GSPN BASICS - Modelling with Generalised Stochastic Petri Nets - page 103

Formally, a GSPN model is an 8-tuple
M GSPN = ( P, T, Π, I, O, H, W, PAR, PRED, MP ) (5.1)
where M π = ( P, T, Π, I, O, H, PAR, PRED, MP ) is a PN model
with priorities and inhibitor arcs as defined in Chapter 4, that is called the
underlying PN model, and W : T → IR is a function defined on the set of
transitions.
Timed transitions are associated with priority zero, whereas all other priority
levels are reserved for immediate transitions.
The underlying PN model constitutes the structural component of a GSPN
model, and it must be confusion-free at priority levels greater than zero (i.e.,
in subnets of immediate transitions).
The function W allows the definition of the stochastic component of a GSPN
model. In particular, it maps transitions into real positive functions of the
GSPN marking. Thus, for any transition t it is necessary to specify a func-
tion W(t,M). In the case of marking independency, the simpler notation w k
is normally used to indicate W(t k ), for any transition t k ∈ T. The quantity
W(t k ,M) (or w k ) is called the “rate” of transition t k in marking M if t k is
timed, and the “weight” of transition t k in marking M if t k is immediate.
Since in any marking all firing delays of timed transitions have a negative
exponential pdf, and all the delays are independent random variables, the
sojourn time in a tangible marking is a random variable with a negative
exponential pdf whose rate is the sum of the rates of all enabled timed tran-
sitions in that marking. This result stems from the fact that the minimum
of a set of independent random variables with negative exponential pdf also
has a negative exponential pdf whose rate is the sum of the rates of the
individual pdfs.
In the case of vanishing markings, the weights of the immediate transitions
enabled in an ECS can be used to determine which immediate transition
will actually fire, if the vanishing marking enables more than one conflicting
immediate transition.
When transitions belonging to several different ECSs are simultaneously
enabled in a vanishing marking, as we already explained, the choice among
these transitions is irrelevant. The irrelevance implies that no choice is
needed among the different ECSs, and that the transitions can fire concur-
rently. It is thus possible to say that either transitions fire simultaneously,
or a fixed ordering among ECSs exists, or the ECSs comprising enabled
transitions are selected with equal probability, or an ECS is chosen with a
probability proportional to the sum of the weights of the enabled immediate
transitions it contains.
It must be emphasized that the irrelevance in the order of transition firings
is an important consequence of the restriction that subnets of immediate
transitions must be confusion-free. The restriction to confusion-free imme-
diate subnets also has a beneficial impact on the model definition, since the