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 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 i
R i
= S i
+ C i
(8.18)
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