TIME IN PETRI NETS - Modelling with Generalised Stochastic Petri Nets - page 71

• For how long is a processor idle waiting for the shared memory ?
• Which is the rate of access of a processor to the shared memory ?
The answers to these questions can be obtained essentially in two different
ways: the first one requires to exercise the net simulating its behaviour,
whereas the second one, possible only under some specific conditions, re-
quires an analysis based on the reachability graph of the untimed PN system,
and on the temporal specifications.
Consider now the first approach: from a significative execution sequence
of the net it is possible to identify all the states (markings) satisfying a
given condition and to add (and possibly average) the times spent in each
of the markings of interest. This is obviously possible because the model
execution generates states in which the PN system sojourns for some time.
With reference to Fig. 3.5, the answer to the first question can be obtained
considering all the markings in which a token is present in p requesting 1 (or
p requesting 2 for the other processor). The answer to the second question
can be obtained in a similar way by considering all the markings in which a
token is present in p accessing 1 or p accessing 2 .
The second approach in not always possible. As we already pointed out,
the introduction of temporal specifications into a PN model or system must
be done so as not to modify the basic behaviour, and specifically the non-
determinism of the choices occurring in the net execution. Only in this
case the reachability set of the untimed PN system contains all execution
sequences possible also in the timed PN system.
If the temporal specification is given in a deterministic way, the behaviour
of the model is deterministically specified, and the choices due to conflicting
transitions may be always solved in the same way. This would make many
of the execution sequences in the untimed PN system impossible.
Instead, if the temporal specifications are defined in a stochastic manner,
by associating independent continuous random variables with transitions to
specify their firing delays, the non-determinism is preserved (in the sense
that all the execution sequences possible in the untimed PN system are also
possible in the timed version of the PN system) under the condition that
the support of the distribution of the random variables is [0, ∞ ). In this
case the temporal specification at the net level can be superimposed over a
reachability graph that can be generated using the untimed version of the
PN system to obtain the characterization of the behaviour of the timed PN
system.