PETRI NETS AND THEIR PROPERTIES - Modelling with Generalised Stochastic Petri Nets - page 30

p p 1 p p 2

t 1 t 2 t 1 t 2

p 1 p 2 p 3 p 4

(a) (b)

Figure 2.9: Two cases of asymmetric conflict

of t 0 , it is also true that the firing of t 0 decreases the enabling degree of
t. This indeed happens for t 2 and t 3 , and for t 4 and t 5 . However, this
is not always the case. Asymmetric conflicts can be easily constructed as
shown in Fig. 2.9. In the net of Fig. 2.9( a) the asymmetry is due to the
structure (the firing of t 2 does not decrease the enabling degree of t 1 , but
the viceversa is clearly false) while in Fig. 2.9( b) it is due to the particular
marking. In the marking depicted in the figure, M = 3p 1 + 2p + p 2 , we
have ED(t 1 ,M) = 2, ED(t 2 ,M) = 1, and the firing of t 1 does not decrease
the enabling degree of t 2 , while the viceversa is false. A specular situation
arises when M = 2p 1 + 3p + p 1 . Notice the similarities between the net in
Fig. 2.9( a) and the portion of the readers & writers example that comprises
transitions t 4 and t 5 : the only difference is the absence of the inhibitor arc.
Formally, we state that transition t l is in effective conflict with transition t m
in marking M, and we write t l EC(M)t m , if and only if t l and t m are both
enabled in M, but the enabling degree of t m in the marking M 0 , produced
by the firing of t l , is strictly smaller than the enabling degree of t m in M.
In formulae:

Definition 2.3.6 (Effective conflict) For any Petri net system, ∀ t l ,t m ∈
T such that t l 6 = t m , ∀ M : P → IN, transition t l is in effective conflict with
t m in marking M (denoted t l EC(M)t m ) iff

M[t l i M 0 and ED(t m ,M) < ED(t m ,M 0 )

Observe the use of the qualifier “effective” to the term conflict: histori-
cally the term “conflict” has been reserved to the situation in which two
transitions have a common input place. In order not to induce any misun-
derstanding we shall use the term structural conflict in the latter case.