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PROSA - Formally Proven Schedulability Analysis
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Marco Maida
PROSA - Formally Proven Schedulability Analysis
Commits
ee05f255
Commit
ee05f255
authored
Dec 21, 2019
by
Björn Brandenburg
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Add periodic task model
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Require
Export
prosa
.
model
.
task
.
arrival
.
sporadic
.
(** * The Periodic Task Model *)
(** In the following, we define the classic Liu & Layland arrival process,
i.e., the arrival process commonly known as the _periodic task model_,
wherein the arrival times of consecutive jobs of a periodic task are
separated by the task's _period_. *)
(** ** Task Model Parameter for Periods *)
(** Under the periodic task model, each task is characterized by its period,
which we denote as [task_period]. *)
Class
PeriodicModel
(
Task
:
TaskType
)
:
=
task_period
:
Task
->
duration
.
(** ** Model Validity *)
(** Next, we define the semantics of the periodic task model. *)
Section
ValidPeriodicTaskModel
.
(** Consider any type of periodic tasks. *)
Context
{
Task
:
TaskType
}
`
{
PeriodicModel
Task
}.
(** A valid periodic task has a non-zero period. *)
Definition
valid_period
tsk
:
=
task_period
tsk
>
0
.
(** Further, in the context of a set of periodic tasks, ... *)
Variable
ts
:
TaskSet
Task
.
(** ... every task in the set must have a valid period. *)
Definition
valid_periods
:
=
forall
tsk
:
Task
,
tsk
\
in
ts
->
valid_period
tsk
.
(** Next, consider any type of jobs stemming from these tasks ... *)
Context
{
Job
:
JobType
}
`
{
JobTask
Job
Task
}
`
{
JobArrival
Job
}.
(** ... and an arbitrary arrival sequence of such jobs. *)
Variable
arr_seq
:
arrival_sequence
Job
.
(** We say that a task respects the periodic task model if the arrivals of
its jobs in the arrival sequence are separated by integer multiples of
the period. *)
Definition
respects_periodic_task_model
(
tsk
:
Task
)
:
=
forall
(
j
j'
:
Job
),
(** Given two different jobs j and j' ... *)
j
<>
j'
->
(** ...that belong to the arrival sequence... *)
arrives_in
arr_seq
j
->
arrives_in
arr_seq
j'
->
(** ... and that stem from the given task, ... *)
job_task
j
=
tsk
->
job_task
j'
=
tsk
->
(** ... if [j] arrives no later than [j'] ..., *)
job_arrival
j
<=
job_arrival
j'
->
(** ... then the arrivals of [j] and [j'] are separated by an integer
multiple of [task_period tsk]. *)
exists
n
,
n
>
0
/\
job_arrival
j'
=
(
n
*
task_period
tsk
)
+
job_arrival
j
.
(** Every task in a set of periodic tasks must satisfy the above
criterion. *)
Definition
taskset_respects_periodic_task_model
:
=
forall
tsk
,
tsk
\
in
ts
->
respects_periodic_task_model
tsk
.
End
ValidPeriodicTaskModel
.
(** ** Treating Periodic Tasks as Sporadic Tasks *)
(** Since the periodic-arrivals assumption is stricter than the
sporadic-arrivals assumption (i.e., any periodic arrival sequence is also a
valid sporadic arrivals sequence), it is sometimes convenient to apply
analyses for sporadic tasks to periodic tasks. We therefore provide an
automatic "conversion" of periodic tasks to sporadic tasks, i.e., we tell
Coq that it may use a periodic task's [task_period] parameter also as the
task's minimum inter-arrival time [task_min_inter_arrival_time]
parameter. *)
Section
PeriodicTasksAsSporadicTasks
.
(** Any type of periodic tasks ... *)
Context
{
Task
:
TaskType
}
`
{
PeriodicModel
Task
}.
(** ... and their corresponding jobs ... *)
Context
{
Job
:
JobType
}
`
{
JobTask
Job
Task
}
`
{
JobArrival
Job
}.
(** ... may be interpreted as a type of sporadic tasks by using each task's
period as its minimum inter-arrival time ... *)
Global
Instance
periodic_as_sporadic
:
SporadicModel
Task
:
=
{
task_min_inter_arrival_time
:
=
task_period
}.
(** ... such that the model interpretation is valid, both w.r.t. the minimum
inter-arrival time parameter ... *)
Remark
valid_period_is_valid_inter_arrival_time
:
forall
tsk
,
valid_period
tsk
->
valid_task_min_inter_arrival_time
tsk
.
Proof
.
trivial
.
Qed
.
(** ... and the separation of job arrivals. *)
Remark
periodic_task_respects_sporadic_task_model
:
forall
arr_seq
tsk
,
respects_periodic_task_model
arr_seq
tsk
->
respects_sporadic_task_model
arr_seq
tsk
.
Proof
.
move
=>
arr_seq
tsk
PERIODIC
j
j'
NOT_EQ
ARR_j
ARR_j'
TSK_j
TSK_j'
ORDER
.
rewrite
/
task_min_inter_arrival_time
/
periodic_as_sporadic
.
have
PERIOD_MULTIPLE
:
exists
n
,
n
>
0
/\
job_arrival
j'
=
(
n
*
task_period
tsk
)
+
job_arrival
j
by
apply
PERIODIC
=>
//.
move
:
PERIOD_MULTIPLE
=>
[
n
[
GT0
->]].
rewrite
addnC
leq_add2r
.
by
apply
leq_pmull
.
Qed
.
(** For convenience, we state these obvious correspondences also at the level
of entire task sets. *)
Remark
valid_periods_are_valid_inter_arrival_times
:
forall
ts
,
valid_periods
ts
->
valid_taskset_inter_arrival_times
ts
.
Proof
.
trivial
.
Qed
.
Remark
periodic_task_sets_respect_sporadic_task_model
:
forall
ts
arr_seq
,
taskset_respects_periodic_task_model
ts
arr_seq
->
taskset_respects_sporadic_task_model
ts
arr_seq
.
Proof
.
move
=>
?
?
PERIODIC
?
?.
apply
periodic_task_respects_sporadic_task_model
.
by
apply
PERIODIC
.
Qed
.
End
PeriodicTasksAsSporadicTasks
.
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