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

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,

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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