Graphics Programs Reference
In-Depth Information
for SPNs, with a difference in the meaning of the parameters w
k
. When the
marking is vanishing, the parameters w
k
are the weights of the immediate
transitions enabled in that marking and define the selection policy used to
make the choice. When the marking is tangible, the parameters w
k
of the
timed transitions enabled in that marking are the rates of their associated
negative exponential distributions. The average sojourn time in vanishing
markings is zero, while the average sojourn time in tangible markings is
given by equation (
6.5)
.
Observing the evolution of the GSPN system, we can notice that the dis-
tribution of the sojourn time in an arbitrary marking can be expressed as
a composition of negative exponential and deterministically zero distribu-
tions: we can thus recognize that the marking process
{
M(t),t
≥
0
}
is a
semi-Markov stochastic process whose analysis is discussed in Appendix A.
When several immediate transitions are enabled in the same vanishing mark-
ing, deciding which transition to fire first makes sense only in the case of
conflicts. If these immediate transitions do not “interfere”, the choice of fir-
ing only one transition at a time becomes an operational rule of the model
that hardly relates with the actual characteristics of the real system. In
this case the selection is inessential from the point of view of the overall be-
haviour of the net. The concept of ECS introduced in Chapter 4 represents
a basic element for identifying these situations and for making sure that
both the specification of the model and its operational rules are consistent
with the actual behaviour of the system we want to analyse.
Assuming, as we have already pointed out in the previous chapter, that the
GSPN is not confused (see Section
2.3.4
for a definition), the computation of
the ECSs of the net corresponds to partitioning the set of immediate tran-
sitions into equivalence classes such that transitions of the same partition
may be in conflict among each other in possible markings of the net, while
transitions of different ECSs behave in a truly concurrent manner.
When transitions belonging to the same ECS are the only ones enabled in
a given marking, one of them (say transition t
k
) is selected to fire with
probability:
w
k
ω
k
(M
i
)
P
{
t
k
|
M
i
}
=
(6.16)
where ω
k
(M
i
) is the weight of ECS(t
k
) in marking M
i
and is defined as
follows:
X
ω
k
(M
i
) =
w
j
(6.17)
t
j
∈[
ECS
(t
k
)∧E(M
i
)]
Within the ECS we may have transitions that are in direct as well as in
indirect conflicts. This means that the distributions of the firing selection
probabilities computed for different markings may assign different probabil-
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