PETRI NETS AND THEIR PROPERTIES - Modelling with Generalised Stochastic Petri Nets - page 28

p 2 p 1 p 3

3 3

t 1 t 2

p 4

Figure 2.7: Implementation of an or

type or can also be expressed with the PN paradigm, but in a more complex
manner. Suppose we want transition t to delete three tokens from p 1 and
add one to p 4 when p 2 ≥ 1 or p 3 is empty. This can be described by means
of two transitions t 1 and t 2 , both changing the marking in the same way,
depending however on which of the two alternative preconditions holds true.
This situation is depicted in Fig. 2.7.

2.3.4 Typical situations in the PN system evolution

Conflicts — In the previous section we stated that PNs are well suited
to the description of conflicts and concurrency. Indeed, a conflict entails a
choice which can be naturally described with the PN notation as in Fig. 2.8.
When transition t fires, a token is deposited in place p, and both t 1 and t 2
become enabled. However, as soon as any one of them fires, it removes the
token from p, thus disabling the other one. Consider now the case of two
tokens in p. If t 1 fires first then t 2 is still enabled: nevertheless something has
changed since now t 2 can fire at most once in a row. We want our definition
of conflict to capture also these situations, and we therefore distinguish how
many times a transition is enabled in a given marking with the following
definition.

Definition 2.3.5 (Enabling degree) For any PN system, the enabling
degree is a function ED : T × [P → IN] → IN such that ∀ t ∈ T,
∀ M : P → IN, ED(t,M) = k iff

•∀ p ∈ • t, M(p) ≥ k · I(p,t), and

•∀ p ∈ ◦ t, M(p) < H(p,t), and