Library prosa.analysis.facts.preemption.rtc_threshold.nonpreemptive

Require Export prosa.analysis.facts.preemption.job.nonpreemptive.
Require Export prosa.model.task.preemption.fully_nonpreemptive.

Task's Run to Completion Threshold

In this section, we prove that instantiation of function task run to completion threshold to the fully non-preemptive model indeed defines a valid run-to-completion threshold function.

Section TaskRTCThresholdFullyNonPreemptive.

Consider any type of tasks ...

Context {Task : TaskType}.
Context `{TaskCost Task}.

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

  Context {Job : JobType}.
  Context `{JobTask Job Task}.
  Context `{JobArrival Job}.
  Context `{JobCost Job}.

We assume a fully non-preemptive task model.

  #[local] Existing Instance fully_nonpreemptive_job_model.
  #[local] Existing Instance fully_nonpreemptive_task_model.
  #[local] Existing Instance fully_nonpreemptive_rtc_threshold.

Consider any arrival sequence with consistent arrivals.

Variable arr_seq : arrival_sequence Job.
Hypothesis H_arrival_times_are_consistent : consistent_arrival_times arr_seq.

Next, consider any ideal non-preemptive uniprocessor schedule of this arrival sequence ...

Variable sched : schedule (ideal.processor_state Job).
Hypothesis H_nonpreemptive_sched : nonpreemptive_schedule sched.

... where jobs do not execute before their arrival or after completion.

Hypothesis H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched.
Hypothesis H_completed_jobs_dont_execute : completed_jobs_dont_execute sched.

First we prove that if the cost of a job j is equal to 0, then job_rtct j = 0 ...

  Fact job_rtc_threshold_is_0:
    ∀ j,
      job_cost j = 0 →
      job_rtct j = 0.
  Proof.
    intros.
    apply/eqP; rewrite eqn_leq; apply/andP; split; last by done.
    unfold job_rtct.
      by rewrite H3; compute.
  Qed.

... and ε otherwise.

  Fact job_rtc_threshold_is_ε:
    ∀ j,
      job_cost j > 0 →
      arrives_in arr_seq j →
      job_rtct j = ε.
  Proof.
    intros ? ARRj POSj; unfold ε in ×.
    unfold job_rtct.
    rewrite job_last_nps_is_job_cost.
      by rewrite subKn.
  Qed.

Consider a task with a positive cost.

Variable tsk : Task.
Hypothesis H_positive_cost : 0 < task_cost tsk.

Then, we prove that task_rtct function defines a valid task's run to completion threshold.

  Lemma fully_nonpreemptive_valid_task_run_to_completion_threshold:
    valid_task_run_to_completion_threshold arr_seq tsk.
  Proof.
    intros; split.
    - by unfold task_rtc_bounded_by_cost.
    - intros j ARR TSK.
      move: TSK ⇒ /eqP <-; rewrite /fully_nonpreemptive_rtc_threshold.
      edestruct (posnP (job_cost j)) as [ZERO|POS].
      + by rewrite job_rtc_threshold_is_0.
      + by erewrite job_rtc_threshold_is_ε; eauto 2.
  Qed.

End TaskRTCThresholdFullyNonPreemptive.
Global Hint Resolve fully_nonpreemptive_valid_task_run_to_completion_threshold : basic_rt_facts.