PETRI NETS AND THEIR PROPERTIES - Modelling with Generalised Stochastic Petri Nets - page 16

Places can contain tokens, that are drawn as black dots within places. The
state of a PN is called marking, and is defined by the number of tokens
in each place. As in classical automata theory, in PN there is a notion of
initial state (initial marking). The number of tokens initially found in the
PN places can be conveniently represented by symbols that are parameters
of the model. An initial marking with one or more parameters represents the
family of markings that can be obtained by assigning different legal values
to the parameters.
A Petri net model can be formally defined in the following way:
Definition 2.1.1 A PN model is an 8-tuple
M = { P,T,I,O,H,PAR,PRED,MP } (2.1)
where
P is the set of places;
T is the set of transitions, T ∩ P = ∅ ;
I,O,H : T → Bag(P), are the input, output and inhibition functions, re-
spectively, where Bag(P) is the multiset on P,
PAR is a set of parameters;
PRED is a set of predicates restricting parameter ranges;
MP : P → IN ∪ PAR is the function that associates with each place either
a natural number or a parameter ranging on the set of natural numbers.
Functions I, O, and H describe the input, output, and inhibitor arcs of
transitions, respectively. Given a transition t ∈ T, we denote with • t =
{ p ∈ P : I(t,p) > 0 } and with t • = { p ∈ P : O(t,p) > 0 } the input
and output sets of transition t, and with ◦ t = { p ∈ P : H(t,p) > 0 } the
inhibition set of transition t. Since Bag(P) can be seen as a function from
P to IN, then both the following are syntactically correct:
• I(t), representing the multiset of input places of transition t
• I(t,p) denoting the multiplicity of element p in the multiset I(t)
For convenience, I, O, and H can also be represented by | T |×| P | matrices
of natural numbers.
Formally, a marking is a function M : P → IN, and we refer to M(p), for
p ∈ P, as the number of tokens in place p in marking M. For convenience,
the marking can also be viewed either as a | P | -component vector, or as an
element of Bag(P).
MP is to be interpreted as the parametric initial marking of the PN model:
by instantiating to actual values all the parameters (in the set PAR), we