Hardware Reference
In-Depth Information
PERIOD RESCALING
If the elastic coefficients are set equal to task nominal utilizations, elastic compression
has the effect of a simple rescaling, where all the periods are increased by the same
percentage. In order to work correctly, however, period rescaling must be uniformly
applied to all the tasks, without restrictions on the maximum period. This means
having
U
f
=0and
U
v
0
=
U
0
. Under this assumption, by setting
E
i
=
U
i
0
, Equation
(9.38) becomes:
U
d
)
U
i
0
U
0
=
U
i
0
U
0
U
i
0
U
0
∀
iU
i
=
U
i
0
−
(
U
0
−
[
U
0
−
(
U
0
−
U
d
)] =
U
d
from which we have
U
0
U
d
T
i
=
T
i
0
.
(9.42)
This means that in overload situations (
U
0
>
1) the compression algorithm causes all
task periods to be increased by a common scale factor
U
0
U
d
η
=
.
Note that after compression is performed, the total processor utilization becomes
n
C
i
ηT
i
0
1
η
U
0
=
U
d
U
=
=
U
0
=
U
d
U
0
i
=1
as desired.
If a maximum period needs to be defined for some task, an online guarantee test can
easily be performed before compression to check whether all the new periods are less
than or equal to the maximum value. This can be done in
O
(
n
) by testing whether
T
max
i
∀
i
=1
,...,n
ηT
i
0
≤
.
By deciding to apply period rescaling, we lose the freedom of choosing the elastic
coefficients, since they must be set equal to task nominal utilizations. However, this
technique has the advantage of leaving the task periods ordered as in the nominal
configuration, which simplifies the compression algorithm in the presence of resource
constraints and enables its usage in fixed priority systems, where priorities are typi-
cally assigned based on periods.
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