Graphics Programs Reference
In-Depth Information
all its components equal to one, used to enforce the normalization condition
typical of all probability distributions.
To keep the notation simple, in the rest of this chapter we will use η i instead
of η(M i ) to denote the steady state probability of marking M i . The sojourn
time is the time spent by the PN system in a given marking M. As we
already observed in Chapter 5, the fact that a CTMC can be associated
with the SPN system ensures that the sojourn time in the ith marking is
exponentially distributed with rate q i . The pdf of the sojourn time in a
marking corresponds to the pdf of the minimum among the firing times
of the transitions enabled in the same marking; it thus follows that the
probability that a given transition T k E(M i ) fires (first) in marking M i
has the expression:
P { T k | M i } = w k
q i
.
(6.4)
Using the same argument, we can observe that the average sojourn time in
marking M i is given by the following expression:
1
q i
SJ i
=
.
(6.5)
6.1.1
SPN performance indices
The steady-state distribution η is the basis for a quantitative evaluation of
the behaviour of the SPN that is expressed in terms of performance indices.
These results can be computed using a unifying approach in which proper
index functions (also called reward functions) are defined over the markings
of the SPN and an average reward is derived using the steady-state proba-
bility distribution of the SPN. Assuming that r(M) represents one of such
reward functions, the average reward can be computed using the following
weighted sum:
X
R =
r(M i ) η i
(6.6)
M i ∈RS(M 0 )
Different interpretations of the reward function can be used to compute
different performance indices. In particular, the following quantities can be
computed using this approach.
(1) The probability of a particular condition of the SPN. Assuming that
condition Υ(M) is true only in certain markings of the PN, we can define
the following reward function:
8
<
: 1
0
Υ(M) = true
r(M) =
(6.7)
otherwise
The desired probability P { Υ } is then computed using equation 6.6.
The
same result can also be expressed as:
P { Υ } =
X
η i
(6.8)
M i ∈A
 
 
 
 
 
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