Graphics Programs Reference
In-Depth Information
Using the method outlined by equation ( 6.3) we obtain the transition prob-
ability matrix of the REMC reported in Table 6.13. Solving the system of
linear equations ( 6.35) , we obtain the steady state probabilities for all the
states of the REMC.
Supposing that the transition weights assume the following values:
λ = 0.2, µ 1 = 2.0, µ 2 = 1.0, θ = 0.1, ν = 5.0, α = 0.99, and β = 0.01
we obtain, for example:
ψ 0 0
= 0.10911, ψ 0 2
= 0.11965, and ψ 0 37
= 0.00002
Assuming marking 1 as a reference, we compute the visits to any other state
using equation (6.36) .
Focusing our attention on markings M 0 , M 2 , and
M 37 , we obtain:
v 0,0
= 1.0, v 2,0
= 1.09666, and v 37,0
= 0.00020.
Knowing the mean sojourn time for the markings of the GSPN:
SJ 0
= 2.5, SJ 2
= 0.3125, and SJ 37
= 10.0
we can compute the mean cycle time with respect to reference marking M 0 :
CY (M 0 ) = 4.31108
and consequently the steady-state probability of each marking. For the three
markings arbitrarily chosen before we have:
η 0
= 0.57990, η 2
= 0.07949, and η 37
= 0.00047
6.3.2
Performance analysis of the example GSPN
The performance of the parallel processing system of Fig. 6.9 can be conve-
niently studied by evaluating the probability of having at least one process
waiting for synchronization.
For this purpose, we can use the following
reward function:
8
<
(M(p 5 ) 1) (M(p 6 ) 1)
: 1
0
r(M) =
(6.41)
otherwise
With this definition the desired probability is computed to have the value
R = 0.23814.
Similarly, the processing power of this system (i.e., the average number of
processors doing useful work accessing local data only) can be obtained using
the following reward function:
r(M) = M(p 1 )
(6.42)
which provides R = 1.50555.

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