Graphics Programs Reference
In-Depth Information
In the GSPN subnet resulting from the reduction of t
1
and t
2
over T
a
, shown
existed in the original GSPN, so that no place could be deleted. The size of
the state space has been reduced, as all vanishing markings of the original
GSPN are not reachable any longer (for example, the marking
{
p
2
,p
3
,p
4
}
is
not generated by the reduced net of Fig.
6.13(
b)). Observe that the subnet
depicted in Fig.
6.13(
a) includes only one of the timed transitions that may
deposit tokens in the input places of the ECS composed by t
1
and t
2
, but
in general there can be more than one. This is the reason why the two
immediate transitions t
1
and t
2
are still present in the reduced subnet of
Fig.
6.13(
b), since they shall also be reduced over all timed transitions that
add tokens to p
3
and/or p
4
.
Let us now apply the two rules to the reduction of the example GSPN in
Fig.
6.9
to an equivalent SPN. First of all, the immediate transition t
start
can be removed. We can view t
start
as a free-choice conflict comprising just
one transition, and thus incorporate it into the two timed transitions T
ndata
and T
check
. As a result, the output set of the two timed transitions T
ndata
transition rates is necessary, since the weight of t
start
is equal to 1.
As a second step, we can remove the immediate transition t
syn
, that can be
viewed as a non-free-choice conflict comprising just one transition. It is thus
necessary to consider the possible situations at the firing of either of the two
timed transitions T
par1
and T
par2
. When T
par1
fires, t
syn
fires if p
6
contains
a token; otherwise, the token produced by the firing of T
par1
waits for the
synchronization in p
5
. The elimination of t
syn
thus requires the introduction
of two timed transitions T
a
and T
b
in place of T
par1
, and T
c
and T
d
for T
par2
.
The resulting GSPN is shown in Fig.
6.15.
Also in this case, no modification
of the transition rates is necessary, since the weight of t
syn
is equal to 1.
Finally, the free-choice conflict formed by transitions T
OK
and T
KO
can be
eliminated by splitting the two timed transitions T
a
and T
d
in two timed
transitions each. From transition T
a
we generate T
a
0
and T
a
00
to account
for the two cases in which either T
OK
or T
KO
fires. The rate of the timed
transition T
a
0
is obtained as the product of the rate of the timed transition
T
a
by the weight of the immediate transition T
OK
, and similarly in the other
cases. The resulting SPN model is shown in Fig.
6.16.
It should be observed that the SPN model in Fig.
6.16
is much more complex
than the original model in Fig.
6.9,
and much less adherent to our natural
description of the system operations. This is usually the case, and provides a
demonstration of the greater expressivness of GSPNs over SPNs. However,
the transformation from GSPNs to SPNs can be easily automated, and
made invisible to the user, as a useful intermediate step instrumental for
the generation of the continuous-time Markov chain underlying the GSPN
system.
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