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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