Library prosa.analysis.abstract.run_to_completion

Require Export prosa.analysis.facts.model.service_of_jobs.
Require Export prosa.analysis.facts.preemption.rtc_threshold.job_preemptable.
Require Export prosa.analysis.abstract.definitions.

Run-to-Completion Threshold of a job

In this module, we provide a sufficient condition under which a job receives enough service to become non-preemptive. Previously we defined the notion of run-to-completion threshold (see file abstract.run_to_completion_threshold.v). Run-to-completion threshold is the amount of service after which a job cannot be preempted until its completion. In this section we prove that if cumulative interference inside a busy interval is bounded by a certain constant then a job executes long enough to reach its run-to-completion threshold and become non-preemptive.

Section AbstractRTARunToCompletionThreshold.

Consider any type of tasks ...

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

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

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

In addition, we assume existence of a function mapping jobs to their preemption points.

Context `{JobPreemptable Job}.

Consider any arrival sequence with consistent arrivals...

Variable arr_seq : arrival_sequence Job.
Hypothesis H_arrival_times_are_consistent : consistent_arrival_times arr_seq.

Next, consider any ideal uni-processor schedule of this arrival sequence.

Variable sched : schedule (ideal.processor_state Job).

Assume that the job costs are no larger than the task costs.

Hypothesis H_jobs_respect_taskset_costs : arrivals_have_valid_job_costs arr_seq.

Let tsk be any task that is to be analyzed.

Variable tsk : Task.

Assume we are provided with abstract functions for interference and interfering workload.

Variable interference : Job → instant → bool.
Variable interfering_workload : Job → instant → duration.

For simplicity, let's define some local names.

  Let work_conserving := work_conserving arr_seq sched tsk.
  Let cumul_interference := cumul_interference interference.
  Let cumul_interfering_workload := cumul_interfering_workload interfering_workload.
  Let busy_interval := busy_interval sched interference interfering_workload.

We assume that the schedule is work-conserving.

Hypothesis H_work_conserving: work_conserving interference interfering_workload.

Let j be any job of task tsk with positive cost.

  Variable j : Job.
  Hypothesis H_j_arrives : arrives_in arr_seq j.
  Hypothesis H_job_of_tsk : job_task j = tsk.
  Hypothesis H_job_cost_positive : job_cost_positive j.

Next, consider any busy interval t1, t2) of job [j].

Variable t1 t2 : instant.
Hypothesis H_busy_interval : busy_interval j t1 t2.

First, we prove that job j completes by the end of the busy interval. Note that the busy interval contains the execution of job j, in addition time instant t2 is a quiet time. Thus by the definition of a quiet time the job should be completed before time t2.

Lemma job_completes_within_busy_interval:
completed_by sched j t2.

In this section we show that the cumulative interference is a complement to the total time where job j is scheduled inside the busy interval.

Section InterferenceIsComplement.

Consider any sub-interval t, t + delta) inside the busy interval [t1, t2).

    Variables (t : instant) (delta : duration).
    Hypothesis H_greater_than_or_equal : t1 ≤ t.
    Hypothesis H_less_or_equal: t + delta ≤ t2.

We prove that sum of cumulative service and cumulative interference in the interval t, t + delta) is equal to delta.

    Lemma interference_is_complement_to_schedule:
      service_during sched j t (t + delta) + cumul_interference j t (t + delta) = delta.

  End InterferenceIsComplement.

In this section, we prove a sufficient condition under which job j receives enough service.

Section InterferenceBoundedImpliesEnoughService.

Let progress_of_job be the desired service of job j.

Variable progress_of_job : duration.
Hypothesis H_progress_le_job_cost : progress_of_job ≤ job_cost j.

Assume that for some delta, the sum of desired progress and cumulative interference is bounded by delta (i.e., the supply).

    Variable delta : duration.
    Hypothesis H_total_workload_is_bounded:
      progress_of_job + cumul_interference j t1 (t1 + delta) ≤ delta.

Then, it must be the case that the job has received no less service than progress_of_job.

    Theorem j_receives_at_least_run_to_completion_threshold:
      service sched j (t1 + delta) ≥ progress_of_job.

  End InterferenceBoundedImpliesEnoughService.

In this section we prove a simple lemma about completion of a job after is reaches run-to-completion threshold.

Section CompletionOfJobAfterRunToCompletionThreshold.

Assume that completed jobs do not execute ...

Hypothesis H_completed_jobs_dont_execute:
completed_jobs_dont_execute sched.

.. and the preemption model is valid.

Hypothesis H_valid_preemption_model:
valid_preemption_model arr_seq sched.

Then, job j must complete in job_cost j - job_run_to_completion_threshold j time units after it reaches run-to-completion threshold.

    Lemma job_completes_after_reaching_run_to_completion_threshold:
      ∀ t,
        job_run_to_completion_threshold j ≤ service sched j t →
        completed_by sched j (t + (job_cost j - job_run_to_completion_threshold j )).

  End CompletionOfJobAfterRunToCompletionThreshold.

End AbstractRTARunToCompletionThreshold.