Library prosa.analysis.facts.priority.fifo

In this section, we prove some fundamental properties of the FIFO policy.

Section BasicLemmas.

We assume the basic (i.e., Liu & Layland) readiness model under which any pending job is ready.

#[local] Existing Instance basic_ready_instance.

Consider any type of jobs with arrival times and execution costs.

Context `{Job : JobType} {Arrival : JobArrival Job} {Cost : JobCost Job}.

Consider any arrival sequence of such jobs ...

Variable arr_seq : arrival_sequence Job.

... and the resulting ideal uniprocessor schedule. We assume that the schedule is valid and work-conserving.

  Variable sched : schedule (ideal.processor_state Job).
  Hypothesis H_schedule_is_valid : valid_schedule sched arr_seq.
  Hypothesis H_work_conservation : work_conserving arr_seq sched.

Suppose jobs have preemption points ...

Context `{JobPreemptable Job}.

...and that the preemption model is valid.

Hypothesis H_valid_preemption_model :
valid_preemption_model arr_seq sched.

Assume that the schedule respects the FIFO scheduling policy whenever jobs are preemptable.

Hypothesis H_respects_policy : respects_JLFP_policy_at_preemption_point arr_seq sched (FIFO Job).

We observe that there is no priority inversion in a FIFO-compliant schedule.

  Lemma FIFO_implies_no_priority_inversion :
    ∀ j t,
      arrives_in arr_seq j →
      pending sched j t →
      ¬ priority_inversion sched j t.

In this section, we prove the cumulative priority inversion for any task is bounded by 0.

Section PriorityInversionBounded.

Consider any kind of tasks.

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

Consider a task tsk.

Variable tsk : Task.

Assume the arrival times are consistent.

Hypothesis H_consistent_arrival_times : consistent_arrival_times arr_seq.

Assume that the schedule follows the FIFO policy at preemption time.

Hypothesis H_respects_policy_at_preemption_point :
respects_JLFP_policy_at_preemption_point arr_seq sched (FIFO Job).

Assume the schedule is valid.

Hypothesis H_valid_schedule : valid_schedule sched arr_seq.

Assume there are no duplicates in the arrival sequence.

Hypothesis H_arrival_sequence_is_a_set : arrival_sequence_uniq arr_seq.

Then we prove that the amount of priority inversion is bounded by 0.

    Lemma FIFO_implies_no_pi :
      priority_inversion_is_bounded_by_constant arr_seq sched tsk 0.
  End PriorityInversionBounded.

We prove that in a FIFO-compliant schedule, if a job j is scheduled, then all jobs with higher priority than j have been completed.

  Lemma scheduled_implies_higher_priority_completed :
    ∀ j t,
      scheduled_at sched j t →
      ∀ j_hp,
        arrives_in arr_seq j_hp →
        ~~hep_job j j_hp →
        completed_by sched j_hp t.

The next lemma considers FIFO schedules in the context of tasks.

Section SequentialTasks.

If the scheduled jobs stem from a set of tasks, ...

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

... then the tasks in a FIFO-compliant schedule necessarily execute sequentially.

Lemma tasks_execute_sequentially : sequential_tasks arr_seq sched.

We also note that the FIFO policy respects sequential tasks.

Fact fifo_respects_sequential_tasks : policy_respects_sequential_tasks.

End SequentialTasks.

Finally, let us further assume that there are no needless preemptions among jobs of equal priority.

Hypothesis H_no_superfluous_preemptions : no_superfluous_preemptions sched.

In the absence of superfluous preemptions and under assumption of the basic readiness model, there are no preemptions at all in a FIFO-compliant schedule.

  Lemma no_preemptions_under_FIFO :
    ∀ j t,
      ~~ preempted_at sched j t.

It immediately follows that FIFO schedules are non-preemptive.

Corollary FIFO_is_nonpreemptive : nonpreemptive_schedule sched.

End BasicLemmas.

We add the following lemmas to the basic facts database

Global Hint Resolve
  fifo_respects_sequential_tasks
  tasks_execute_sequentially
  : basic_rt_facts.