Library prosa.analysis.facts.busy_interval.carry_in

Busy Interval From Workload Bound

In the following, we derive an alternative condition for the existence of a busy interval based on the total workload. If the total workload generated by the task set is bounded, then there necessarily exists a moment without any carry-in workload, from which it follows that a busy interval has ended.
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}.

Consider any valid arrival sequence.
Allow for any fully-consuming uniprocessor model.
Next, consider any schedule of the arrival sequence ...
... where jobs do not execute before their arrival or after completion.
Assume a given reflexive JLFP policy.
Further, allow for any work-bearing notion of job readiness ...
  Context `{!JobReady Job PState}.
  Hypothesis H_job_ready : work_bearing_readiness arr_seq sched.

... and assume that the schedule is work-conserving.

Times without Carry-In

As the first step, we derive a sufficient condition for the existence of a no-carry-in instant for uniprocessor for JLFP schedulers.
We start by noting that, trivially, the processor has no carry-in at the beginning (i.e., has no carry-in at time instant 0).
  Lemma no_carry_in_at_zero :
    no_carry_in arr_seq sched 0.
  Proof. by movej _ +; rewrite /arrived_before ltn0. Qed.

Next, we relate idle times to the no-carry-in condition. First, the presence of a pending job implies that the processor isn't idle due to work-conservation.
  Lemma pending_job_not_idle :
     j t,
      arrives_in arr_seq j
      pending sched j t
      ¬ is_idle arr_seq sched t.
    movej t ARR PEND IDLE.
    have [jhp [ARRhp [READYhp _]]] :
       j_hp : Job, arrives_in arr_seq j_hp job_ready sched j_hp t hep_job j_hp j
      by apply: H_job_ready.
    move: IDLE; rewrite is_idle_iff ⇒ /eqP; rewrite scheduled_job_at_none ⇒ // IDLE.
    have [j_other SCHED]: j_other : Job, scheduled_at sched j_other t
      by apply: H_work_conserving ARRhp _; apply/andP.
    by move: (IDLE j_other) ⇒ /negP.

Second, an idle time implies no carry in at this time instant.
  Lemma idle_instant_no_carry_in :
      is_idle arr_seq sched t
      no_carry_in arr_seq sched t.
    movet IDLE j ARR HA.
    apply: (pending_job_not_idle j t) ⇒ //.
    by apply/andP; split; rewrite // /has_arrived ltnW.

Moreover, an idle time implies no carry in at the next time instant, too.
  Lemma idle_instant_next_no_carry_in :
      is_idle arr_seq sched t
      no_carry_in arr_seq sched t.+1.
    movet IDLE j ARR HA.
    apply: (pending_job_not_idle j t) ⇒ //.
    apply/andP; split; rewrite // /has_arrived.
    by apply: incompletion_monotonic NCOMP.

Bounded-Workload Assumption

We now introduce the central assumption from which we deduce the existence of a busy interval.
To this end, recall the notion of workload of all jobs released in a given interval [t1, t2)...
... and total service of jobs within some time interval [t1, t2).
We assume that, for some positive Δ, the sum of the total blackout and the total workload generated in any interval of length Δ starting with no carry-in is bounded by Δ. Note that this assumption bounds the total workload of jobs released in a time interval [t, t + Δ) regardless of their priorities.
  Variable Δ : duration.
  Hypothesis H_delta_positive : Δ > 0.
  Hypothesis H_workload_is_bounded :
      no_carry_in arr_seq sched t
      blackout_during sched t (t + Δ) + total_workload t (t + Δ) Δ.

In the following, we also require a unit-speed processor.

Existence of a No-Carry-In Instant

Next we prove that, if for any time instant t there is a point where the total workload generated since t is upper-bounded by the length of the interval, there must exist a no-carry-in instant.
  Section ProcessorIsNotTooBusy.

As a stepping stone, we prove in the following section that for any time instant t there exists another time instant t' ∈ (t, t + Δ] such that the processor has no carry-in at time t'.
Consider an arbitrary time instant t ...
      Variable t : duration.

... such that there is no carry-in at time t.
      Hypothesis H_no_carry_in : no_carry_in arr_seq sched t.

First, recall that the total service is bounded by the total workload. Therefore the sum of the total blackout and the total service of jobs in the interval [t, t + Δ) is bounded by Δ.
      Lemma total_service_is_bounded_by_Δ :
        blackout_during sched t (t + Δ) + total_service t (t + Δ) Δ.
        have EQ: \sum_(t x < t + Δ) 1 = Δ.
        { by rewrite big_const_nat iter_addn mul1n addn0 -{2}[t]addn0 subnDl subn0. }
        rewrite -{3}EQ {EQ}.
        rewrite /total_service /blackout_during /supply.blackout_during.
        rewrite /service_of_jobs/service_during/service_at exchange_big //=.
        rewrite -big_split //= leq_sum //; movet' _.
        have [BL|SUP] := blackout_or_supply sched t'.
        { rewrite -[1]addn0; apply leq_add; first by case: (is_blackout).
          rewrite leqn0; apply/eqP; apply big1j _.
          eapply no_service_during_blackout in BL.
          by apply: BL. }
        { rewrite /is_blackout SUP add0n.
          exact: service_of_jobs_le_1. }

Next we consider two cases: (1) The case when the sum of blackout and service is strictly less than Δ, and (2) the case when the sum of blackout and service is equal to Δ.
In the first case, we use the pigeonhole principle to conclude that there is an idle time instant; which in turn implies existence of a time instant with no carry-in.
      Lemma low_total_service_implies_existence_of_time_with_no_carry_in :
        blackout_during sched t (t + Δ) + total_service t (t + Δ) < Δ
          δ < Δ no_carry_in arr_seq sched (t.+1 + δ).
        rewrite /total_service-{3}[Δ]addn0 -{2}(subnn t) addnBA // [Δ + t]addnCLTS.
        have [t_idle [/andP [LEt GTe] IDLE]]: t0 : nat,
                                                t t0 < t + Δ
                                                 is_idle arr_seq sched t0.
        { apply: low_service_implies_existence_of_idle_time_rs =>//.
          rewrite !subnKC in LTS; try by apply leq_addr.
          by rewrite addKn. }
        move: LEt; rewrite leq_eqVlt; move ⇒ /orP [/eqP EQ|LT].
        { 0; split ⇒ //.
          rewrite addn0 EQs ARR BEF.
          by apply: idle_instant_next_no_carry_in. }
        have EX: γ, t_idle = t + γ.
        { by (t_idle - t); rewrite subnKC // ltnW. }
        move: EX ⇒ [γ EQ].
        move : GTe LT; rewrite EQ ltn_add2l -{1}[t]addn0 ltn_add2lGTe LT.
         (γ.-1); split.
        - apply leq_trans with γ.
          + by rewrite prednK.
          + by rewrite ltnW.
        - rewrite -subn1 -addn1 -addnA subnKC // ⇒ s ARR BEF.
          exact: idle_instant_no_carry_in.

In the second case, the sum of blackout and service within the time interval [t, t + Δ) is equal to Δ. We also know that the total workload is lower-bounded by the total service and upper-bounded by Δ. Therefore, the total workload is equal to the total service, which implies completion of all jobs by time t + Δ and hence no carry-in at time t + Δ.
      Lemma completion_of_all_jobs_implies_no_carry_in :
        blackout_during sched t (t + Δ) + total_service t (t + Δ) = Δ
        no_carry_in arr_seq sched (t + Δ).
        rewrite /total_serviceEQserv s ARR BEF.
        move: (H_workload_is_bounded t) ⇒ WORK.
        have EQ: total_workload 0 (t + Δ)
                 = service_of_jobs sched predT (arrivals_between arr_seq 0 (t + Δ)) 0 (t + Δ);
          last exact: workload_eq_service_impl_all_jobs_have_completed.
        have CONSIST: consistent_arrival_times arr_seq by [].
        have COMPL := all_jobs_have_completed_impl_workload_eq_service
                        _ arr_seq CONSIST sched
                        predT 0 t t.
        feed_n 2 COMPL ⇒ //.
        { movej A B; apply: H_no_carry_in.
          - apply: in_arrivals_implies_arrived =>//.
          - by have : arrived_between j 0 t
                        by apply: (in_arrivals_implies_arrived_between arr_seq). }
        apply/eqP; rewrite eqn_leq; apply/andP; split;
          last by apply: service_of_jobs_le_workload.
        rewrite /total_workload (workload_of_jobs_cat arr_seq t);
          last by apply/andP; split; [|rewrite leq_addr].
        - rewrite (service_of_jobs_cat_scheduling_interval _ _ _ _ _ _ _ t) //;
            last by apply/andP; split; [|rewrite leq_addr].
          + rewrite COMPL -addnA leq_add2l.
            rewrite -service_of_jobs_cat_arrival_interval;
              last by apply/andP; split; [|rewrite leq_addr].
            by evar (b : nat); rewrite -(leq_add2l b) EQserv.

    End ProcessorIsNotTooBusyInduction.

Finally, we show that any interval of length Δ contains a time instant with no carry-in.
    Lemma processor_is_not_too_busy :
       t, δ,
        δ < Δ no_carry_in arr_seq sched (t + δ).
      elim⇒ [|t [δ [LE FQT]]];
        first by 0; split; [ | rewrite addn0; apply: no_carry_in_at_zero].
      move: (posnP δ) ⇒ [Z|POS]; last first.
      - (δ.-1); split.
        + by apply: leq_trans LE; rewrite prednK.
        + by rewrite -subn1 -addn1 -addnA subnKC //.
      - move: FQT; rewrite Z addn0FQT {LE}.
        move: (total_service_is_bounded_by_Δ t); rewrite leq_eqVlt ⇒ /orP [/eqP EQ | LT].
        + (Δ.-1); split; first by rewrite prednK.
          rewrite addSn -subn1 -addn1 -addnA subnK //.
          by apply: completion_of_all_jobs_implies_no_carry_in.
        + by apply:low_total_service_implies_existence_of_time_with_no_carry_in.

  End ProcessorIsNotTooBusy.

Busy Interval Existence

Consider an arbitrary job j with positive cost.
  Variable j : Job.
  Hypothesis H_from_arrival_sequence : arrives_in arr_seq j.
  Hypothesis H_job_cost_positive : job_cost_positive j.

We show that there must exist a busy interval [t1, t2) that contains the arrival time of j.
  Theorem busy_interval_from_total_workload_bound :
     t1 t2,
      t1 job_arrival j < t2
       t2 t1 + Δ
       busy_interval arr_seq sched j t1 t2.
    rename H_from_arrival_sequence into ARR, H_job_cost_positive into POS.
    edestruct (exists_busy_interval_prefix
                 arr_seq H_valid_arr_seq
                 sched j ARR H_priority_is_reflexive (job_arrival j))
      as [t1 [PREFIX GE]]; first by apply: job_pending_at_arrival.
    move: GE ⇒ /andP [GE1 _].
    have [δ [LE nCAR]]: δ : nat, δ < Δ no_carry_in arr_seq sched (t1.+1 + δ)
      by apply: processor_is_not_too_busy ⇒ //.
    have QT : quiet_time arr_seq sched j (t1.+1 + δ) by apply: no_carry_in_implies_quiet_time.
    have EX: t2, ((t1 < t2 t1.+1 + δ) && quiet_time_dec arr_seq sched j t2).
    { (t1.+1 + δ); apply/andP; split.
      - by apply/andP; split; first rewrite addSn ltnS leq_addr.
      - exact/quiet_time_P. }
    move: (ex_minnP EX) ⇒ [t2 /andP [/andP [GTt2 LEt2] QUIET] MIN]; clear EX.
    have NEQ: t1 job_arrival j < t2.
    { apply/andP; split⇒ [//|].
      rewrite ltnNge; apply/negPCONTR.
      have [_ [_ [NQT _]]] := PREFIX.
      have {}NQT := NQT t2.
      feed NQT; first by (apply/andP; split; [|rewrite ltnS]).
      by apply: NQT; apply/quiet_time_P. }
     t2; split⇒ [//|]; split.
    { by apply: (leq_trans LEt2); rewrite addSn ltn_add2l. }
    { move: PREFIX ⇒ [_ [QTt1 [NQT _]]]; repeat split⇒ //; last by exact/quiet_time_P.
      movet /andP [GEt LTt] QTt.
      feed (MIN t);
        last by move: LTt; rewrite ltnNge; move ⇒ /negP LTt; apply: LTt.
      apply/andP; split.
      + by apply/andP; split; last (apply leq_trans with t2; [apply ltnW |]).
      + exact/quiet_time_P. }

End BusyIntervalExistence.