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n
1
þ
j
k
k
k
such that n
i
1
þ
n
1
¼
s
n
1
; s
1
; s
2
; ...; s
n
1
g
n
1
. These real-time tasks support
different functionalities. Let
X
Sys
be the set of all tasks that can possibly implement
the system, and let us denote by Current
Sys
(t) the set of tasks implementing the
system Sys at t time units. These tasks should meet all deadlines de
ned in user
requirements. In this case, we note that Feasibility
ð
Current
Sys
ð
t
ÞÞ
True.
Problem
Now, we suppose the arrival of n
2
new aperiodic tasks at run-time at time t
1
for each
recon
guration scenario
w
h
. By considering a feasible System Sys before the
application of the recon
guration scenario
w
h
, each one of the tasks of
n
old
is
feasible, e.g. the execution of each instance is
finished before the corresponding
deadline. When the recon
guration scenario is applied at time t
1
, two cases exist,
If tasks of Current
Sys
ð
t
1
Þ
¼
n
new
[ n
old
are feasible, then no reaction should be
done by the agent,
Otherwise, the agent should provide two solutions for users (
nd the new
parameters
a
,
b
and
c
of tasks of
n
new
and
n
old
as the
first solution and
nd the
parameter
of the Poisson distribution as a second solution) in order to re-obtain
λ
the system
'
s feasibility.
The Poisson law with parameter
λ
, or law of the rare events, corresponds to the
following model:
Over a period T, an event arrives on average
time. We call X the random
variable determining the number of times when the event occurs for the period T.
X takes whole values: 0, 1, 2
λ
…
This random variable follows a law of probability de
ned by:
k
¼
k
e
k
P(k)
¼
P(X
¼
k)
k
!
where,
is a strictly positive real number
λ
1/
is a rate of the occurrences of aperiodic tasks per hyperperiod.
λ
Characterization
In our proposed and original work, the occurrence of aperiodic tasks (events)
follows a Poisson law with a constant parameter
called its
occurrence rate per time unit. In our model, we assume that the time unit is equal to
an hyperperiod hp
k
, and a variable 1
=k
2
LCM
, where LCM is the well-known Least
Common Multiple and (a
i,1
) is the earliest release time (activation time) of each
¼½
0
;
þ
max
k
ð
a
i
;
1
Þ
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