GSPN BASICS - Modelling with Generalised Stochastic Petri Nets - page 98

5.2 Mixing Exponentially Distributed and Null De-

lays

Several reasons suggest the introduction of the possibility of using immediate
transitions into PN models together with timed transitions. As we observed
in Chapter 3, the firing of a transition may describe either the completion
of a time-consuming activity, or the verification of a logical condition. It
is thus natural to use timed transitions in the former case, and immediate
transitions in the latter. Moreover, as we noted in the previous example,
when all transitions are timed the temporal specification of the model must
in some cases consider at one time both the timing and the probability
inherent in a choice. It seems natural to separate the two aspects in the
modelling paradigm, to simplify the model specification. Furthermore, by
allowing the use of immediate transitions, some important benefits can be
obtained in the model solution. They will be described in detail in the
next chapter; we only mention here the fact that the use of immediate
transitions may significantly reduce the cardinality of the reachability set,
and may eliminate the problems due to the presence in the model of timed
transitions with rates that differ by orders of magnitude. The latter situation
results in so-called “stiff” stochastic processes, that are quite di cult to
handle from a numerical viewpoint. On the other hand, the introduction of
immediate transitions in an SPN does not raise any significant complexity
in the analysis, as we shall see in the next chapter.
SPN models in which immediate transitions coexist with timed transitions
with race policy and random firing delays with negative exponential pdf are
known by the name generalized SPNs (GSPNs).
In the graphical representation of GSPNs, immediate transitions are drawn
as bars or segments and are denoted by a name that is normally of the form
t x , where x is either a number or a mnemonic string; timed transitions are
drawn as (white or black) rectangular boxes, and are denoted by a name
that is normally of the form T x .
Immediate transitions are fired with priority over timed transitions. Thus,
if timing is disregarded, the resulting PN model comprises transitions at dif-
ferent priority levels. The adoption of the race policy may seem to implicitly
provide the priority of immediate over timed transitions; this is indeed the
case in most situations, but the explicit use of priority simplifies the de-
velopment of the theory. We shall return to this subtle point later in this
chapter.
Recall that markings in the reachability set can be classified as tangible or
vanishing. A tangible marking is a marking in which (only) timed transitions
are enabled. A vanishing marking is a marking in which (only) immediate
transitions are enabled (the “only” is in parenthese since the different pri-
ority level makes it impossible for timed and immediate transitions to be