PETRI NETS WITH sPRIORITY - Modelling with Generalised Stochastic Petri Nets - page 74

• Π : T → IN is the priority function that maps transitions onto natural
numbers representing their priority level.

For notational convenience we use π j instead of Π(t j ), whenever this does
not generate ambiguities. The priority definition that we assume in the topic
is global: the enabled transitions with a given priority k always fire before
any other enabled transition with priorit y 1 j < k.
This kind of priority definition can be used for two different modelling pur-
poses: (1) it allows the partition of the transition set into classes representing
actions at different logical levels, e.g. actions that take time versus actions
corresponding to logical choices that occur instantaneously; (2) it gives the
possibility of specifying a deterministic conflict resolution criterion.

Enabling and firing — The firing rule in PN models with priority

requires the following new definitions:

• a transition t j is said to have concession in marking M if the numbers
of tokens in its input and inhibitor places verify the usual enabling
conditions for PN models without priority

• a transition t j is said to be enabled in marking M if it has concession
in the same marking, and if no transition t k ∈ T of priority π k > π j
exists that has concession in M. As a consequence, two transitions
may be simultaneously enabled in a given marking only if they have
the same priority level

• a transition t j can fire only if it is enabled. The effect of transition
firing is identical to the case of PN models without priority

Observe that in this new framework all the transitions enabled in a given
marking M have the same priority level, say j. In the GSPN formalism,
priority level zero is assigned to all timed transitions, and priority levels
greater than zero are associated with immediate transitions: this leads to
a classification of markings based on the priority level j of the transitions
enabled in it. The tangible markings defined in the previous chapter cor-
respond to states characterized by a priority level j = 0, plus all the dead
states (i.e., all states M such that E(M) = ∅ ); vanishing markings instead
are characterized by a priority level j > 0: we may partition the set of
vanishing markings into subsets of j-priority level vanishing markings.
Note that the presence of priority only restricts the set of enabled transitions
(and therefore the possibilities of firing) with respect to the same PN model

1 Without loss of generality, we also assume that all lower priority levels are not empty,
i.e.:

π j > 0 = ⇒∃t k ∈T

: π k = π j − 1

∀t j ∈T,