Hardware Reference
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As another example, if tasks are allowed to change their frequency and τ 3 reduces its
period to 50, no feasible schedule exists, since the utilization would be greater than 1:
10
20 + 10
40 + 15
50 =1 . 05 > 1 .
Note that a feasible schedule exists for T 1 =22, T 2 =45, and T 3 =50. Hence, the
system can accept the higher request rate of τ 3 by slightly decreasing the rates of τ 1
and τ 2 . Task τ 3 can even run with a period T 3 =40, since a feasible schedule exists
with periods T 1 and T 2 within their range. In fact, when T 1 =24, T 2 =50, and
T 3 =40, U p =0 . 992. Finally, note that if τ 3 requires to run at its minimum period
( T 3 =35), there is no feasible schedule with periods T 1 and T 2 within their range,
hence the request of τ 3 to execute with a period T 3 =35must be rejected.
U p =
Clearly, for a given value of T 3 , there can be many different period configurations
that lead to a feasible schedule; thus one of the possible feasible configurations must
be selected. The elastic approach provides an efficient way for quickly selecting a
feasible period configuration among all the possible solutions.
THE ELASTIC MODEL
The basic idea behind the elastic model is to consider each task as flexible as a spring
with a given rigidity coefficient and length constraints. In particular, the utilization of
a task is treated as an elastic parameter, whose value can be modified by changing the
period within a specified range.
Each task is characterized by four parameters: a computation time C i , a nominal pe-
riod T i 0 (considered as the minimum period), a maximum period T i max , and an elastic
coefficient E i
0, which specifies the flexibility of the task to vary its utilization for
adapting the system to a new feasible rate configuration. The greater E i , the more
elastic the task. Thus, an elastic task is denoted as
τ i ( C i ,T i 0 ,T i max ,E i ) .
In the following, T i will denote the actual period of task τ i , which is constrained to be
in the range [ T i 0 ,T i max ]. Any task can vary its period according to its needs within
the specified range. Any variation, however, is subject to an elastic guarantee and is
accepted only if there is a feasible schedule in which all the other periods are within
their range.
Under the elastic model, given a set of n periodic tasks with utilization U p >U max ,
the objective of the guarantee is to compress tasks' utilization factors to achieve a new
desired utilization U d
U max such that all the periods are within their ranges.
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