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that is
2
−
P
U
s
≤
.
P
Hence,
2
−
P
U
max
s
=
.
(5.8)
P
Thus,
U
s
must be set to be less than or equal to
U
ma
s
. For a given
U
s
, however, there
is an infinite number of pairs (
C
s
,
T
s
) leading to the same utilization, so how can we
select the pair that enhances aperiodic responsiveness? A simple solution is to assign
the server the highest priority; that is, the smallest period, under Rate Monotonic.
However, it is not useful to set
T
s
<T
1
, since a smaller
T
s
implies a smaller
C
s
,
which would cause higher fragmentation (i.e., higher runtime overhead) in aperiodic
execution. Hence, assuming that priority ties between periodic tasks and the server are
broken in favor of the server, then the highest priority of the server can be achieved by
setting
T
s
=
T
1
, and then
C
s
=
U
s
T
s
.
5.3.3
APERIODIC GUARANTEE
This section shows how to estimate the response time of an aperiodic job handled
by a Polling Server, in order to possibly perform an online guarantee of firm aperiodic
requests characterized by a deadline. To do that, consider the case of a single aperiodic
job
J
a
, arrived at time
r
a
, with computation time
C
a
and deadline
D
a
. Since, in the
worst case, the job can wait for at most one period before receiving service, if
C
a
≤
C
s
the request is certainly completed within two server periods. Thus, it is guaranteed if
2
T
s
≤
D
a
.
For arbitrary computation times, the aperiodic request is certainly completed in
C
a
/C
s
server periods; hence, it is guaranteed if
T
s
+
C
a
C
s
T
s
≤
D
a
.
This schedulability test is only sufficient because it does not consider when the server
executes within its period.
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