Library prosa.analysis.facts.blocking_bound_fp

Require Export prosa.analysis.definitions.blocking_bound_fp.
Require Export prosa.analysis.facts.busy_interval.pi.

Lower-Priority Non-Preemptive Segment is Bounded

In this file, we prove that, under the FP scheduling policy, the length of the maximum non-preemptive segment of a lower-priority job (w.r.t. a job of a task tsk) is bounded by blocking_bound.

Section MaxNPSegmentIsBounded.

Consider any type of tasks ...

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

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

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

Consider any kind of processor state model.

Context `{PState : ProcessorState Job}.

Consider an FP policy.

Context {FP : FP_policy Task}.

Consider any arrival sequence ...

Variable arr_seq : arrival_sequence Job.

... and any schedule of this arrival sequence.

Variable sched : schedule PState.

In addition, we assume the existence of a function mapping jobs to their preemption points ...

Context `{JobPreemptable Job}.

... and assume that it defines a valid preemption model with bounded non-preemptive segments.

Hypothesis H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched.

Consider an arbitrary task set ts, ...

Variable ts : list Task.

... assume that all jobs come from the task set, ...

Hypothesis H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts.

... and let tsk be any task in ts that is to be analyzed.

Variable tsk : Task.
Hypothesis H_tsk_in_ts : tsk \in ts.

Consider any job j of tsk.

Variable j : Job.
Hypothesis H_job_of_tsk : job_of_task tsk j.

Then, the maximum length of a nonpreemptive segment among all lower-priority jobs (w.r.t. the given job j) arrived so far is bounded by blocking_bound.

  Lemma priority_inversion_is_bounded_by_blocking :
    ∀ t,
      max_lp_nonpreemptive_segment arr_seq j t ≤ blocking_bound ts tsk.
  Proof.
    move⇒ t; move: H_job_of_tsk ⇒ /eqP TSK.
    apply: leq_trans; first exact: max_np_job_segment_bounded_by_max_np_task_segment.
    apply: (@leq_trans (\max_(j_lp <- arrivals_between arr_seq 0 t
                             | (~~ hep_task (job_task j_lp) tsk ) && (0 < job_cost j_lp ))
                         (task_max_nonpreemptive_segment (job_task j_lp) - ε ))).
    { rewrite /hep_job /FP_to_JLFP TSK.
      apply: leq_big_max ⇒ j' JINB NOTHEP.
      by rewrite leq_sub2r //. }
    { apply /bigmax_leq_seqP ⇒ j' JINB /andP[NOTHEP POS].
      apply leq_bigmax_cond_seq with
        (x := (job_task j')) (F := fun tsk ⇒ task_max_nonpreemptive_segment tsk - 1);
        last by done.
      apply: H_all_jobs_from_taskset.
      by apply: in_arrivals_implies_arrived (JINB). }
  Qed.

End MaxNPSegmentIsBounded.