Graphics Programs Reference
In-Depth Information
parallel (unless otherwise specified by the PN structure) until they complete.
At completion, activities induce a local change of the system state that is
specified with the interconnection of the transition to input and output
places. Note that no special mechanism is necessary for the resolution of
timed conflicts: the temporal information provides a metric that allows the
conflict resolution (the probability that two timers expire at the same instant
is zero — a fine point on which we shall return).
It must be observed that the memoryless characteristics of the negative
exponential pdf render immaterial the distinction among the three policies
[ 2] described in Chapter 3, i.e.:
race policy with resampling
race policy with enabling memory
race policy with age memory
Indeed, whether the timer is reset or not, the pdf of the remaining time to
fire is always distributed according to a negative exponential pdf, with the
same mean, due to the memoryless property of such a pdf. Going back to
the example of a timed PN system with age memory policy presented in
Fig. 3.12, consider the timed transition t end , whose timing describes the
duration of the execution of a local process that may be interrupted by
external requests. The exponential assumption implies that the whole du-
ration of the execution of a local process is a random variable with negative
exponential pdf with a given mean, and that also the remaining execution
time after an interruption is a random variable with negative exponential pdf
with the same mean (again, for more details see Appendix A). This aspect
is quite relevant for the transformation of an SPN model into its underlying
stochastic process, and for its analysis.
As an example of an SPN model, consider the timed version of the PN model
in Fig. 1.8, describing a simple parallel computation. For the readers' con-
venience, the model, in its timed version, is repeated in Fig. 5.1. Observe
that the only difference between the model in Fig. 1.8 and that in Fig. 5.1
is in the graphical look of transitions, that are depicted as bars in the for-
mer, and as boxes in the latter. This is due to the formalism adopted for
the representation of timed transitions throughout the topic, as we already
mentioned in Chapter 3.
To associate a meaningful temporal specification with the model, transition
T newdata , which describes the activity of reading new data, is associated with
a random delay, with a negative exponential pdf, whose mean matches the
average time for the new data read operation. If the observed average time
for the new data read is 10 time units, the negative exponential pdf has rate
0.1. Note that the observation of the real system might reveal that the data
read activity is far from having a negative exponentially distributed dura-
tion. However, with the SPN modelling paradigm we have no flexibility on
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