PETRI NETS WITH sPRIORITY - Modelling with Generalised Stochastic Petri Nets - page 78

Let us now consider the conflict due to the presence of both readers and
writers competing for the shared variable (corresponding to a conflict be-
tween t 4 and t 5 ). The readers and writers might have the same probability
of being granted the access to the variable or this probability might depend
on the number of users of each type waiting to access. In the latter case,
some more complex type of annotation should be added into the model de-
scription to take into account all the different possible conflict situations.
It is thus very important to be able to clearly identify by inspection of the
net structure the sets of potentially conflicting transitions. The SC relation
defined in Chapter 2 may seem appropriate to identify sets of potentially
conflicting transitions; however, we shall see that this is not always a correct
choice.
Conflict — The notion of conflict is drastically influenced by the intro-
duction of a priority structure in PN models. The definition of effective
conflict has to be modified with respect to the new notion of concession.
Instead, the definition of enabling degree given in Chapter 2 remains un-
changed for PN models with priority. Observe that this implies that both
transitions that have concession and enabled transitions have enabling de-
gree greater than zero. Conflict resolution causes the enabling degree of
some transition to be reduced, and this may happen both for transitions
with concession and for enabled ones. Hence the definition of the effective
conflict relation is modified as follows.
Definition 4.2.1 Transition t i is in effective conflict relation with transi-
tion t j in marking M, (t i EC(M) t j ) iff t j has concession in M, t i is enabled
in M, and the enabling degree of t j decreases after the firing of t i .
Observe that a necessary condition for the EC relation to hold is that π i ≥
π j , otherwise t i would not be enabled in M.
The definition of different priority levels for transitions introduces a further
complication, since it destroys the locality of conflicts typical of PN models
without priority.
Let us consider the net in Fig. 4.2. Transitions t 1 and t 2 are both enabled in
the marking represented in the figure (since they both have concession, and
no higher priority transition has concession), and apparently they are not in
conflict, since they do not share input or inhibition places. According to the
definition of concurrent transitions given in Chapter 2, one might conclude
that t 1 and t 2 are concurrent. However, the firing of t 1 enables t 3 , which
has higher priority than t 2 , so that:
1. transition t 2 becomes disabled while keeping its concession
2. transition t 3 is certainly the next transition to fire