Biology Reference
In-Depth Information
Fig. 9. Schematization of conflict-free Petri net model where a place (
p
) possessing multiple input transitions (
t
k
1
,t
k
2
,...,t
k
l
)
is included.
Delay times of transitions of case
(
2
)
For the transitions on each path of Fig. 9, the delay times can be determined according to inequality
(2). And the token amount flowed into place
p
can be computed as discussed above and the maximum
firing frequency of
t
can be determined according to the following inequality:
β
l
...β
k
l
α
2
l
...α
kl
β
1
...β
k
1
α
l
+1
...α
k
1
β
2
...β
k
2
α
l
+2
...α
k
2
f
1
·
+
f
2
·
+
...
+
f
l
·
f
·
α
(3)
The left-hand of inequality (3) is designed to calculate total token amount flowed into
p
from its multiple
input transitions per time unit.
Strategy for determining delay time in the case of conflict
Here, we propose a new firing rule of a timed Petri net by introducing a stochastic approach to
determine firings for a series of transitions in conflict. Suppose a place
p
possesses output transitions,
t
1
,
t
2
,
...t
k
then the firing rule is as follows.
New firing rule of timed Petri nets
TPN
*:
1. Each unreserved token deposited to input place
p
is assigned to be reserved by the transition
t
i
that satisfies the following mathematical expression:
⎧
⎨
⎫
⎬
⎭
k
n
i
α
i
n
j
α
j
min
−
s
i
(4)
⎩
j
=1
2. When the number of reserved tokens of
t
i
is not less than the required token number for the
firing, the firing of
t
i
is decided.
3. After the delay time
d
i
of
t
i
passed,
t
i
fires to remove the reserved tokens from the input place
of
t
i
and deposit unreserved tokens into the output places of
t
i
.
In the above expression (4),
α
i
is the arc weight of
e
(
p
,
t
i
);
s
i
is the firing probability of transition
t
i
,
which represents the proportion of the firing frequency of each transition in the total firing frequency of
the transitions in conflict. As shown in Fig. 10,
s
i
is assigned to corresponding transition
t
i
, which is