# Library prosa.analysis.definitions.schedulability

Require Export prosa.analysis.facts.behavior.completion.

Require Import prosa.model.task.absolute_deadline.

Require Import prosa.model.task.absolute_deadline.

Consider any type of tasks, ...

... any type of jobs associated with these tasks, ...

Context {Job: JobType}.

Context `{JobArrival Job}.

Context `{JobCost Job}.

Context `{JobDeadline Job}.

Context `{JobTask Job Task}.

Context `{JobArrival Job}.

Context `{JobCost Job}.

Context `{JobDeadline Job}.

Context `{JobTask Job Task}.

... and any kind of processor state.

Consider any job arrival sequence...

...and any schedule of these jobs.

Let tsk be any task that is to be analyzed.

Then, we say that R is a response-time bound of tsk in this schedule ...

Definition task_response_time_bound :=

∀ j,

arrives_in arr_seq j →

job_of_task tsk j →

job_response_time_bound sched j R.

∀ j,

arrives_in arr_seq j →

job_of_task tsk j →

job_response_time_bound sched j R.

We say that a task is schedulable if all its jobs meet their deadline

Definition schedulable_task :=

∀ j,

arrives_in arr_seq j →

job_of_task tsk j →

job_meets_deadline sched j.

End Task.

∀ j,

arrives_in arr_seq j →

job_of_task tsk j →

job_meets_deadline sched j.

End Task.

In this section we infer schedulability from a response-time bound
of a task.

Consider any type of tasks, ...

... any type of jobs associated with these tasks, ...

Context {Job: JobType}.

Context `{JobArrival Job}.

Context `{JobCost Job}.

Context `{JobTask Job Task}.

Context `{JobArrival Job}.

Context `{JobCost Job}.

Context `{JobTask Job Task}.

... and any kind of processor state.

Consider any job arrival sequence...

...and any schedule of these jobs.

Assume that jobs don't execute after completion.

Let tsk be any task that is to be analyzed.

Given a response-time bound of tsk in this schedule no larger than its deadline, ...

Variable R: duration.

Hypothesis H_R_le_deadline: R ≤ task_deadline tsk.

Hypothesis H_response_time_bounded: task_response_time_bound arr_seq sched tsk R.

Hypothesis H_R_le_deadline: R ≤ task_deadline tsk.

Hypothesis H_response_time_bounded: task_response_time_bound arr_seq sched tsk R.

...then tsk is schedulable.

Lemma schedulability_from_response_time_bound:

schedulable_task arr_seq sched tsk.

Proof.

intros j ARRj JOBtsk.

rewrite /job_meets_deadline.

apply completion_monotonic with (t := job_arrival j + R);

[ | by apply H_response_time_bounded].

rewrite /job_deadline leq_add2l.

move: JOBtsk ⇒ /eqP →.

by erewrite leq_trans; eauto.

Qed.

End Schedulability.

schedulable_task arr_seq sched tsk.

Proof.

intros j ARRj JOBtsk.

rewrite /job_meets_deadline.

apply completion_monotonic with (t := job_arrival j + R);

[ | by apply H_response_time_bounded].

rewrite /job_deadline leq_add2l.

move: JOBtsk ⇒ /eqP →.

by erewrite leq_trans; eauto.

Qed.

End Schedulability.

We further define two notions of "all deadlines met" that do not
depend on a task abstraction: one w.r.t. all scheduled jobs in a
given schedule and one w.r.t. all jobs that arrive in a given
arrival sequence.

Consider any given type of jobs...

Context {Job : JobType}.

Context `{JobArrival Job}.

Context `{JobCost Job}.

Context `{JobDeadline Job}.

Context `{JobArrival Job}.

Context `{JobCost Job}.

Context `{JobDeadline Job}.

... any given type of processor states.

We say that all deadlines are met if every job scheduled at some
point in the schedule meets its deadline. Note that this is a
relatively weak definition since an "empty" schedule that is idle
at all times trivially satisfies it (since the definition does
not require any kind of work conservation).

Definition all_deadlines_met (sched: schedule PState) :=

∀ j t,

scheduled_at sched j t →

job_meets_deadline sched j.

∀ j t,

scheduled_at sched j t →

job_meets_deadline sched j.

To augment the preceding definition, we also define an alternate
notion of "all deadlines met" based on all jobs included in a
given arrival sequence.

Given an arbitrary job arrival sequence ...

... we say that all arrivals meet their deadline if every job
that arrives at some point in time meets its deadline. Note
that this definition does not preclude the existence of jobs in
a schedule that miss their deadline (e.g., if they stem from
another arrival sequence).

Definition all_deadlines_of_arrivals_met (sched: schedule PState) :=

∀ j,

arrives_in arr_seq j →

job_meets_deadline sched j.

End DeadlinesOfArrivals.

∀ j,

arrives_in arr_seq j →

job_meets_deadline sched j.

End DeadlinesOfArrivals.

We observe that the latter definition, assuming a schedule in
which all jobs come from the arrival sequence, implies the
former definition.

Lemma all_deadlines_met_in_valid_schedule:

∀ arr_seq sched,

jobs_come_from_arrival_sequence sched arr_seq →

all_deadlines_of_arrivals_met arr_seq sched →

all_deadlines_met sched.

Proof.

move⇒ arr_seq sched FROM_ARR DL_ARR_MET j t SCHED.

apply DL_ARR_MET.

by apply (FROM_ARR _ t).

Qed.

End AllDeadlinesMet.

∀ arr_seq sched,

jobs_come_from_arrival_sequence sched arr_seq →

all_deadlines_of_arrivals_met arr_seq sched →

all_deadlines_met sched.

Proof.

move⇒ arr_seq sched FROM_ARR DL_ARR_MET j t SCHED.

apply DL_ARR_MET.

by apply (FROM_ARR _ t).

Qed.

End AllDeadlinesMet.