Hardware Reference
In-Depth Information
that is,
=
U
s
+
n
(
K
1
/n
U
lub
−
1)
.
(5.1)
Now, noting that
C
s
T
s
T
1
−
T
s
U
s
=
=
=
R
s
−
1
T
s
we have
R
s
=(
U
s
+1)
.
Thus,
K
can be rewritten as
2
R
s
2
U
s
+1
,
K
=
=
and finally
U
lub
=
U
s
+
n
2
U
s
+1
1
.
1
/n
−
(5.2)
Taking the limit of Equation (5.1) as
n
→∞
, we find the worst-case bound as a
function of
U
s
to be given by
=
U
s
+ln(
K
)=
U
s
+ln
2
U
s
+1
.
lim
n→∞
U
lub
(5.3)
Thus, given a set of
n
periodic tasks and a Polling Server with utilization factors
U
p
and
U
s
, respectively, the schedulability of the periodic task set is guaranteed under
RM if
U
s
+
n
K
1
/n
1
;
U
p
+
U
s
≤
−
that is, if
n
2
U
s
+1
1
.
1
/n
U
p
≤
−
(5.4)
A plot of Equation (5.3) as a function of
U
s
is shown in Figure 5.5. For comparison,
the RM bound is also reported in the plot. Note that the schedulability test expressed
in Equation (5.4) is also valid for all servers that behave like a periodic task.
Using the Hyperbolic Bound, the guarantee test for a task set in the presence of a
Polling Server can be performed as follows:
n
2
U
s
+1
.
(
U
i
+1)
≤
(5.5)
i
=1
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