Hardware Reference
In-Depth Information
1. if tasks are periodic and are simultaneously activated at time
t
=0, then the
schedule repeats itself every hyperperiod
H
; thus condition (4.26) needs to be
verified only for values of
L
less than or equal to
H
.
2.
g
(0
,L
) is a step function whose value increases when
L
crosses a deadline
d
k
and remains constant until the next deadline
d
k
+1
. This means that if condition
g
(0
,L
)
<L
holds for
L
=
d
k
, then it also holds for all
L
such that
d
k
≤
L<
d
k
+1
. As a consequence, condition (4.26) needs to be verified only for values of
L
equal to absolute deadlines.
The number of testing points can be reduced further by noting that
L
+
T
i
−
L
+
T
i
−
.
D
i
D
i
≤
T
i
T
i
and defining
n
n
L
+
T
i
−
D
i
T
i
−
D
i
L
T
i
G
(0
,L
)=
C
i
=
C
i
+
C
i
T
i
T
i
i
=1
i
=1
we have that
∀
L>
0
,
(0
,L
)
≤
G
(0
,L
)
,
where
n
G
(0
,L
)=
(
T
i
−
D
i
)
U
i
+
LU.
i
=1
From Figure 4.19, we can note that
G
(0
,L
) is a function of
L
increasing as a straight
line with slope
U
. Hence, if
U<
1, there exists an
L
=
L
∗
for which
G
(0
,L
)=
L
.
Clearly, for all
L
L
∗
, we have that
g
(0
,L
)
L
, meaning that the
schedulability of the task set is guaranteed. As a consequence, there is no need to
verify condition (4.26) for values of
L
≥
≤
G
(0
,L
)
≤
L
∗
.
≥
The value of
L
∗
is the time at which
G
(0
,L
∗
)=
L
∗
; that is,
n
D
i
)
U
i
+
L
∗
U
=
L
∗
,
(
T
i
−
i
=1
which gives
i
=1
(
T
i
−
D
i
)
U
i
L
∗
=
.
1
−
U
Considering that the task set must be tested at least until the largest relative dead-
line
D
max
, the results of the previous observations can be combined in the following
theorem.
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