Library prosa.analysis.definitions.busy_interval

Busy Interval for JLFP-models

In this file we define the notion of busy intervals for uniprocessor for JLFP schedulers.
Section BusyIntervalJLFP.

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

Consider any kind of processor state model.
  Context {PState : ProcessorState Job}.

Consider any arrival sequence with consistent arrivals ...
... and a schedule of this arrival sequence.
  Variable sched : schedule PState.

Assume a given JLFP policy.
  Context `{JLFP_policy Job}.

In this section, we define the notion of a busy interval.
  Section BusyInterval.

Consider any job j.
    Variable j : Job.
    Hypothesis H_from_arrival_sequence : arrives_in arr_seq j.

We say that t is a quiet time for j iff every higher-priority job from the arrival sequence that arrived before t has completed by that time.
    Definition quiet_time (t : instant) :=
       (j_hp : Job),
        arrives_in arr_seq j_hp
        hep_job j_hp j
        arrived_before j_hp t
        completed_by sched j_hp t.

Based on the definition of quiet time, we say that interval [t1, t_busy) is a (potentially unbounded) busy-interval prefix iff the interval starts with a quiet time where a higher or equal priority job is released and remains non-quiet. We also require job j to be released in the interval.
    Definition busy_interval_prefix (t1 t_busy : instant) :=
      t1 < t_busy
      quiet_time t1
      ( t, t1 < t < t_busy ¬ quiet_time t)
      t1 job_arrival j < t_busy.

Next, we say that an interval [t1, t2) is a busy interval iff [t1, t2) is a busy-interval prefix and t2 is a quiet time.
    Definition busy_interval (t1 t2 : instant) :=
      busy_interval_prefix t1 t2
      quiet_time t2.

  End BusyInterval.

In this section we define the computational version of the notion of quiet time.
  Section DecidableQuietTime.

We say that t is a quiet time for j iff every higher-priority job from the arrival sequence that arrived before t has completed by that time.
    Definition quiet_time_dec (j : Job) (t : instant) :=
      all
        (fun j_hphep_job j_hp j ==> (completed_by sched j_hp t))
        (arrivals_before arr_seq t).

We also show that the computational and propositional definitions are equivalent.
    Lemma quiet_time_P :
       j t, reflect (quiet_time j t) (quiet_time_dec j t).
    Proof.
      intros; apply/introP.
      - intros QT s ARRs HPs BEFs.
        move: QT ⇒ /allP QT.
        specialize (QT s); feed QT.
        eapply arrived_between_implies_in_arrivals; eauto 2.
          by move: QT ⇒ /implyP Q; apply Q in HPs.
      - move ⇒ /negP DEC; intros QT; apply: DEC.
        apply/allP; intros s ARRs.
        apply/implyP; intros HPs.
        apply QT; try done.
        + by apply in_arrivals_implies_arrived in ARRs.
        + by eapply in_arrivals_implies_arrived_between in ARRs; eauto 2.
    Qed.

  End DecidableQuietTime.

End BusyIntervalJLFP.