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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
 
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