Library prosa.analysis.definitions.schedulability


(* ----------------------------------[ coqtop ]---------------------------------

Welcome to Coq 8.11.2 (June 2020)

----------------------------------------------------------------------------- *)

Require Export prosa.analysis.facts.behavior.completion.
Require Import prosa.model.task.absolute_deadline.

Schedulability

In the following section we define the notion of schedulable task.
Section Task.

Consider any type of tasks, ...
  Context {Task : TaskType}.

... any type of jobs associated with these tasks, ...
  Context {Job: JobType}.
  Context `{JobArrival Job}.
  Context `{JobCost Job}.
  Context `{JobDeadline Job}.
  Context `{JobTask Job Task}.

... and any kind of processor state.
  Context {PState : Type}.
  Context `{ProcessorState Job PState}.

Consider any job arrival sequence...
  Variable arr_seq: arrival_sequence Job.

...and any schedule of these jobs.
  Variable sched: schedule PState.

Let [tsk] be any task that is to be analyzed.
  Variable tsk: Task.

Then, we say that R is a response-time bound of [tsk] in this schedule ...
  Variable R: duration.

... iff any job [j] of [tsk] in this arrival sequence has completed by [job_arrival j + R].
  Definition task_response_time_bound :=
    ∀ j,
      arrives_in arr_seq j →
      job_task j = tsk →
      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_task j = tsk →
      job_meets_deadline sched j.

End Task.

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

Consider any type of tasks, ...
  Context {Task : TaskType}.
  Context `{TaskDeadline Task}.

... any type of jobs associated with these tasks, ...
  Context {Job: JobType}.
  Context `{JobArrival Job}.
  Context `{JobCost Job}.
  Context `{JobTask Job Task}.

... and any kind of processor state.
  Context {PState : Type}.
  Context `{ProcessorState Job PState}.

Consider any job arrival sequence...
  Variable arr_seq: arrival_sequence Job.

...and any schedule of these jobs.
  Variable sched: schedule PState.

Assume that jobs don't execute after completion.
  Hypothesis H_completed_jobs_dont_execute: completed_jobs_dont_execute sched.

Let [tsk] be any task that is to be analyzed.
  Variable tsk: Task.

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.

...then [tsk] is schedulable.
  Lemma schedulability_from_response_time_bound:
    schedulable_task arr_seq sched tsk.

(* ----------------------------------[ coqtop ]---------------------------------

1 subgoal (ID 352)
  
  Task : TaskType
  H : TaskDeadline Task
  Job : JobType
  H0 : JobArrival Job
  H1 : JobCost Job
  H2 : JobTask Job Task
  PState : Type
  H3 : ProcessorState Job PState
  arr_seq : arrival_sequence Job
  sched : schedule PState
  H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
  tsk : Task
  R : duration
  H_R_le_deadline : R <= task_deadline tsk
  H_response_time_bounded : task_response_time_bound arr_seq sched tsk R
  ============================
  schedulable_task arr_seq sched tsk

----------------------------------------------------------------------------- *)

  Proof.
    intros j ARRj JOBtsk.

(* ----------------------------------[ coqtop ]---------------------------------

1 subgoal (ID 356)
  
  Task : TaskType
  H : TaskDeadline Task
  Job : JobType
  H0 : JobArrival Job
  H1 : JobCost Job
  H2 : JobTask Job Task
  PState : Type
  H3 : ProcessorState Job PState
  arr_seq : arrival_sequence Job
  sched : schedule PState
  H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
  tsk : Task
  R : duration
  H_R_le_deadline : R <= task_deadline tsk
  H_response_time_bounded : task_response_time_bound arr_seq sched tsk R
  j : Job
  ARRj : arrives_in arr_seq j
  JOBtsk : job_task j = tsk
  ============================
  job_meets_deadline sched j

----------------------------------------------------------------------------- *)

    rewrite /job_meets_deadline.

(* ----------------------------------[ coqtop ]---------------------------------

1 subgoal (ID 363)
  
  Task : TaskType
  H : TaskDeadline Task
  Job : JobType
  H0 : JobArrival Job
  H1 : JobCost Job
  H2 : JobTask Job Task
  PState : Type
  H3 : ProcessorState Job PState
  arr_seq : arrival_sequence Job
  sched : schedule PState
  H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
  tsk : Task
  R : duration
  H_R_le_deadline : R <= task_deadline tsk
  H_response_time_bounded : task_response_time_bound arr_seq sched tsk R
  j : Job
  ARRj : arrives_in arr_seq j
  JOBtsk : job_task j = tsk
  ============================
  completed_by sched j (job_deadline j)

----------------------------------------------------------------------------- *)

    apply completion_monotonic with (t := job_arrival j + R);
      [ | by apply H_response_time_bounded].

(* ----------------------------------[ coqtop ]---------------------------------

1 subgoal (ID 370)
  
  Task : TaskType
  H : TaskDeadline Task
  Job : JobType
  H0 : JobArrival Job
  H1 : JobCost Job
  H2 : JobTask Job Task
  PState : Type
  H3 : ProcessorState Job PState
  arr_seq : arrival_sequence Job
  sched : schedule PState
  H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
  tsk : Task
  R : duration
  H_R_le_deadline : R <= task_deadline tsk
  H_response_time_bounded : task_response_time_bound arr_seq sched tsk R
  j : Job
  ARRj : arrives_in arr_seq j
  JOBtsk : job_task j = tsk
  ============================
  job_arrival j + R <= job_deadline j

----------------------------------------------------------------------------- *)

    rewrite /job_deadline leq_add2l JOBtsk.

(* ----------------------------------[ coqtop ]---------------------------------

1 subgoal (ID 385)
  
  Task : TaskType
  H : TaskDeadline Task
  Job : JobType
  H0 : JobArrival Job
  H1 : JobCost Job
  H2 : JobTask Job Task
  PState : Type
  H3 : ProcessorState Job PState
  arr_seq : arrival_sequence Job
  sched : schedule PState
  H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
  tsk : Task
  R : duration
  H_R_le_deadline : R <= task_deadline tsk
  H_response_time_bounded : task_response_time_bound arr_seq sched tsk R
  j : Job
  ARRj : arrives_in arr_seq j
  JOBtsk : job_task j = tsk
  ============================
  R <= task_deadline tsk

----------------------------------------------------------------------------- *)

      by erewrite leq_trans; eauto.

(* ----------------------------------[ coqtop ]---------------------------------

No more subgoals.

----------------------------------------------------------------------------- *)

  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.
Section AllDeadlinesMet.

Consider any given type of jobs...
  Context {Job : JobType}.
  Context `{JobArrival Job}.
  Context `{JobCost Job}.
  Context `{JobDeadline Job}.

... any given type of processor states.
  Context {PState: eqType}.
  Context `{ProcessorState Job PState}.

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.

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.
  Section DeadlinesOfArrivals.

Given an arbitrary job arrival sequence ...
    Variable arr_seq: arrival_sequence Job.

... 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.

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.

(* ----------------------------------[ coqtop ]---------------------------------

1 subgoal (ID 347)
  
  Job : JobType
  H : JobArrival Job
  H0 : JobCost Job
  H1 : JobDeadline Job
  PState : eqType
  H2 : ProcessorState Job PState
  ============================
  forall (arr_seq : arrival_sequence Job) (sched : schedule PState),
  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.

(* ----------------------------------[ coqtop ]---------------------------------

1 subgoal (ID 355)
  
  Job : JobType
  H : JobArrival Job
  H0 : JobCost Job
  H1 : JobDeadline Job
  PState : eqType
  H2 : ProcessorState Job PState
  arr_seq : arrival_sequence Job
  sched : schedule PState
  FROM_ARR : jobs_come_from_arrival_sequence sched arr_seq
  DL_ARR_MET : all_deadlines_of_arrivals_met arr_seq sched
  j : Job
  t : instant
  SCHED : scheduled_at sched j t
  ============================
  job_meets_deadline sched j

----------------------------------------------------------------------------- *)

    apply DL_ARR_MET.

(* ----------------------------------[ coqtop ]---------------------------------

1 subgoal (ID 356)
  
  Job : JobType
  H : JobArrival Job
  H0 : JobCost Job
  H1 : JobDeadline Job
  PState : eqType
  H2 : ProcessorState Job PState
  arr_seq : arrival_sequence Job
  sched : schedule PState
  FROM_ARR : jobs_come_from_arrival_sequence sched arr_seq
  DL_ARR_MET : all_deadlines_of_arrivals_met arr_seq sched
  j : Job
  t : instant
  SCHED : scheduled_at sched j t
  ============================
  arrives_in arr_seq j

----------------------------------------------------------------------------- *)

    by apply (FROM_ARR _ t).

(* ----------------------------------[ coqtop ]---------------------------------

No more subgoals.

----------------------------------------------------------------------------- *)

  Qed.

End AllDeadlinesMet.