Library prosa.analysis.definitions.priority_inversion

Cumulative Priority Inversion for JLFP-models

In this module we define the notion of cumulative priority inversion for uni-processor for JLFP schedulers.
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}.

Consider any arrival sequence with consistent arrivals.
Next, consider any ideal uniprocessor schedule of this arrival sequence.
Assume a given JLFP policy.
  Context `{JLFP_policy Job}.

In this section, we define a notion of bounded priority inversion experienced by a job.
Consider an arbitrary task tsk.
    Variable tsk : Task.

Consider any job j of tsk.
    Variable j : Job.
    Hypothesis H_from_arrival_sequence : arrives_in arr_seq j.
    Hypothesis H_job_task : job_of_task tsk j.

We say that the job incurs priority inversion if it has higher priority than the scheduled job. Note that this definition implicitly assumes that the scheduler is work-conserving. Therefore, it cannot be applied to models with jitter or self-suspensions.
    Definition is_priority_inversion (t : instant) :=
      if sched t is Some jlp then
        ~~ hep_job jlp j
      else false.

Then we compute the cumulative priority inversion incurred by a job within some time interval t1, t2).
    Definition cumulative_priority_inversion (t1 t2 : instant) :=
      \sum_(t1 t < t2) is_priority_inversion t.

We say that priority inversion of job j is bounded by a constant B iff cumulative priority inversion within any busy interval prefix is bounded by B.
    Definition priority_inversion_of_job_is_bounded_by (B : duration) :=
       (t1 t2 : instant),
        busy_interval_prefix arr_seq sched j t1 t2
        cumulative_priority_inversion t1 t2 B.

  End JobPriorityInversionBound.

In this section, we define a notion of the bounded priority inversion for task.
Consider an arbitrary task tsk.
    Variable tsk : Task.

We say that task tsk has bounded priority inversion if all its jobs have bounded cumulative priority inversion.
    Definition priority_inversion_is_bounded_by (B : duration) :=
       (j : Job),
        arrives_in arr_seq j
        job_task j = tsk
        job_cost j > 0
        priority_inversion_of_job_is_bounded_by j B.

  End TaskPriorityInversionBound.

End CumulativePriorityInversion.