Library prosa.results.edf.rta.bounded_nps

RTA for EDF with Bounded Non-Preemptive Segments

In this section we instantiate the Abstract RTA for EDF-schedulers with Bounded Priority Inversion to EDF-schedulers for ideal uni-processor model of real-time tasks with arbitrary arrival models and bounded non-preemptive segments.
Recall that Abstract RTA for EDF-schedulers with Bounded Priority Inversion does not specify the cause of priority inversion. In this section, we prove that the priority inversion caused by execution of non-preemptive segments is bounded. Thus the Abstract RTA for EDF-schedulers is applicable to this instantiation.
Consider any type of tasks ...
  Context {Task : TaskType}.
  Context `{TaskCost Task}.
  Context `{TaskDeadline Task}.
  Context `{TaskRunToCompletionThreshold Task}.
  Context `{TaskMaxNonpreemptiveSegment Task}.

... and any type of jobs associated with these tasks.
  Context {Job : JobType}.
  Context `{JobTask Job Task}.
  Context `{Arrival : JobArrival Job}.
  Context `{Cost : JobCost Job}.

We assume the classic (i.e., Liu & Layland) model of readiness without jitter or self-suspensions, wherein pending jobs are always ready.
  #[local] Existing Instance basic_ready_instance.

For clarity, let's denote the relative deadline of a task as D.
  Let D tsk := task_deadline tsk.

Consider the EDF policy that indicates a higher-or-equal priority relation. Note that we do not relate the EDF policy with the scheduler. However, we define functions for Interference and Interfering Workload that actively use the concept of priorities.
  Let EDF := EDF Job.

Consider any arrival sequence with consistent, non-duplicate arrivals.
Next, consider any valid ideal uni-processor schedule of this arrival sequence ...
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.
Next, we assume that the schedule is a work-conserving schedule...
... and the schedule respects the scheduling policy at every preemption point.
Consider an arbitrary task set ts, ...
  Variable ts : list Task.

... assume that all jobs come from the task set, ...
... and the cost of a job cannot be larger than the task cost.
Let max_arrivals be a family of valid arrival curves, i.e., for any task tsk in ts max_arrival tsk is (1) an arrival bound of tsk, and (2) it is a monotonic function that equals 0 for the empty interval delta = 0.
Let tsk be any task in ts that is to be analyzed.
  Variable tsk : Task.
  Hypothesis H_tsk_in_ts : tsk \in ts.

Consider a valid preemption model...
...and a valid task run-to-completion threshold function. That is, task_rtct tsk is (1) no bigger than tsk's cost, (2) for any job of task tsk job_rtct is bounded by task_rtct.
We introduce as an abbreviation rbf for the task request bound function, which is defined as task_cost(T) × max_arrivals(T,Δ) for a task T.
Next, we introduce task_rbf as an abbreviation for the task request bound function of task tsk.
  Let task_rbf := rbf tsk.

Using the sum of individual request bound functions, we define the request bound function of all tasks (total request bound function).
Next, we define an upper bound on interfering workload received from jobs of other tasks with higher-than-or-equal priority.
Let's define some local names for clarity.
For a job with the relative arrival offset A within its busy window, we define the following blocking bound. Only other tasks that potentially release non-zero-cost jobs are relevant, so we define a predicate to exclude pathological cases.

Search Space

If priority inversion is caused exclusively by non-preemptive sections, then we do not need to consider the priority-inversion bound in the search space. Hence we define the following search space, which refines the more general bounded_pi.is_in_search_space for our specific setting.
  Definition is_in_search_space (L A : duration) :=
    (A < L) && (task_rbf_changes_at tsk A
                || bound_on_total_hep_workload_changes_at ts tsk A).

For the following proof, we exploit the fact that the blocking bound is monotonically decreasing in A, which we note here.
  Fact blocking_bound_decreasing :
     A1 A2,
      A1 A2
      blocking_bound A1 blocking_bound A2.
    moveA1 A2 LEQ.
    rewrite /blocking_bound.
    apply: bigmax_subsettsk_o IN /andP[/andP[OTHER LT] ARR].
    by repeat (apply /andP; split) ⇒ //; lia.

To use the refined search space with the abstract theorem, we must show that it still includes all relevant points. To this end, we first observe that a step in the blocking bound implies the existence of a task that could release a job with an absolute deadline equal to the absolute deadline of the job under analysis.
  Lemma task_with_equal_deadline_exists :
      priority_inversion_changes_at blocking_bound A
       tsk_o, (tsk_o \in ts)
                 && (blocking_relevant tsk_o)
                 && (tsk_o != tsk)
                 && (D tsk_o == D tsk + A).
    moveA. rewrite /priority_inversion_changes_atNEQ.
    have LEQ: blocking_bound A blocking_bound (A - ε) by apply: blocking_bound_decreasing; lia.
    have LT: blocking_bound A < blocking_bound (A - ε) by lia.
    move: LT; rewrite /blocking_boundLT {LEQ} {NEQ}.
    move: (bigmax_witness_diff LT) ⇒ [tsk_o [IN [NOT HOLDS]]].
    move: HOLDS ⇒ /andP[REL LTeps].
     tsk_o; repeat (apply /andP; split) ⇒ //;
      first by apply /eqPEQ; move: LTeps; rewrite EQ; lia.
    move: NOT; rewrite negb_and ⇒ /orP[/negP // |].
    by move: LTeps; rewrite /εLTeps; lia.

With the above setup in place, we can show that the search space defined above by is_in_search_space covers the the more abstract search space defined by bounded_pi.is_in_search_space.
  Lemma search_space_inclusion :
      {A L},
       bounded_pi.is_in_search_space ts tsk blocking_bound L A
       is_in_search_space L A.
     moveA L /andP[BOUND STEP].
     apply /andP; split ⇒ //; apply /orP.
     move: STEP ⇒ /orP[/orP[STEP|RBF] | IBF]; [right| by left| by right].
     move: (task_with_equal_deadline_exists STEP) ⇒ [tsk_o /andP[/andP[/andP[IN REL] OTHER] EQ]].
     rewrite /bound_on_total_hep_workload_changes_at.
     apply /hasP; tsk_o ⇒ //.
     apply /andP; split; first by rewrite eq_sym.
     move: EQ. rewrite /D ⇒ /eqP EQ.
     rewrite /task_request_bound_function EQ.
     move: REL; rewrite /blocking_relevant ⇒ /andP [ARRIVES COST].
     rewrite eqn_pmul2l //.
     have → : A + task_deadline tsk - (task_deadline tsk + A)
              = 0 by lia.
     have → : A + ε + task_deadline tsk - (task_deadline tsk + A)
              = ε by lia.
     by move: (H_valid_arrival_curve tsk_o IN) ⇒ [-> _]; lia.

Priority inversion is bounded

In this section, we prove that a priority inversion for task tsk is bounded by the maximum length of non-preemptive segments among the tasks with lower priority.
First, we observe that the maximum non-preemptive segment length of any task that releases a job with an earlier absolute deadline (w.r.t. a given job j) and non-zero execution cost upper-bounds the maximum possible length of priority inversion (of said job j).
    Lemma priority_inversion_is_bounded_by_max_np_segment :
       {j t1},
        max_length_of_priority_inversion j t1
         \max_(j_lp <- arrivals_between arr_seq 0 t1 | (~~ EDF j_lp j)
                                                         && (job_cost j_lp > 0))
           (task_max_nonpreemptive_segment (job_task j_lp) - ε).
      movej t1.
      rewrite /max_length_of_priority_inversion /max_length_of_priority_inversion.
      apply: leq_big_maxj' JINB NOTHEP.
      rewrite leq_sub2r //.
      apply in_arrivals_implies_arrived in JINB.
      by apply H_valid_model_with_bounded_nonpreemptive_segments.

Second, we prove that the maximum length of a priority inversion of a given job j is indeed bounded by defined the blocking bound.
    Lemma priority_inversion_is_bounded_by_blocking:
       j t1 t2,
        arrives_in arr_seq j
        job_of_task tsk j
        busy_interval_prefix arr_seq sched j t1 t2
        max_length_of_priority_inversion j t1 blocking_bound (job_arrival j - t1).
      intros j t1 t2 ARR TSK BUSY; unfold max_length_of_priority_inversion, blocking_bound.
      destruct BUSY as [TT [QT [_ LE]]]; move: LE ⇒ /andP [GE LT].
      apply: leq_trans; first by apply: priority_inversion_is_bounded_by_max_np_segment.
      apply /bigmax_leq_seqPj' JINB NOTHEP.
      have ARR': arrives_in arr_seq j'
        by apply: in_arrivals_implies_arrived; exact: JINB.
      apply leq_bigmax_cond_seq with (x := (job_task j')) (F := fun tsktask_max_nonpreemptive_segment tsk - 1);
        first by apply H_all_jobs_from_taskset.
      eapply in_arrivals_implies_arrived_between in JINB; last by rt_eauto.
      move: JINB; move ⇒ /andP [_ TJ'].
      repeat (apply/andP; split); last first.
      { rewrite /EDF -ltnNge in NOTHEP.
        move: TSK ⇒ /eqP <-.
        have ARRLE: job_arrival j' < job_arrival j by apply leq_trans with t1.
        move: NOTHEP; rewrite /job_deadline /absolute_deadline.job_deadline_from_task_deadline /D.
        by lia. }
      { move: NOTHEP ⇒ /andP [_ NZ].
        move: (H_valid_job_cost j' ARR'); rewrite /valid_job_cost.
        by lia. }
      { apply: non_pathological_max_arrivals; last first.
          - exact: ARR'.
          - by rewrite /job_of_task.
          - by apply H_is_arrival_curve, H_all_jobs_from_taskset, ARR'. }

Using the lemma above, we prove that the priority inversion of the task is bounded by the maximum length of a nonpreemptive section of lower-priority tasks.
    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_sched_valid ⇒ [COARR MBR].
      destruct (leqP (t2 - t1) (blocking_bound (job_arrival j - t1))) as [NEQ|NEQ].
      { apply leq_trans with (t2 - t1); last by done.
        rewrite /cumulative_priority_inversion.
        rewrite -[X in _ X]addn0 -[t2 - t1]mul1n -iter_addn -big_const_nat.
        by rewrite leq_sum //; intros t _; destruct (priority_inversion_dec).
      edestruct @preemption_time_exists as [ppt [PPT NEQ2]]; rt_eauto.
      move: NEQ2 ⇒ /andP [GE LE].
      apply leq_trans with (cumulative_priority_inversion arr_seq sched j t1 ppt);
        last apply leq_trans with (ppt - t1).
      - rewrite /cumulative_priority_inversion.
        rewrite (@big_cat_nat _ _ _ ppt) //=; last first.
        { rewrite ltn_subRL in NEQ.
          apply leq_trans with (t1 + blocking_bound (job_arrival j - t1)); last by apply ltnW.
          apply leq_trans with (t1 + max_length_of_priority_inversion j t1); first by done.
          by rewrite leq_add2l; eapply priority_inversion_is_bounded_by_blocking; eauto 2; apply/eqP. }
        rewrite -[X in _ X]addn0 leq_add2l leqn0.
        rewrite big_nat_cond big1 //; movet /andP [/andP [GEt LTt] _ ].
        edestruct @not_quiet_implies_exists_scheduled_hp_job
          with (K := ppt - t1) (t := t) as [j_hp [ARRB [HP SCHEDHP]]]; rt_eauto.
        { by ppt; split; [done | rewrite subnKC //; apply/andP; split]. }
        { by rewrite subnKC //; apply/andP; split. }
        apply/eqP; rewrite eqb0; apply/negP; move ⇒ /priority_inversion_P INV.
        feed_n 3 INV; rt_eauto; last move: INV ⇒ [_ [j_lp /andP[SCHED PRIO]]].
        enough (EQ : j_lp = j_hp); first by subst; rewrite HP in PRIO.
        by eapply ideal_proc_model_is_a_uniprocessor_model; rt_eauto.
      - rewrite /cumulative_priority_inversion.
        rewrite -[X in _ X]addn0 -[ppt - t1]mul1n -iter_addn -big_const_nat.
        by rewrite leq_sum //; intros t _; destruct (priority_inversion_dec).
      - rewrite leq_subLR.
        apply leq_trans with (t1 + max_length_of_priority_inversion j t1); first by done.
        by rewrite leq_add2l; eapply priority_inversion_is_bounded_by_blocking; eauto 2; apply/eqP.

  End PriorityInversionIsBounded.

Response-Time Bound

In this section, we prove that the maximum among the solutions of the response-time bound recurrence is a response-time bound for tsk.
  Section ResponseTimeBound.

Let L be any positive fixed point of the busy interval recurrence.
    Variable L : duration.
    Hypothesis H_L_positive : L > 0.
    Hypothesis H_fixed_point : L = total_rbf L.

Consider any value R, and assume that for any given arrival offset A in the search space, there is a solution of the response-time bound recurrence which is bounded by R.
    Variable R : duration.
    Hypothesis H_R_is_maximum:
       (A : duration),
        is_in_search_space L A
         (F : duration),
          A + F blocking_bound A
                  + (task_rbf (A + ε) - (task_cost tsk - task_rtct tsk))
                  + bound_on_total_hep_workload A (A + F)
          R F + (task_cost tsk - task_rtct tsk).

Then, using the results for the general RTA for EDF-schedulers, we establish a response-time bound for the more concrete model of bounded nonpreemptive segments. Note that in case of the general RTA for EDF-schedulers, we just assume that the priority inversion is bounded. In this module we provide the preemption model with bounded nonpreemptive segments and prove that the priority inversion is bounded.
    Theorem uniprocessor_response_time_bound_edf_with_bounded_nonpreemptive_segments:
      response_time_bounded_by tsk R.
      eapply uniprocessor_response_time_bound_edf; rt_eauto.
      - by apply priority_inversion_is_bounded.
      - moveA BPI_SP.
        by apply H_R_is_maximum, search_space_inclusion.

  End ResponseTimeBound.

End RTAforEDFwithBoundedNonpreemptiveSegmentsWithArrivalCurves.