Library prosa.analysis.facts.sporadic.arrival_bound

Sporadic Arrival Bound

In the following, we upper bound the number of jobs that can arrive in any interval as constrained by the sporadic task model's minimum inter-arrival time task_min_inter_arrival_time.
Consider any sporadic tasks ...
  Context {Task : TaskType} `{SporadicModel Task}.

... and their jobs.
  Context {Job : JobType} `{JobTask Job Task} `{JobArrival Job}.

We define the classic "ceiling of the interval length divided by minimum inter-arrival time", which we prove to be correct in the following.
To establish the bound's soundness, consider any well-formed arrival sequence, ...
... and any valid sporadic task tsk to be analyzed.
Before we can establish the bound, we require two auxiliary bounds, which we derive next. First, we consider minimum offset of the n-th job of the task that arrives in a given interval.
  Section NthJob.

For technical reasons, we require a "dummy" job in scope to use the nth function. In the proofs, we establish that the dummy job is never used, i.e., it is an irrelevant artifact induced by the ssreflect API. It may be safely ignored.
    Variable dummy : Job.

We observe that the i-th job to arrive in an interval [t1,t2) arrives no earlier than (task_min_inter_arrival_time tsk) ×i time units after the beginning of the interval due the minimum inter-arrival time of the sporadic task.
    Lemma arrival_of_nth_job :
       t1 t2 n i j,
        n = number_of_task_arrivals arr_seq tsk t1 t2
        i < n
        j = nth dummy (task_arrivals_between arr_seq tsk t1 t2) i
        job_arrival j t1 + (task_min_inter_arrival_time tsk) × i.
      movet1 t2 n i j. rewrite /number_of_task_arrivals.
      case ARR : (task_arrivals_between arr_seq tsk t1 t2) ⇒ [|j' js'] → // LIM JOB.
      elim: i LIM j JOB ⇒ [LIM j JOB|i IH LIM j JOB].
      { rewrite muln0 addn0.
        apply: job_arrival_between_ge; rt_eauto.
        apply: (task_arrivals_between_subset _ tsk _ t2).
        by rewrite JOB ARR; apply mem_nth. }
      { rewrite mulnSr addnA.
        pose prev_j := nth dummy (j' :: js') i.
        have prev_LIM : t1 + task_min_inter_arrival_time tsk × i + task_min_inter_arrival_time tsk
                         job_arrival prev_j + task_min_inter_arrival_time tsk
          by rewrite leq_add2r; apply IH ⇒ //; lia.
        apply: (leq_trans prev_LIM).
        have IN_j : j \in task_arrivals_between arr_seq tsk t1 t2
          by rewrite JOB ARR; apply mem_nth.
        have IN_prev : prev_j \in task_arrivals_between arr_seq tsk t1 t2
          by rewrite /prev_j ARR; apply mem_nth; lia.
        apply: H_sporadic_model ⇒ //=.
        { rewrite JOB /prev_j ⇒ /eqP.
          rewrite nth_uniq; try lia; rewrite -ARR.
          by apply: task_arrivals_between_uniq;rt_eauto. }
        { by apply/in_arrivals_implies_arrived/(task_arrivals_between_subset _ tsk t1 t2). }
        { by apply/in_arrivals_implies_arrived/(task_arrivals_between_subset _ tsk t1 t2). }
        { apply: in_task_arrivals_between_implies_job_of_task.
          exact IN_prev. }
        { apply: in_task_arrivals_between_implies_job_of_task.
          exact: IN_j. }
        { rewrite /prev_j JOB.
          have SORTED : sorted by_arrival_times (j' :: js')
            by rewrite -ARR; apply task_arrivals_between_sorted; rt_eauto.
          eapply (sorted_leq_nth _ _ _ SORTED); try lia.
          - rewrite unfold_in simpl_predE; lia.
          - rewrite unfold_in simpl_predE; lia. } }
      Unshelve. all: by rewrite /by_arrival_times/transitive/reflexive; lia.

  End NthJob.

As a second auxiliary lemma, we establish a minimum length on the interval for a given number of arrivals by applying the previous lemma to the last job in the interval. We consider only the case of "many" jobs, i.e., n 2, which ensures that the interval [t1, t2) spans at least one inter-arrival time.
  Lemma minimum_distance_for_n_sporadic_arrivals:
     t1 t2 n,
      number_of_task_arrivals arr_seq tsk t1 t2 = n
      n 2
      t2 > t1 + (task_min_inter_arrival_time tsk) × n.-1.
    movet1 t2 n H_num_arrivals H_many_jobs.
    (* First, let us get ourselves a job so we can discharge the dummy job parameter. *)
    destruct (task_arrivals_between arr_seq tsk t1 t2) as [|j js] eqn:ARR;
      first by move: ARR H_num_arrivals H_many_jobs; rewrite /number_of_task_arrivals ⇒ → //= ->; lia.
    (* Now that we can use nth, let's proceed with the actual proof. *)
    set j_last := (nth j (task_arrivals_between arr_seq tsk t1 t2) n.-1).
    have LAST : job_arrival j_last < t2.
    { apply: job_arrival_between_lt; rt_eauto.
      apply: task_arrivals_between_subset.
      apply mem_nth.
      by move: H_num_arrivals; rewrite /number_of_task_arrivals ⇒ ->; lia. }
    have DIST : t1 + task_min_inter_arrival_time tsk × n.-1 job_arrival j_last.
    { apply: arrival_of_nth_job; auto;
        first by rewrite [number_of_task_arrivals arr_seq tsk t1 _]H_num_arrivals; lia.
      by rewrite /j_last //=. }
    by lia.

Based on the above lemma, it is easy to see that max_sporadic_arrivals is indeed a correct upper bound on the maximum number of arrivals in a given interval.
  Theorem sporadic_task_arrivals_bound:
     t1 t2,
      number_of_task_arrivals arr_seq tsk t1 t2 max_sporadic_arrivals tsk (t2 - t1).
    movet1 t2.
    case COUNT: (number_of_task_arrivals arr_seq tsk t1 t2) ⇒ // [n'].
    case COUNT: n' COUNT ⇒ // [|n] NARR.
    { (* one arrival *)
      apply div_ceil_gt0 ⇒ //; rewrite subn_gt0.
      by eapply number_of_task_arrivals_nonzero; eauto. }
    { (* two or more arrivals *)
      clear n' COUNT.
      move: NARR. set n' := n.+2NARR.
      have SEP: t2 > t1 + (task_min_inter_arrival_time tsk) × n'.-1
        by apply: minimum_distance_for_n_sporadic_arrivals.
      move: SEP. rewrite -ltn_subRLSEP.
      by apply: div_ceil_multiple. }

End SporadicArrivalBound.