ANALYSIS OF GSPN MODELS - Modelling with Generalised Stochastic Petri Nets - page 120

while in marking M s = (p 0 + p 1 + p 2 + p 3 ) we have:

w a
(w a + w b + w c + w d )

P { t a | M s } =

w b
(w a + w b + w c + w d )

P { t b | M s } =

w c
(w a + w b + w c + w d )

P { t c | M r } =

and

P { t d | M s } = w d
(w a + w b + w c + w d )
The corresponding probabilities within the two markings are different, but
their relative values, when the transitions are enabled, remain constant and
completely characterized by the initial specification of the model. For in-

stance,

P { t a | M r }

P { t d | M r } = P { t a | M s }

P { t d | M s } = w a

w d
During the evolution of a GSPN, it may happen that several ECSs are
simultaneously enabled in a vanishing marking. According to the usual
firing mechanism of Petri nets, we should select the transition to fire by first
non-deterministically choosing one of the ECSs and then a transition within
such ECS. The assumption that the GSPN is not confused guarantees that
the way in which the choice of the ECS is performed is irrelevant with respect
to the associated stochastic process. One possibility is that of computing the
weight of the ECS by adding the parameters of all the enabled transitions
that belong to that ECS and of using this weight to select the ECS with a
method suggested by equation ( 6.15) . A simple derivation shows that the
use of this method implies that the selection of the immediate transition to
be fired in a vanishing marking can be performed with the general formula
( 6.15) that was originally derived for timed transitions (and thus for tangible
markings) only [3, 14] . Moreover, it is possible to show that if we consider
the probabilities associated to the many different sequences of immediate
transitions whose firings lead from a given vanishing marking to a target
tangible one, they turn out to be all equal [ 3] .
This last property strongly depends on the absence of confusion in the
GSPN; the fact that the presence of confused subnets of immediate tran-
sitions within a GSPN is an undesirable feature of the model can also be
explained considering the following example.
Suppose that a subnet is identified as confused using the structural confusion
condition of Section 2.3.4. Suppose also that the subnet has the structure
depicted in Fig. 2.11 of Chapter 2 and repeated here in Fig. 6.6 for con-

venience. Given the initial marking M 0 = (p 0 + p 2 ) and the parameters