Library prosa.analysis.facts.model.arrival_curves

Require Export prosa.util.epsilon.
Require Export prosa.model.task.arrival.curves.
Require Export prosa.analysis.facts.model.task_arrivals.

Facts About Arrival Curves

We observe that tasks that exhibit job arrivals must have non-pathological arrival curves. This allows excluding pathological cases in higher-level proofs.

Section NonPathologicalCurve.

Consider any kind of tasks characterized by an upper-bounding arrival curve...

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

... and the corresponding type of jobs.

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

Consider an arbitrary task ...

Variable tsk : Task.

... and an arrival sequence ...

Variable arr_seq : arrival_sequence Job.

... that is compliant with the upper-bounding arrival curve.

Hypothesis H_curve_is_valid : respects_max_arrivals arr_seq tsk (max_arrivals tsk).

If we have evidence that a job of the task tsk ...

Variable j : Job.
Hypothesis H_job_of_tsk : job_of_task tsk j.

... arrives at any point in time, ...

Hypothesis H_arrives : arrives_in arr_seq j.

... then the maximum number of job arrivals in a non-empty interval cannot be zero.

  Lemma non_pathological_max_arrivals :
    max_arrivals tsk ε > 0.
  Proof.
    move: (H_arrives) ⇒ [t ARR].
    move: (H_curve_is_valid t t.+1 ltac:(by done)).
    rewrite -addn1 -addnBAC // addnBl_leq // ⇒ VALID.
    apply: (leq_trans _ VALID).
    rewrite /number_of_task_arrivals size_of_task_arrivals_between.
    rewrite big_ltn addn1 // big_geq // addn0.
    rewrite /task_arrivals_at size_filter //= -has_count.
    by apply /hasP; ∃ j.
  Qed.

End NonPathologicalCurve.