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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 2 times per time unit.
= f 1
·
1
β 2 =( K p 1 2 ) β 2 ,...,K p i− 1
In this way, K p 2
β i− 1 =( K pi− 2
2 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.
 
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