Graphics Programs Reference
In-Depth Information
transitions.
The rates associated with timed transitions that represent a PAR statement
can be obtained from measurements on a real system. This is not true,
however, for the other timed transitions of the model as they may depend
on input data. Consider for example the case of a timed transition that
represents a whole sequence of statements; we can have two cases: either
the number of executed statements is fixed (the sequence does not include
any branching) or it is not. In the first case it is possible to estimate the
execution time; in the second case some information is needed to estimate
the mean number of times a loop is executed. If this is not possible, the only
solution is to perform an analysis that depends on the mean values of the
inputs. Similar problems exist for the assignment of weights to immediate
transitions; and it may happen that a model must be evaluated for different
combinations of weights values. It should be clear by this quick introduction
to the problem that the assignment of weights and rates to transitions is an
important step in the quantitative analysis, and it cannot be completely
automated.
We shall use the example of Fig.
10.10
to show the type of results that can
be derived from the GSPN system.
Probability of two communications at the same time — The CTMC
derived from the GSPN system provides the probability that there is more
than one communication place marked at the same time. The possibility of
concurrent communication is given by: P
{
M(c
1
)+M(c
2
)+M(c
3
)+M(c
4
) >
1
}
. This probability is identically equal to zero, which is not surprising if we
consider that c
1
,c
2
,c
3
and c
4
are part of the Spooler process and therefore
are covered by a P-semiflow with token count equal to one. An analogous
relation, based on the P-invariants of processes P
1
and P
2
, holds for c
1
,c
3
and c
2
,c
4
considered two at a time: this indicate that the same channel can
be used for transmitting the number of packets and the packets themselves,
and, more importantly, that even if we want to keep the two channels sepa-
rate for clarity, they can nevertheless be mapped on the same physical link
without affecting the system performance.
Joint distribution of p
k
and p
n
— We want to study the probability
of having k = α and n = β where α and β are natural values. From the
solution of the model we get the distribution of tokens in the different places,
and we can compute all the desired joint probabilities. For our example we
have obtained: P
{
M(p
k
) > 0
∧
M(p
n
) > 0
}
= 0 and in a similar way
P
{
M(p
k
) > 0
∧
M(p
x
) = 0
}
= 0. This result confirms what was remarked
about the dependence of the two variables.
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