Graphics Programs Reference
In-Depth Information
any of the transitions comprised in their ECS could lead to a new form of
“confusion” that is not captured by the definitions contained in Chapter 2
and discussed in the previous section.
For this reason, a restriction of the type of marking-dependency allowed
in GSPN models was informally proposed in [ 14] . A definition of marking
dependency that satisfies these restrictions can be obtained by allowing the
specification of marking dependent parameters as the product of a nominal
rate (or weight in the case of immediate transitions) and of a dependency
function defined in terms of the marking of the places that are connected to
a transition through its input and inhibition functions.
Denote with M /t the restriction of a generic marking M to the input and
inhibition sets of transition t:
[
M /t
=
S
t) M(p)
(6.18)
p∈( t
Let f(M /t ) be the marking dependency function that assumes positive values
every time transition t is enabled in M; using this notation, the marking
dependent parameters may be defined in the following manner:
8
<
: µ i (M) = f(M /T i ) w i
ω j (M) = f(M /t j ) w j
in the case of marking dependent firing rates
in the case of marking dependent weights
(6.19)
Multiple-servers and infinite-servers can be represented as special cases of
timed transitions with marking dependent firing rates that are consistent
with this restriction. In particular, a timed transition T i with a negative-
exponential delay distribution with parameter w i and with an infinite-server
policy, has a marking dependency function of the following form:
f(M /T i ) = ED(T i ,M)
(6.20)
where ED(T i ,M) is the enabling degree of transition T i in marking M (see
Section 2.3.4) . Similarly, a timed transition T i with multiple-server policy of
degree K has a marking dependency function defined in the following way:
f(M /T i ) = min (ED(T i ,M), K)
(6.21)
Other interesting situations can be represented using the same technique,
Fig. 6.8 depicts two such cases. In Fig. 6.8( a), we have a situation of two
competing infinite servers: transition T 1 fires with rate w 1 M(p 1 ) if place
p 0 is marked; similarly for transition T 2 . Obviously both transitions are
interrupted when the first of the two fires removing the token from place p 0 .
Using δ(x) to represent the following step function:
8
<
: 0
1
x = 0
δ(x) =
(6.22)
x > 0

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