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arc and adding a token to each place connected with an output arc. The transitions can be divided into
immediate transitions, firing without delay, and timed transitions, firing after a certain delay. A Petrinet
is a bipartite directed graph, which can be represented graphically. The places contain indistinguishable
tokens, which can be fired by the transitions. The vector representing the number of tokens in each
place is called the marking of the net. The reachability graph contains all markings reachable from an
initial marking. Formal definitions of elementary Petrinets and theoretical results about their structural
properties can be found in [22]. Several extensions of elementary Petrinets were proposed for more
compact models and a higher level of abstraction. In colored Petrinets tokens can be distinguished, and
in predicate-transition nets tokens are expressions, which are manipulated by the firing of transitions.
Initially, Petrinets model only qualitative aspects of a system. In order to include quantitative aspects,
every place can hold a well defined number of tokens. The capacity of the place defines the maximum of
tokens which can be held by this place. However, the definition of the firing rule has to be extended. The
input and output arc will be labeled by integer values. In the case of the input arc, this value states that
the transition can fire, if each input place represents equal or more tokens than the input arc will specify.
The firing process of a transition will delete tokens and will produce tokens into the output places. The
number of deleting and producing tokens will be specified by the arrow weight.
More formally we will define some basic terms.
A
net
N
=
(P,T,F) is defined as
-
P and T are finite disjunct sets and
-
F is a subset of the set (P
×
T)
∪
(T
×
P).
A
place-transition net
consists of
-
places (represented as circles);
-
transitions (represented as boxes);
-
arrows from places to transitions and from transitions to places;
-
a capacity indication for every place;
-
a weight for every arrow (represented as a number);
-
an initial marking, defining the initial number of tokens for every place.
In a place-transition net
-
a marking is indicated by the number of tokens in every place;
-
a place p is in the pre-set (or post-set) of a transition t, if there is an arrow from p to t (or an arrow
from t to p);
-
a transition t is activated, if
1. for every place p from the pre-set of t the weight of the arrow from p to t is not greater than the
number of tokens indicated at p;
2. for every place p in the post-set of t the number of tokens at p increased by the weight of the
arrow from t to p is not greater than the capacity of p;
-
an activated transition t will occur in the number of tokens at every place p is decreased by g, if g is
the arrow weight of (p
→
t) and in that the number of tokens at every place p' is increased by g', if
g' is the arrow weight of (t
→
p').
