Hardware Reference
In-Depth Information
The following definitions are also used in this section:
U
i
min
=
C
i
/T
i
max
n
U
min
=
U
i
min
i
=1
U
i
0
=
C
i
/T
i
0
n
U
0
=
U
i
0
i
=1
Clearly, a solution can always be found if
U
min
≤
U
d
; hence, this condition has to be
verified a priori.
It is worth noting that the elastic model is more general than the classical Liu and
Layland's task model, so it does not prevent a user from defining hard real-time tasks.
In fact, a task having
T
i
max
is equivalent to a hard real-time task with fixed
period, independently of its elastic coefficient. A task with
E
i
=0can arbitrarily vary
its period within its specified range, but it cannot be varied by the system during load
reconfigurations.
=
T
i
0
To understand how an elastic guarantee is performed in this model, it is convenient to
compare an elastic task
τ
i
with a linear spring
S
i
characterized by a rigidity coefficient
k
i
, a nominal length
x
i
0
, and a minimum length
x
i
min
. In the following,
x
i
will denote
the actual length of spring
S
i
, which is constrained to be greater than or equal to
x
i
min
.
In this comparison, the length
x
i
of the spring is equivalent to the task's utilization
factor
U
i
=
C
i
/T
i
, and the rigidity coefficient
k
i
is equivalent to the inverse of the
task's elasticity (
k
i
=1
/E
i
). Hence, a set of
n
periodic tasks with total utilization
factor
U
p
=
i
=1
U
i
can be viewed as a sequence of
n
springs with total length
L
=
i
=1
x
i
.
In the linear spring system, this is equivalent to compressing the springs so that the
new total length
L
d
is less than or equal to a given maximum length
L
max
.
More
formally, in the spring system the problem can be stated as follows.
Given a set of
n
springs with known rigidity and length constraints, if the
total length
L
0
=
i
=1
x
i
0
>L
max
, find a set of new lengths
x
i
such that
and
i
=1
x
i
x
i
≥
x
i
min
=
L
d
, where
L
d
is any arbitrary desired length
such that
L
d
<L
max
.
Search WWH ::
Custom Search