Library prosa.analysis.facts.busy_interval.pi_bound

Bounded Priority Inversion Due to Non-Preemptive Sections

In the following, we relate the maximum cumulative priority inversion with a given blocking bound, assuming that priority version is caused (only) by non-preemptive segments.
Consider any type of tasks ...
  Context {Task : TaskType}.
  Context `{TaskCost Task}.

In addition, we assume that each task has a maximal non-preemptive segment ...
... and any type of preemptable jobs associated with these tasks.
  Context {Job : JobType}.
  Context `{JobTask Job Task}.
  Context `{Arrival : JobArrival Job}.
  Context `{Cost : JobCost Job}.
  Context `{JobPreemptable Job}.

Consider any valid arrival sequence,...
... and any schedule of this arrival sequence.
  Context {PState : ProcessorState Job}.
  Variable sched : schedule PState.

Consider a JLFP policy that indicates a higher-or-equal priority relation, and assume that the relation is reflexive and transitive.
And assume that they define a valid preemption model with bounded nonpreemptive segments.
Further, allow for any work-bearing notion of job readiness.
We assume that the schedule is valid.
Next, we assume that the schedule is a work-conserving schedule...
...,it respects the scheduling policy at every preemption point,...
... and complies with the preemption model.
Now, we consider an arbitrary task set ts...
  Variable ts : seq Task.

... and assume that all jobs stem from tasks in this task set.
Let tsk be any task in ts that is to be analyzed.
  Variable tsk : Task.
  Hypothesis H_tsk_in_ts : tsk \in ts.

Let blocking_bound be a bound on the priority inversion caused by jobs with lower priority.
The following argument assumes an ideal uniprocessor.
We show that, if the maximum length of a priority inversion of a given job j is bounded by the blocking bound,...
... then the priority inversion incurred by any job is bounded by the blocking bound.
  Lemma priority_inversion_is_bounded :
    priority_inversion_is_bounded_by arr_seq sched tsk blocking_bound.
    movej ARR TSK POS t1 t2 PREF.
    move: (PREF) ⇒ [_ [_ [_ /andP [T _]]]].
    move: H_valid_schedule ⇒ [COARR MBR].
    have [NEQ|NEQ] := boolP ((t2-t1) blocking_bound (job_arrival j - t1)).
    { apply leq_trans with (t2 -t1) ⇒ [|//].
      rewrite /cumulative_priority_inversion.
      rewrite -[X in _ X]addn0 -[t2 - t1]mul1n -iter_addn -big_const_nat.
      by rewrite leq_sum //; movet _; destruct (priority_inversion). }
    have [ppt [PPT' /andP[GE LE]]]: ppt : instant,
                                      preemption_time arr_seq sched ppt
                                      t1 ppt
                                      t1 + max_length_of_priority_inversion arr_seq j t1
      by exact: preemption_time_exists.
    apply leq_trans with (cumulative_priority_inversion arr_seq sched j t1 ppt).
    - by apply: priority_inversion_occurs_only_till_preemption_point =>//.
    - apply leq_trans with (ppt - t1).
      + rewrite /cumulative_priority_inversion -[X in _ X]addn0 -[ppt - t1]mul1n
                -iter_addn -big_const_nat.
        by rewrite leq_sum //; movet _; case: priority_inversion.
      + rewrite leq_subLR.
        apply leq_trans with (t1 + max_length_of_priority_inversion arr_seq j t1) ⇒ [//|].
        by rewrite leq_add2l; apply: (H_priority_inversion_is_bounded_by_blocking j t1 t2).

End PriorityInversionIsBounded.