Graphics Programs Reference
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where A =
{
M
i
∈
RS(M
0
) : Υ(M
i
) = true
}
.
(2) The expected value of the number of tokens in a given place. In this case
the reward function r(M) is simply the value of the marking of that place
(say place j):
r(M) = n iff M(p
j
) = n
(6.9)
Again this is equivalent to identifying the subset A(j,n) of RS(M
0
) for
which the number of tokens in place p
j
is n (A(j,n)
=
{
M
i
∈
RS(M
0
) :
= n
}
); the expected value of the number of tokens in p
j
is given
M
i
(p
j
)
by:
X
[n P
{
A(j,n)
}
]
E[M(p
j
)] =
(6.10)
n>0
where the sum is obviously limited to values of n
≤
k, if the place is k-
bounded.
(3) The mean number of firings per unit of time of a given transition. As-
sume that we want to compute the firing frequency of transition T
j
(the
throughput of T
j
); observing that a transition may fire only when it is en-
abled, we have that the reward function assumes the value w
j
in every
marking that enables T
j
:
8
<
T
j
∈
E(M)
:
w
j
0
r(M) =
(6.11)
otherwise
The same quantity can also be computed using the more traditional ap-
proach of identifying the subset A
j
of RS(M
0
) in which a given transition
T
j
is enabled (A
j
=
{
M
i
∈
RS(M
0
) : T
j
∈
E(M
i
)
}
). The mean number of
firings of T
j
per unit of time is then given by:
X
f
j
=
w
j
η
i
(6.12)
M
i
∈A
j
These results show that indeed, Petri nets can be used not only as a for-
malism for describing the behaviour of distributed/parallel systems and for
assessing their qualitative properties, but also as a tool for computing per-
formance indices that allow the e
ciency of these systems to be evaluated.
To illustrate the details of this last analysis step, a simple example is pre-
sented in the following subsection, with the explicit derivation of the CTMC
infinitesimal generator, of the steady-state probability distribution of the
different markings, and of some performance indices.
6.1.2
An example SPN system
Consider the SPN version of the example of a shared memory system in-
troduced in Chapter
1
(Fig.
1.5)
, representing two processors accessing a
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