Biology Reference
In-Depth Information
Fig. 8. Schematization of a conflict-free Petri net model where the number of input transitions of each place is one.
Here we propose a new method of dealing with the delay time of transitions for a subclass of Petri nets,
retention-free Petri nets, under two conditions. These are the conflict-free condition and the normal one
including conflicts. We introduce a stochastic approach to determine the firings of transitions in conflict.
Note that inhibitory arcs and cyclic structures (e.g., feedback loop, self-loop) are not taken into account
in this contribution. That is, the Petri net dealt with here is an acyclic one without inhibitory arcs.
As defined above, in a times Petri net, the delay time
d
i
of
t
i
is the reciprocal of the maximum
firing frequency
f
i
, and it is obvious that the token amount flowed-in per time unit is no more than that
flowed-out with
f
i
. We thus decide
d
i
by calculating
f
i
.
Strategy for determining delay time in the case of conflict-free
In a conflict-free Petri net, each place may have one or more input arcs but at most one output arc.
Without loss of generality, we consider two cases: (1) each transition has exact one input and one output
arc and they construct a path from a source transition to a sink transition, as shown in Fig. 8; (2) There
are
l
paths as in case (1) and these paths merge into a place
p
that is connected to a sink transition
t
,as
shown in Fig. 9.
Delay times of transitions of case
(
1
)
To keep retention-free behavior, each transition
t
i
on the path of Fig. 8 mu
st
satisfy
K
p
i−
1
f
i
·
α
i
,
i
−
1
where
K
p
i−
1
1
is the token amount flowed into place
p
i−
1
per time unit and
f
i
is the maximum
firing
frequency of
t
i
. It is obvious that
K
p
i−
1
i
−
β
i−
1
is determined by firing frequency
f
i−
1
(not
f
i−
1
)
that is further determined by the firing frequencies of previous transitions. Since the source transition
t
1
can fire
f
1
times per time unit,
K
p
1
=
f
i−
1
·
i−
1
β
1
that enables
t
2
to fire
f
2
=
K
p
1
/α
2
times per time unit.
=
f
1
·
1
β
2
=(
K
p
1
/α
2
)
β
2
,...,K
p
i−
1
In this way,
K
p
2
β
i−
1
=(
K
pi−
2
2
/α
i−
1
)
β
i−
1
. Resultantly
the following condition related to the maximum firing frequency of
t
i
can be obtained:
=
f
2
·
=
f
i−
1
·
2
i
−
1
i
−
β
1
...β
i−
1
α
2
...α
i−
1
f
1
·
f
i
·
α
i
(2)
where
f
1
is the maximum firing frequency of the source transition
t
1
,
f
i
is the maximum firing frequency
of
t
i
that is an output transition of place
p
i−
1
,
β
j
(
j
=
1,
...
,
i
−
1) and
α
j
(
j
=
2,
...
,
i
) are the weights
of arcs connected from the input transitions and to the out transition as shown in Fig. 8.