Library prosa.analysis.facts.priority.fifo

We first make some trivial observations about the FIFO priority policy to avoid having to re-reason these steps repeatedly in the subsequent proofs.

Section PriorityFacts.

Consider any type of jobs.

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

Under FIFO scheduling, ...

Context {JLFP : JLFP_policy Job}.
Hypothesis H_policy_is_FIFO : policy_is_FIFO JLFP.

... higher-or-equal priority implies no later arrival.

  Fact FIFO_policy_arrival_order :
    ∀ j j',
      hep_job j j' → job_arrival j ≤ job_arrival j'.

The FIFO policy is reflexive by assumption.

Fact FIFO_policy_is_reflexive :
reflexive_job_priorities JLFP.

The FIFO policy is transitive by assumption.

Fact FIFO_policy_is_transitive :
transitive_job_priorities JLFP.

The FIFO policy is total by assumption.

Fact FIFO_policy_is_total :
total_job_priorities JLFP.

Hence, a strictly earlier-arriving job has higher-or-equal priority.

  Fact FIFO_policy_earlier_arrival :
    ∀ j j',
      job_arrival j < job_arrival j' → hep_job j j'.

Conversely, if a job does not have higher-or-equal priority, then the other job arrives no later.

  Fact FIFO_policy_not_hep_job_arrival_order :
    ∀ j j',
      ~~ hep_job j j' → job_arrival j' ≤ job_arrival j.

This opposite priority relation is sometimes needed in terms of always_higher_priority.

  Fact FIFO_policy_always_higher_priority :
    ∀ j j',
      ~~ hep_job j j' →
      always_higher_priority j' j.

End PriorityFacts.

We add the following lemmas to the basic facts database

Global Hint Resolve
  FIFO_policy_is_reflexive
  FIFO_policy_is_transitive
  FIFO_policy_is_total
  FIFO_policy_not_hep_job_arrival_order
  FIFO_policy_always_higher_priority
  : basic_rt_facts.

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

Section BasicLemmas.

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

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

Assume FIFO scheduling.

Consider any valid arrival sequence of such jobs ...

Variable arr_seq : arrival_sequence Job.
Hypothesis H_valid_arrivals : valid_arrival_sequence arr_seq.

... and the resulting uniprocessor schedule.

  Context {PState : ProcessorState Job}.
  Hypothesis H_uniproc : uniprocessor_model PState.
  Variable sched : schedule PState.

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

Context {RM : JobReady Job PState}.
Hypothesis H_basic_readiness : basic_readiness RM.

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

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.

  Context {JLFP : JLFP_policy Job}.
  Hypothesis H_policy_is_FIFO : policy_is_FIFO JLFP.
  Hypothesis H_respects_policy : respects_JLFP_policy_at_preemption_point arr_seq sched JLFP.

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 arr_seq sched j t.

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.

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 JLFP.

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 arr_seq sched tsk (constant 0).

As a corollary, FIFO implies the absence of service inversion.

    Corollary FIFO_implies_no_service_inversion :
      service_inversion_is_bounded_by arr_seq sched tsk (constant 0).

  End PriorityInversionBounded.

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.