Graphics Programs Reference

In-Depth Information

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

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