nitions which are denoted by the non terminal ML
DEFINITION
Pro ductions
ELEMENT empty
j PLACE ELEMENT
j TRANSITION ELEMENT
j ARC ELEMENT
j TYPEDEF ELEMENT
j seeML FILENAME ELEMENT
FILENAME STRING
PLACE place ID NAME INIT CAP COLOUR
j place ID like ID
INIT empty j init MULTISET
CAP empty j capacity MULTISET
COLOUR empty j colour COLOURSET
COLOURSET like ID j ML DEFINITION
MULTISET INTEGER j INTEGER STRING MSLIST
MSLIST empty j MULTISET
TRANSITION transition ID NAME GUARD
j transition ID like ID
GUARD empty j guard BOOLEXPR
BOOLEXPR like ID j ML EXPRESSION
ARC arc ID from ID to ID WEIGHT
WEIGHT empty j weight MULTISETEXPR
MULTISETEXPR like ID j ML EXPRESSION
TYPEDEF typedef ID ML DEFINITION
It is p ossible to omit the guard in a transition sp eci cation as de ned by
the grammar In this case the guard expression defaults to true Since
Place Transition Nets are considered as a sp ecial case of CPNs we de ne a
default colour set f g such that a multiset can b e a simple integer value n
which is equivalentton If the colour set consists of a single colour x a
missing arc weight defaults to x and an integer arc weight n is equivalent
to n x
func x y z
t
x y
z
z
C E F C F
T
C T E
z
x y
z
t
func x y z
Figure Example Coloured Petri Net
Example Consider the simple Coloured Petri Net il lustratedinFig
which models the wel l known dining philosophers problem The example is
for philosophers so that
fp p g ifp T or p E
C p
ff f g ifp F
The guard function func x y z in that gure is de nedtobe
func x y z if z p x f y f
i i i
then TRUE
el se F ALS E
and i means i modulo
The following is the description of that net in the APN notation
beginnet Coloured Petri Net
seeML MLSpecification
place T init p p p p p colour with p jp jp jp jp
place E colour like T
place F init f f f f f colour with f jf jf jf jf
transition t guard func x y z
transition t like t
arc a from T to t weight z
arc a from t to E weight z
arc a from E to t weight z
arc a from t to T weight z
arc a from t to F weight x y
arc a from F to t weight x y
endnet
a
The style macros for the corresp onding L T X description given b elow di er
E
somewhat from those used for Place Transition Nets
Coloured Petri Net
for ML function specification see MLSp eci cation
f p p p p p g Place T marking
coloursetf with p jp jp jp jp g
Place E
coloursetf is defined as for T g
Place F marking f f f f f f g
coloursetf with f jf jf jf jf g
Transition t guardf func x y z g
Transition t is defined as for t
Arc a from T to t weight z
Arc a from t to E weight z
Arc a from E to t weight z
Arc a from t to T weight z
Arc a from t to F weight x y
Arc a from F to t weight x y
Further APN Notation for Coloured Petri Nets
The de nition of CPNs as given by Jensen in employs ML expressions
for the description of multisets arc expressions and guards An earlier
de nition of CPNs explicitly sp eci es all ring mo des of a transition
by a nite set of transition colours This representation known as the
functional representation has to b e used for the purp ose of analysis It
is also the approach used for Coloured GSPNs de ned in and QPNs
b oth of whichusea ow oriented view of a system
When all ring mo des of a transition explicitly sp eci ed the de nition of
CPNs b ecomes
De nition A Coloured Petri Net CPN is a tuple
CP N P T C I I M where
is a set of non empty colour sets
P is a nite and non empty set of places
T is a nite and non empty set of transitions with P T
C is a colour function P T
I and I are the backward and forward incidence functions de ned
on P T such that
I p t I p t C t C p p t P T
MS
M is a function de nedon P describing the initial marking such that
M p C p p P
MS
This representation can b e describ ed in the APNN for CPNs of Sec
by mo difying the following pro ductions
COLOURSET like ID
j with STRING COLOUR REST
COLOUR REST empty
j j STRING COLOUR REST
GUARD empty j guard GUARDEXPR
GUARDEXPR like ID
j MODE VAR MODE CONST GUARD REST
GUARD REST empty
j orelse MODE VAR MODE CONST GUARD REST
MODE VAR STRING
MODE CONST STRING
MULTISETEXPR like ID
j MULTISET
j case MODE VAR of MODE CONST
MULTISET MSEXPR REST
j if MODE VAR MODE CONST
then MULTISET MSEXPR ELSE
MSEXPR REST empty
j j MODE CONST MULTISET MSEXPR REST
MSEXPR ELSE empty
j else MULTISET
j else MULTISETEXPR
MULTISET INTEGER
j INTEGER STRING MSLIST
MSLIST empty j INTEGER STRING MSLIST
Note that a guard expression lists all p ossible ring mo des by a disjunction
over all p ossible bindings of a mo de variable e g a transition with ring
mo des black and blue would have a guard guard mode black orelse mode
blue
Obviously this is not a straightforward solution in terms of a functional
representation but it integrates b oth kinds of CPNs into a single notational
framework
Sto chastic Petri Nets
In this section we consider Petri Nets to which time has b een added There
are a lot of concepts for intro ducing time e g deterministic and sto chastic
times timing information attached to places or transitions etc see e g
Here we consider time augmented nets to show the extensibilityof
the APNN The most well known of such nets are Generalised Sto chastic
Petri Nets GSPNs
Generalised Sto chastic Petri Nets
As the name indicates Generalised Sto chastic Petri Nets GSPNs are timed
nets An exp onentially distributed ring time is asso ciated with timed tran
ta sitions Such nets also contain immediate transitions which re instan
neously and with priorityover timed transitions Furthermore the de ni
tion of a GSPN p ermits inhibitor arcs and di erent levels of priority for
immediate transitions
De nition A GSPN cf is a tuple GS P N PN Pr H W
where
PN P T I I M is the underlying Place Transition Net
H P T is a set of inhibitor arcs
Pr is an assignment of priorities to transitions with priority for
timedtransitions and higher priorities for immediate transitions
and
W w w is an array whose entry w R
m i
is a rate of a negative exponential distribution specifying the ring
delay when transition t is a timedtransition or
i
is a weight specifying the relative ring frequency when transition
t is an immediate transition
i
APN Notation for GSPNs
The default value for arc weights and ring rates or frequencies is Unless
otherwise sp eci ed a transition is considered to b e timed Note that if
all priorities equal zero then the de nition of a GSPN coincides with the
de nition of a Sto chastic Petri Net SPN
The grammar for GSPNs extends the grammar of Place Transition Nets
by the concepts of priority weights and inhibitor arcs This leads to the
following mo di cations
TRANSITION transition ID NAME PRIO T WEIGHT
j transition ID like ID
PRIO empty j prio INTEGER
T WEIGHT empty j weight REAL
ARC arc ID from ID to ID WEIGHT A TYPE
A TYPE empty j type ordinary j type inhibitor
A missing typ e sp eci cation of an arc defaults to an ordinary arc We assume
the usual pro duction rules for the derivation of real constants from REAL
Note that marking dep endent ring rates are not considered for now p1 p2 p3
2
t1t2 t3
2
p4 p5 p6
Figure Example GSPN
Example Consider the simple GSPN il lustrated in Fig The fol lowing
is the de nition of that net using the APN notation
beginnet gspnexample
place p init place p
place p like p place p
place p init place p
transition t prio weight
transition t prio weight
transition t
arc a from p to t arc a from t to p
arc a from p to t arc a from t to p
arc a from t to p arc a from p to t
arc a from p to t arc a from t to p
arc a from p to t weight
arc a from t to p weight
endnet
a
The L T X Output is
E
gspnexample
Place p marking f g
Place p
Place p is defined as for p
Place p
Place p marking f g
Place p
Transition t priority weight
Transition t priority weight
Transition t
Arc a from p to t
Arc a from t to p
Arc a from p to t
Arc a from t to p
Arc a from t to p
Arc a from p to t
Arc a from p to t
Arc a from t to p
Arc a from p to t weight
Arc a from t to p weight
Coloured Generalised Sto chastic Petri Nets
In the case of Generalised Sto chastic Petri Nets tokens do not have colours
In order to merge the two concepts of coloured tokens and timed nets tokens
have colours and an execution rate is now asso ciated with every ring mode
of a transition We call the resultant net a Coloured Generalised Sto chastic
Petri Net CGSPN
Such nets are however not yet considered to b e hierarchical Hierarc hical
nets are describ ed in Sec
De nition Using the same convention as that in Sec we de ne a
CGSPN as a tuple CGSPN P T A C Pr H G W E I where
is a set of non empty colour sets
P is a nite and non empty set of places
T is a nite and non empty set of transitions
A isa nitesetofarcs such that
P T P A T A
C is a colour function P
Pr is an assignment of priorities to transitions with priority for
timedtransitions and higher priorities for immediate transitions
H P T is a set of inhibitor arcs
G is the guard function from T into expressions such that
ype Var G t t T Type G t B and T
The semantics of inhibitor arcs in CGSPNs follows that of
W is the weight function from the transitions T into expressions such
that t T
Type W t R and Type Var W t which speci es the
ring delay when transition t is a timedtransition and the relative
ring frequency when transition t is an immediate transition
E is the arc function from A into expressions such that
a A Type E a C p a and Type Var E a
MS
wherewe p a denotes the placeconnectedbyarc a
I is the initialisation function de nedfrom P into closedexpressions
such that
p P Type I p C p
MS
APN Notation for CGSPNs
The notation clearly has to provide the sp eci cation of colour sets for places
and typ es of colour sets Furthermore transitions can re according to di er
ent ring mo des which results in generalising arc weights to expressions that
evaluate to multisets and a guard expression whichevaluates to a b o olean
value Therefore transition descriptions are extended by guards arc weights
are generalised to contain expressions whichevaluate to a multiset over the
colour set attached to the input place Place descriptions haveacolourset
description and its attributes for initialisation and capacity are generalised
to expressions of the appropriate typ e
In the following the complete set of pro ductions for CGSPNs is given
NET beginnet ID ELEMENT endnet
ELEMENT empty
j PLACE ELEMENT
j TRANSITION ELEMENT
j ARC ELEMENT
j TYPEDEF ELEMENT
j seeML FILENAME ELEMENT
FILENAME STRING
ID STRING
PLACE place ID NAME INIT CAP COLOUR
j place ID like ID
name STRING NAME empty j
INIT empty j init MULTISET
CAP empty j capacity MULTISET
COLOUR empty j colour COLOURSET
COLOURSET like ID j ML DEFINITION
TRANSITION transition ID NAME PRIO T WEIGHT GUARD
j transition ID like ID
PRIO empty j prio INTEXPR
T WEIGHT empty j weight REALEXPR
GUARD empty j guard BOOLEXPR
ARC arc ID from ID to ID WEIGHT A TYPE
WEIGHT empty j weight MULTISETEXPR
A TYPE empty j type ordinary j type inhibitor
MULTISET INTEGER STRING MSLIST
MSLIST empty j MULTISET
INTEXPR like ID j ML EXPRESSION
REALEXPR like ID j ML EXPRESSION
MULTISETEXPR like ID j ML EXPRESSION
BOOLEXPR like ID j ML EXPRESSION
TYPEDEF typedef ID ML DEFINITION
De nitions of integer b o olean real and multiset expressions follow the ML
syntax Arcs default to those of typ e ordinary
Example Consider the familiar Producer Consumer problem modi edto
represent a gizmo assembly process The associated CGSPN is il lustratedin
Fig Gizmos areonlyassembledasneededandrequirethreewheels a
body and a door It takes di erent exponential ly distributed times to build
els body and door respectively and assembling a gizmo takes an the whe
exponential time with a di erent mean rate
In other words transition CONSUME transition CO is enabledwhenthree
of the tokens with colour WHEEL and one token eachofcolour BODY
and DOOR areonplace B A token of colour GIZMO on place C indi
cates that a gizmo needs to be assembled if possible Parts areonlyproduced
as they areconsumed
The various functions are de ned as fol lows
fWHEEL BODY DOORg if p fB P g
C p
fGIZMOg if p fC C P g
if t fI I g
Pr t
otherwise
H
x WHEEL y BODY z DOOR g GIZMO if t fI COg
G t
TRUE otherwise
case x of WHEEL
j BODY
W t j DOOR if t PR
if t CO
otherwise
x if a f P PR PR B g
x y z if a f B C O I P g E a
g if a f CO P P I CO C C I I C C CO g
fGIZMOg if p fC P g
I p
otherwise
P
x
C
PR x y z
PRODUCE
I
x
B
g
g
g
x y z
CO
I
g
P CONSUME
g
g
C
Figure Coloured GSPN of a gizmo assembly pro cess
beginnet cgspnexample
place P colour with WHEELjBODYjDOOR
place P init GIZMO colour with GIZMO
place B like P
place C init GIZMO colour with GIZMO
place C colour with GIZMO
transition I prio
guard x WHEEL and y BODY and z DOOR and g GIZMO
transition I prio
transition PR name PRODUCE prio
weight case x of WHEEL j BODY j DOOR
transition CO name CONSUME prio
guard like I weight
arc from P to PR weight x
arc from PR to B weight x
arc from B to CO weight x y z
arc from CO to P weight g
arc from P to I weight g
arc from I to P weight x y z
arc from C to CO weight g
arc from CO to C weight g
arc from C to I weight g
arc from I to C weight g
endnet
Hierarchies
Petri Nets tend to b ecome very large if used for mo deling real world prob
lems One wayofavoiding the di culty of representing such nets graphically
is to de ne one or more subnets and to p ossibly build a hierarchyofsub
nets In this way large nets can b e represented in a structured way which
moreover allows one to use a top down design for complex mo dels as well
as the reuse of net descriptions or identical submo dels
Hub er et al describ e ve concepts of hierarchical CPNs substitution
transitions substitution places invo cation transitions fusion sets for places
and fusion sets for transitions They de ne the semantics of these hierarchy
concepts by giving a translation into an equivalent non hierarchical CPN
In the following we demonstrate how these concepts can b e expressed in
an extended APNN but give only a very brief informal description of the
semantics of these hierarchical concepts for details the reader is referred to
Substitution transitions
We use the notation in and consider a hierarchical CPN as a set of
related pages If a page has a re ned no de n this page is called sup erpage
in relation to the subpage whichgives the re nement of n In this section
we consider the re nement of transitions by substitution Substitution of
transitions is very much like macro expansion in programming languages
Figure gives an example with a page N containing a substitution transition
n and the subpage N which corresp onds to n No des in the preset or p ostset
of n are called so cket no des Their counterparts inside the subpage are called
p ort no des and a binding function b must b e given which assigns a p ort
no de to eachsocket no de Note that this function need not b e injectiveor
surjective e g b p b p and p is not b ound to anysocket no de of n
p1 Superpage N
p2 np3
b b b
Substitution transition
Subpage N’
p1’ t1 p3’
p2’ t2 p4’
Figure Trivial example of substitution transition
An APNN description has to provide the following information
for the substitution transition
the ID of the subpage given by substitute ID
the binding function b sp eci ed in the arc description which
connects the transition with its input and output places
The binding b of an input place is given within an arc descrip
tion by a sequence of bind SOCKETID with TRANSITIONID
cont PORTID arc attributes
cont PORTID means that a no de with identity PORTID is an
element of the subpage contained in the no de of the preceding
attribute namely the no de with TRANSITIONID
For an output place the corresp onding arc attributes are
bind TRANSITIONID cont PORTID with SOCKETID
for no des of a subpage
an attribute which denotes input and output p orts i e port in
port out and port io
Since the preset and p ostset of a substitution transition de ne the set of
so cket no des a sp ecial notation to distinguish so cket no des is not necessary
The new elements of an APNN description for the example pages N and N
are
beginnet N
transition n substitute N
arc p n from p to n bind p with n cont p
arc p n from p to n bind p with n cont p
arc np from n to p bind n cont p with p
endnet
beginnet N
place p port in place p port in
place p port out place p port out
endnet
The reason for distributing the binding information over arcs connecting
the preset transition and p ostset is that the existence of such a binding
dep ends on the existence of suchanarc Furthermore in case of re ning
adjacent no des the binding information dep ends on b oth re nements and
is not exclusive to one of its no des
Substitution places
Substitution places are analogous to substitution transitions apart from a
di erent semantics i e the translation into a non hierarchical net is di erent
from the translation of substitution transitions Nevertheless the APNN
description can b e handled analogously a substitution place sp eci cation
contains the subpage ID by substitute ID and the binding function is
distributed over the arc description just as for substitution transitions Port
no des of the subpage have an additional attribute port in port out
or port io
Invo cation transitions
Invo cation transitions are like function calls in fact they even allow recur
sive re use of subpages The binding of so cket and p ort no des is equivalent
to substitution transitions Wechange substitute ID to invoke ID
to distinguish invo cation transitions from substitution transitions in the
APNN Since subpages of invo cation transitions allow for explicit exit
statements to denote termination conditions of an invo cation place and
transition sp eci cations can b e extended byan exit attribute in the APN
notation
Fusion sets for places and fusion sets for transitions
The main idea ab out fusion is to fold a set of no des into a single no de
Consequently a fusion set contains an arbitrary numb er of places or an
arbitrary numb er of transitions
Hub er et al distinguish three typ es of fusion sets
Page fusion set This is just a drawing convenience in graphical user
interfaces in case of a single page instance Incaseofseveral page
instances a page fusion set folds all memb ers from all page instances
into a single no de
Instance fusion set This means that all memb ers p er instance of a page
are folded into a single instance sp eci c no de
Global fusion set This allows fusion set memb ers from all pages and all
page instances to b e merged into a single no de
Fusion sets are integrated into the APNN by extending a net description
with fuse ID TYPE IDLIST where ID provides a unique identi er for
a fusion set TYPE is a string constant page inst or global IDLIST is a
j separated list of unique no de identi er This is not critical in case of page
or instance fusion sets but multiple reuse of a page can lead to di culties
in global fusion sets In order to handle this we assume an appropriate
naming convention as mentioned in e g for Design CPN to obtain
unique identi er
The information contained in a fusion set de nition is aggregated into a
new net element in APNN instead of distributing the information over its
memb ers One reason for this is to avoid rep etitive sp eci cation of typ e and
ID But mainly b ecause conceptually a new no de is created by a fusion set
i e the fusion of e g transitions results into a new transition whose guard
is a conjunction of all guards of the fusion set members
Using indep endent copies of subpages lead to the notion of instances of a page a x y
Substitution place Substitution transition y1 px x1 py
x2 y2 Substitution transition Substitution place Port refinement Port refinement ppx1 py1 px1 ppy1
px2 ppy2
ppx2 py2
Figure Pathological example for re ned adjacent no des with p ort re ne
ment
Re nement of adjacentnodes
The hierarchical concepts regarded so far reveal certain di culties to handle
the re nement of adjacent no des a case which is esp ecially considered in
We try to demonstrate the di culties by means of an arti cial example as
shown in Fig whicheven contains re nement of p ort no des
In Chapter R Fehling suggests to sp ecify a binding b etween
re ned p orts e g b etween ppx and ppy Here for consistency we suggest
instead an explicit binding b etween so cket no des e g x and x and re ned
p ort no des e g ppy and ppy The binding information is describ ed with
arc a x y in APNN by the following statement
arc a from x to y
bind x cont px cont ppx with y cont y
bind x cont px cont ppx with y cont y
bind x cont x with y cont py cont ppy
bind x cont x with y cont py cont ppy
In p ort re nements do not contain p ort no des to match their environ
ment We decided to supp ort this as well in order to avoid arcs b etween
separate page descriptions and to keep the descriptions homogeneous In
our example arcs x px and x px carry the binding for the p ort re ne
mentofpx
arc x px from x to px
bind x with px cont px
arc x px from x to px
bind x with px cont px
Note that all implicit bindings like the binding b etween px and y can b e
extracted from the given binding sp eci cation by shortening the sequences
of cont statements e g bind x cont px with y Implicit bindings
and implicit arcs are regarded in detail in
All these concepts of a hierarchyarenow used for the notation of Hierarchical
CGSPNs which serve as an example The grammar of CGSPNs is extended
by the following to accommo date Hierarchical CGSPNs
APN Notation for Hierarchical CGSPNs
Mo di ed Pro ductions
NET empty
j inputnet ID NET
j beginnet ID ELEMENT endnet NET
j beginnet ID like ID endnet NET
ELEMENT empty
j PLACE ELEMENT
j TRANSITION ELEMENT
j ARC ELEMENT
j TYPEDEF ELEMENT
j seeML FILENAME ELEMENT
j FUSION ELEMENT
FUSION fuse ID TYPE ID IDLIST
TYPE page j inst j global
IDLIST empty j j ID IDLIST
PLACE place ID NAME INIT CAP COLOUR P TYPE PORT EXIT
j place ID like ID
P TYPE empty j substitute ID
j port io PORT empty j port in j port out
EXIT empty j exit
TRANSITION transition ID NAME T TYPE PRIO T WEIGHT GUARD
PORT EXIT
j transition ID like ID
T TYPE empty j substitute ID j invoke ID
ARC arc ID from ID to ID WEIGHT A TYPE BIND
BIND empty j bind ID CONT with ID CONT
CONT empty j cont ID CONT
Hierarchical Queueing Petri Nets
Hierarchical Queueing Petri Nets HQPNs are a sp ecial kind of hierarchi
cal Petri Nets with certain extensions to integrate queueing networks The
main motivation is to combine hierarchical description techniques with cor
resp onding hierarchical solution techniques We describ e the APNN for
this mo deling formalism for several reasons
to demonstrate the extensibility of the APNN to further Petri net
formalisms
currently workisunderway to supp ort HQPNs by an adequate soft
ware to ol which supp orts the APNN
In Bause et al intro duced the concept of a queued place into
net theory for the purp ose of p erformance analysis see Fig A queued
place combines the notion of a queue where tokens are queued and served
according to a certain scheduling strategy with a dep ository which is iden
tical to an ordinary place Input transitions of a queued place re tokens
into the queue where they are not available to the output transitions until
service completion On service completion the tokens are put into the de
p ository whence they are available to output transitions in the conventional
way They called the new typ e of net a Queueing Petri Net QPN
QPNs additionally allow the re nement of places leading to Hierarc hical
QPNs see Fig
dep ository
queue
Figure Example queued place of a HQPN and its shorthand notation
Figure Example subnet place of a HQPN and its shorthand notation
De nition HQPN A Hierarchically Queueing Petri Net HQPN is a
triple HQPN CGSPN QS RF where
CGSPN P T A C Pr H G W E I as de nedbefore is the
underlying coloured GSPN
QS is a function de nedonP P giving an assignment of a queue
Q
ing speci cation to a queuedplace p P
Q
RF is a function de nedon P P n P giving the re nement of a
R Q
place p P as a HQPN
R
Hence HQPNs are CGSPNs where some places are queued places and some
places are substitution places Note that by de nition no des cannot b e b oth
a substitution place and a queued place Fusion sets are supp orted with the
constraint that only substitution places can b e fused and that all substitution
places of a fusion set share the same subpage The page hierarchy which
results from a HQPN forms a directed acyclic graph
The description of the underlying CPN is based on the APNN of the func
tional representation as given in Sec such that all ring mo des are
explicitly describ ed and hence no external ML de nitions are necessary
APN Notation for HQPNs
Pro ductions
NET empty
j inputnet ID NET
j beginnet ID ELEMENT endnet NET
like ID endnet NET j beginnet ID
ELEMENT empty
j PLACE ELEMENT
j TRANSITION ELEMENT
j ARC ELEMENT
j TYPEDEF ELEMENT
j FUSION ELEMENT
ID STRING
TYPEDEF typedef ID COLOURSET
FUSION fuse ID TYPE ID IDLIST
TYPE page j inst j global
IDLIST empty j j ID IDLIST
PLACE place ID NAME INIT CAP COLOUR P TYPE
j place ID like ID
NAME empty j name STRING
INIT empty j init MULTISET
CAP empty j capacity MULTISET
COLOUR empty j colour COLOURSET
P TYPE empty
j substitute ID
j queue immediate SCHED weight TUPLEEXPR
j queue timed SCHED weight TUPLEEXPR
SCHED empty
j sched STRING SCHED
j noserver INTEGER SCHED
j rank INTEXPR SCHED
j prio INTEXPR SCHED
TRANSITION transition ID NAME PRIO T WEIGHT GUARD PORT
j transition ID like ID
PRIO empty j prio INTEXPR
T WEIGHT empty j weight REALEXPR
GUARD empty j guard GUARDEXPR
PORT empty j port in j port out j port io
ARC arc ID from ID to ID WEIGHT A TYPE BIND
WEIGHT empty j weight MULTISETEXPR
j type inhibitor A TYPE empty j type ordinary
BIND empty j bind ID CONT with ID CONT
CONT empty j cont ID CONT
var TUPLEEXPR denotes an ML expression whichevaluates to a tuple mean coe
of real values where mean is the mean and coe var the co e cientofvari
ation for the service time of a customer in the queue The rank expression
allows the sp eci cation of priorities for di erent colours of tokens to deter
mine the state of the queue in case of bulk arrivals
subnetblue
queuedplace t1 t2
subnetblack
Figure Example net with a queued place and two re ned places
port node p1 p2 p3 in
2
t5 t4 t3
port node 2 out
p4 p5 p6
Figure Subpage for substitution places
Fig is an example intended merely to demonstrate the APNN for HQPNs
The queued place has a FCFS scheduling strategy and its colour set contains
only two colours black and blue Transitions t and t eachhavetwo ring
mo des reblack and reblue The transitions merely transp ort a token to
or from the queued place to subnetblack or to subnetblue The re nement
of the places subnetblack and subnetblue are instances of the single coloured
GSPN given in Fig with input p ort in and output p ort out
Sp eci cation of the example
inputnet subpagespecification
beginnet hqpnexample
place queuedplace init black blue colour with black j blue
queue timed sched FCFS noserver
rank case colour of black j blue
weight case colour of black j blue
place subnetblack substitute subpagespecification
place subnetblue substitute subpagespecification
transition t prio
weight case mode of fireblack j fireblue
guard mode fireblack orelse mode fireblue
transition t like t
arc a from t to queuedplace
weight case mode of fireblack black j fireblue blue
arc b from queuedplace to t weight like a
arc c from t to subnetblue
weight if mode fireblue else empty
bind t with subnetblue cont in
arc d from t to subnetblack
weight if mode fireblack else empty
bind t with subnetblack cont in
arc e from subnetblue to t
weight if mode fireblue else empty
bind subnetblue cont out with t
arc f from subnetblack to t
weight if mode fireblack else empty
bind subnetblack cont out with t
endnet
In this case the keyword empty in the weight expressions of arcs denotes
the emptymultiset Note that substitute ID creates a fresh instance
from the net with this ID If two substitution places shall share the same
instance they must b e memb ers of the same fusion set
a
The L T X output of the example is
E
Net description imported from subpagesp eci cation
hqpnexample
Place queuedplace
marking f black blueg
coloursetf with black j blue g
timed queueing place with FCFS scheduling server s
order of arrival in case of bulk arrivals
j blue case colour of black
weight case colour of black j blue
Place subnetblack is an instance of subpagesp eci cation
Place subnetblue is an instance of subpagesp eci cation
Transition t priority
weight case mo de of reblack j reblue
guardfmo de black orelse mo de blueg
Transition t is defined as for t
Arc a from t to queuedplace
weight case mo de of reblack black j reblue blue
Arc b from queuedplace to t
weight is defined as for a
Arc c from t to subnetblue
weight if mo de reblue else empty
binding t with subnetblue wherein it binds in
Arc d from t to subnetblack
weight if mo de reblack else empty
binding t with subnetblack whereinitbindsin
Arc e from subnetblue to t
weight if mo de reblue else empty
binding subnetblue wherein it binds out with t
Arc f from subnetblack to t
weight if mo de reblack else empty
out with t binding subnetblack wherein it binds
Summary
The rst attempt at de ning a universal notation for describing Petri Net
mo dels was made in The authors could nd no other or any subse
quent reference to the sub ject when the need arose for suchaninterface in
the collab oration b etween their resp ective research groups In their partic
ular case they wanted to exchange mo del descriptions for analysis by the
QPN analyser develop ed and b eing develop ed by one group with the
DNA net develop ed by the other
The notation describ ed in this rep ort prop oses an uniform extensible for
malism which should accommo date a large class of nets For Coloured Petri
Nets the APNN employs ML so far The APNN can easily b e mo di ed to
supp ort other programming languages The notation moreover allows for
ahuman readable mo del description which can b e derived directly using
a
L T X commands known to a wide audience allowing a single format for the
E
incorp oration of the net description
At the same time the APNN allows di erent researchers to discuss and ex
p eriment with variations of mo dels of actual systems using p ossibly di erent
analysis algorithms implemented in to ols which for di erent reasons they
do not share
Comments ab out the APNN are clearly most welcome and should b e ad
dressed to anyone of the authors We trust that while almost certainly
not acceptable in its entirety at present it will b e the start of a discussion
leading to a generally accepted abstract formalism for describing high level
Petri Nets
References
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A Complete APNN grammar
In the following we present the pro ductions of a complete APNN for CGSPNs
including hierarchical description and queued places As throughout the
rep ort terminal symb ols are given bylower case letters non terminals by
upp er case letters For conciseness we use the APNN for CPNs with gen
eral ML expressions as given in Sec Note that one can easily mo dify
the APNN to supp ort other programming languages empty denotes the
empty string
With this APNN a wide range of Petri Nets is covered Possible extensions
of the notation can b e incorp orated as demonstrated in this rep ort
NET empty
j inputnet ID NET
j beginnet ID ELEMENT endnet NET
j beginnet ID like ID endnet NET
ELEMENT empty
j PLACE ELEMENT
j TRANSITION ELEMENT
j ARC ELEMENT
j TYPEDEF ELEMENT
j seeML FILENAME ELEMENT
j FUSION ELEMENT
ID STRING
FILENAME STRING
TYPEDEF typedef ID COLOURSET
FUSION fuse ID TYPE ID IDLIST
TYPE page j inst j global
IDLIST empty j j ID IDLIST
PLACE place ID NAME INIT CAP COLOUR P TYPE PORT EXIT
j place ID like ID
NAME empty j name STRING
INIT empty j init MULTISETEXPR
CAP empty j capacity MULTISETEXPR
COLOUR empty j colour COLOURSET
P TYPE empty
j substitute ID
j queue immediate SCHED weight TUPLEEXPR
j queue timed SCHED weight TUPLEEXPR
SCHED empty
j sched STRING SCHED
j noserver INTEGER SCHED
j rank INTEXPR SCHED
j prio INTEXPR SCHED
PORT empty j port in j port out j port io
EXIT empty j exit
TRANSITION transition ID NAME T TYPE PRIO T WEIGHT GUARD
PORT EXIT
j transition ID like ID
empty j substitute ID j invoke ID T TYPE
PRIO empty j prio INTEXPR
T WEIGHT empty j weight REALEXPR
GUARD empty j guard BOOLEXPR
ARC arc ID from ID to ID WEIGHT A TYPE BIND
WEIGHT empty j weight MULTISETEXPR
A TYPE empty j type ordinary j type inhibitor
BIND empty j bind ID CONT with ID CONT
CONT empty j cont ID CONT
INTEXPR like ID j ML EXPRESSION
REALEXPR like ID j ML EXPRESSION
BOOLEXPR like ID j ML EXPRESSION
TUPLEEXPR like ID j ML EXPRESSION
MULTISETEXPR like ID j ML EXPRESSION
COLOURSET like ID j ML DEFINITION
TYPEDEF typedef ID ML DEFINITION
Start symbol NET