ANALYSIS OF GSPN MODELS - Modelling with Generalised Stochastic Petri Nets - page 123

Table 6.6: Specification of the transitions of the SPN of Fig. 6.7
transition weight priority ECS

t 0 w 0 1 -

t 1 β 1 -

t 2 α 1 -

t a w a 1 -

and

α
(w 0 + α)

w 0
(w 0 + α)

α
(w a + α)

P { M 0 ,M 2 } = + .

From a modelling point of view, there are good reasons to consider the two
subnets of Figs. 6.7( a) and 6.7( b) equivalent since the sequence of immediate
actions represented by t 0 and t a of Fig. 6.7( b) should be reducible to transi-
tion t 0 of Fig. 6.7( a) without affecting the behaviour of the model. Instead,
the difference among the total probabilities that we have just computed
shows that, in the case of confused models, the trivial action of splitting an
atomic (and instantaneous) action in two has drastic effects not only on the
graphical description of the model, but also (and more important) on the
values of the results obtained from its quantitative evaluation.

6.2.1 Marking dependency

The firing times and the weights that we have considered so far were as-
sumed to be independent of the marking of the GSPN system. In principle,
however, it is possible to work with transition parameters that are marking-
dependent. When one or more timed transitions are enabled in a given
marking, we can compute the distribution of the sojourn time in that mark-
ing, as well as the probability of the transition that fires first, using negative
exponential distributions whose firing rates may depend on that specific
marking. Similarly, the selection of the immediate transition that fires in a
vanishing marking enabling several immediate transitions can be computed
using weighting factors that may be marking-dependent. In all these cases,
equations ( 6.5) , ( 6.15) , ( 6.16) , and ( 6.17) can be generalized assuming that
all the parameters are functions of the marking for which they are computed.
In practice, the generality allowed by this extension (which was indeed as-
sumed by the original GSPN proposal [4] ) is in contrast with the claim of
GSPNs as a high-level language for the description of complex systems. In-
deed, while the construction of a GSPN model requires a local view of the be-
haviour of the real system, the specification of (general) marking-dependent
parameters requires the analyst to be aware of the possible (global) states of
the system. Moreover, a dependency of the weights of conflicting immediate
transitions on the marking of places that are not part of the input set of