Library prosa.analysis.facts.sporadic.arrival_times

Require Export prosa.model.task.arrival.sporadic.
Require Export prosa.analysis.facts.job_index.

Job Arrival Times in the Sporadic Model

In this file, we prove basic facts about the arrival times of jobs in the sporadic task model.

Section ArrivalTimes.

Consider sporadic tasks ...

Context {Task : TaskType} `{SporadicModel Task}.

... and any type of jobs associated with these tasks.

Context {Job : JobType} `{JobTask Job Task} `{JobArrival Job}.

Consider any unique arrival sequence with consistent arrivals, ...

  Variable arr_seq : arrival_sequence Job.
  Hypothesis H_consistent_arrivals: consistent_arrival_times arr_seq.
  Hypothesis H_uniq_arrseq: arrival_sequence_uniq arr_seq.

... and any sporadic task tsk that is to be analyzed.

  Variable tsk : Task.
  Hypothesis H_sporadic_model: respects_sporadic_task_model arr_seq tsk.
  Hypothesis H_valid_inter_min_arrival: valid_task_min_inter_arrival_time tsk.

We first show that for any two jobs j1 and j2, j2 arrives after j1 provided job_index of j2 strictly exceeds the job_index of j1.

  Lemma lower_index_implies_earlier_arrival:
    ∀ j1 j2,
      arrives_in arr_seq j1 →
      arrives_in arr_seq j2 →
      job_task j1 = tsk →
      job_task j2 = tsk →
      job_index arr_seq j1 < job_index arr_seq j2 →
      job_arrival j1 < job_arrival j2.
  Proof.
    move⇒ j1 j2 ARR1 ARR2 TSK1 TSK2 LT_IND.
    move: (H_sporadic_model j1 j2) ⇒ SPORADIC; feed_n 6 SPORADIC ⇒ //.
    - rewrite → diff_jobs_iff_diff_indices ⇒ //; eauto; first by lia.
      by subst.
    - apply (index_lte_implies_arrival_lte arr_seq); try eauto.
      by subst.
    - have POS_IA : task_min_inter_arrival_time tsk > 0 by auto.
      by lia.
  Qed.

In the following, consider (again) any two jobs from the arrival sequence that stem from task tsk.

NB: The following variables and hypotheses match the premises of the preceding lemma. However, we cannot move these declarations before the prior lemma because we need lower_index_implies_earlier_arrival to be ∀-quantified in the next proof.

  Variable j1 : Job.
  Variable j2 : Job.
  Hypothesis H_j1_from_arrseq: arrives_in arr_seq j1.
  Hypothesis H_j2_from_arrseq: arrives_in arr_seq j2.
  Hypothesis H_j1_task: job_task j1 = tsk.
  Hypothesis H_j2_task: job_task j2 = tsk.

We prove that jobs j1 and j2 are equal if and only if they arrive at the same time.

  Lemma same_jobs_iff_same_arr:
    j1 = j2 ↔
    job_arrival j1 = job_arrival j2.
  Proof.
    split; first by move⇒ →.
    move⇒ EQ_ARR.
    case: (boolP (j1 == j2)) ⇒ [/eqP EQ | /eqP NEQ] //; exfalso.
    rewrite → diff_jobs_iff_diff_indices in NEQ ⇒ //; eauto; last by rewrite H_j1_task.
    move /neqP: NEQ; rewrite neq_ltn ⇒ /orP [LT|LT].
    all: by apply lower_index_implies_earlier_arrival in LT ⇒ //; lia.
  Qed.

As a corollary, we observe that distinct jobs cannot have equal arrival times.

  Corollary uneq_job_uneq_arr:
      j1 ≠ j2 →
      job_arrival j1 ≠ job_arrival j2.
  Proof. by rewrite -same_jobs_iff_same_arr. Qed.

End ArrivalTimes.