Hardware Reference
In-Depth Information
L
i
T
h
C
h
.
+
h
:
P
h
≥P
i
L
i
=
B
i
(8.11)
This means that the response time of task
τ
i
must be computed for all the jobs
τ
i,k
(
k
=1
,
2
,...
) within the longest Level-
i
active period. That is, for all
k
∈
[1
,K
i
],
where
K
i
=
L
i
T
i
.
(8.12)
For a generic job
τ
i,k
, the start time
s
i,k
can be computed considering the blocking
time
B
i
, the computation time of the preceding (
k
1) jobs, and the interference of
the tasks with priority higher than
P
i
. Hence,
s
i,k
can be computed with the following
recurrent relation:
⎧
⎨
−
=
B
i
+
h
:
P
h
>P
i
s
(0)
i,k
C
h
s
(
−
1)
i,k
T
h
+1
C
h
.
(8.13)
+
h
:
P
h
>P
i
⎩
s
(
)
i,k
=
B
i
+(
k
−
1)
C
i
For the same job
τ
i,k
, the finishing time
f
i,k
can be computed by summing to the start
time
s
i,k
the computation time of job
τ
i,k
, and the interference of the tasks that can
preempt
τ
i,k
(those with priority higher than
θ
i
). That is,
⎧
⎨
f
(0)
i,k
=
s
i,k
+
C
i
f
(
−
1)
i,k
T
h
+1
C
h
.
s
i,k
T
h
=
s
i,k
+
C
i
+
h
:
P
h
>θ
i
(8.14)
f
(
)
i,k
⎩
−
Hence, the response time of task
τ
i
is given by
R
i
=max
k∈
[1
,K
i
]
{
f
i,k
−
(
k
−
1)
T
i
}
.
(8.15)
Once the response time of each task is computed, the task set is feasible if
∀
i
=1
,...,n
R
i
≤
D
i
.
(8.16)
The feasibility analysis under preemption thresholds can also be simplified under the
conditions of Theorem 8.1. In this case, we have that the worst-case start time is
S
i
T
h
+1
C
h
+
h
:
P
h
>P
i
S
i
=
B
i
(8.17)
and the worst-case response time of task
τ
i
can be computed as
R
i
T
h
S
i
T
h
+1
C
h
.
+
h
:
P
h
>θ
i
R
i
=
S
i
+
C
i
−
(8.18)
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