Graphics Programs Reference
In-Depth Information
p
a
p
a
T
a
p
b
t
1
t
2
t
3
T
1
T
2
T
3
p
3
p
1
p
2
p
1
p
2
p
3
(a)
(b)
Figure 6.12: Removing immediate transitions: the free-choice case
transition T
a
, and w
k
is the weight of immediate transition t
k
, so that
w
k
(6.44)
P
j=1,2,3
w
j
is the probability that t
k
is the transition that actually fires when t
1
,t
2
, and
t
3
are all enabled.
The timed transition T
k
in the SPN model represents the firing of timed
transition T
a
followed by immediate transition t
k
. If we define its firing rate
induces a movement of a token from p
a
to p
k
, given that transition T
a
is
enabled, is exactly the same as for the GSPN subnet, and therefore the
rates of the underlying Markov processes are the same. Note that place p
b
was deleted in the reduction process: this is not surprising because p
b
is a
vanishing place, a place that is never marked in a tangible marking.
The elimination procedure is somewhat more complex if the set of immediate
transitions enabled by the firing of a timed transition does not form a free-
choice conflict. Indeed, in this case the probability of firing an immediate
transition may depend also on the other simultaneously enabled immediate
transitions. We are thus forced to consider all possible cases. As an example,
consider the subnet in Fig.
6.13(
a).
The firing of timed transition T
a
may enable either only t
1
, or only t
2
, or
both t
1
and t
2
, or neither of them, depending on the marking of p
3
and p
4
.
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