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enabled in the same marking, according to the definition of the enabling
condition for PN models with priority given in Chapter 4). A marking in
which no transition is enabled is tangible. The time spent in any vanishing
marking is deterministically equal to zero, while the time spent in tangible
markings is positive with probability one. Similarly, a place is said to be
vanishing if it can contain tokens only in vanishing markings.
To describe the GSPN dynamics, we separately observe the timed and the
immediate behaviour, hence referring to tangible and vanishing markings,
respectively. Let us start with the timed dynamics (hence with tangible
markings); this is identical to the dynamics in SPNs, that was described
before. We can assume that each timed transition possesses a timer. The
timer is set to a value that is sampled from the negative exponential pdf
associated with the transition, when the transition becomes enabled for the
first time after firing. During all time intervals in which the transition is
enabled, the timer is decremented. Transitions fire when their timer reading
goes down to zero.
With this interpretation, each timed transition can be used to model the
execution of some activity in a distributed environment; all enabled activities
execute in parallel (unless otherwise specified by the PN structure) until they
complete. At completion, activities induce a change of the system state, only
as regards their local environment. No special mechanism is necessary for
the resolution of timed conflicts: the temporal information provides a metric
that allows the conflict resolution.
In the case of vanishing markings, the GSPN dynamics consumes no time:
everything takes place instantaneously. This means that if only one imme-
diate transition is enabled, it fires, and the following marking is produced.
If several immediate transitions are enabled, a metric is necessary to iden-
tify which transition will produce the marking modification. Actually, the
selection of the transition to be fired is relevant only in those cases in which
a conflict must be resolved: if the enabled transitions are concurrent, they
can be fired in any order. For this reason, GSPNs associate weights with
immediate transitions belonging to the same conflict set.
For the time being, let us consider only free-choice conflict sets; the case of
non-free-choice conflict sets will be considered later on, but we can anticipate
at this point that it can be tackled in a similar manner by exploiting the
definition of ECS introduced in Chapter 4. The transition weights are used
to compute the firing probabilities of the simultaneously enabled transitions
comprised within the conflict set. The restriction to free-choice conflict sets
guarantees that transitions belonging to different conflict sets cannot disable
each other, so that the selection among transitions belonging to different
conflict sets is not necessary.
We can thus observe a difference between the specification of the tempo-
ral information for timed transitions and the specification of weights for
immediate transitions. The temporal information associated with a timed
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