ANALYSIS OF GSPN MODELS - Modelling with Generalised Stochastic Petri Nets - page 124

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