Library prosa.results.fixed_priority.rta.bounded_nps

RTA for FP-schedulers with Bounded Non-Preemptive Segments

In this section we instantiate the Abstract RTA for FP-schedulers with Bounded Priority Inversion to FP-schedulers for ideal uni-processor model of real-time tasks with arbitrary arrival models and bounded non-preemptive segments.
Recall that Abstract RTA for FP-schedulers with Bounded Priority Inversion does not specify the cause of priority inversion. In this section, we prove that the priority inversion caused by execution of non-preemptive segments is bounded. Thus the Abstract RTA for FP-schedulers is applicable to this instantiation.
Consider any type of tasks ...
  Context {Task : TaskType}.
  Context `{TaskCost Task}.
  Context `{TaskRunToCompletionThreshold Task}.
  Context `{TaskMaxNonpreemptiveSegment Task}.

... and any type of jobs associated with these tasks.
  Context {Job : JobType}.
  Context `{JobTask Job Task}.
  Context `{Arrival : JobArrival Job}.
  Context `{Cost : JobCost Job}.

Consider an FP policy that indicates a higher-or-equal priority relation, and assume that the relation is reflexive and transitive.
Consider any arrival sequence with consistent, non-duplicate arrivals.
Next, consider any ideal uni-processor schedule of this arrival sequence, ...
... allow for any work-bearing notion of job readiness, ...
... and assume that the schedule is valid.
In addition, we assume the existence of a function mapping jobs to their preemption points ...
  Context `{JobPreemptable Job}.

... and assume that it defines a valid preemption model with bounded non-preemptive segments.
Next, we assume that the schedule is a work-conserving schedule...
... and the schedule respects the scheduling policy.
Assume we have sequential tasks, i.e, jobs from the same task execute in the order of their arrival.
Consider an arbitrary task set ts, ...
  Variable ts : list Task.

... assume that all jobs come from the task set, ...
... and the cost of a job cannot be larger than the task cost.
Let max_arrivals be a family of valid arrival curves, i.e., for any task tsk in ts max_arrival tsk is (1) an arrival bound of tsk, and (2) it is a monotonic function that equals 0 for the empty interval delta = 0.
Let tsk be any task in ts that is to be analyzed.
  Variable tsk : Task.
  Hypothesis H_tsk_in_ts : tsk \in ts.

Consider a valid preemption model...
...and a valid task run-to-completion threshold function. That is, task_rtct tsk is (1) no bigger than tsk's cost, (2) for any job of task tsk job_rtct is bounded by task_rtct.
Let's define some local names for clarity.

Priority inversion is bounded

In this section, we prove that a priority inversion for task tsk is bounded by the maximum length of non-preemptive segments among the tasks with lower priority.
First, we prove that the maximum length of a priority inversion of a job j is bounded by the maximum length of a non-preemptive section of a task with lower-priority task (i.e., the blocking term).
    Lemma priority_inversion_is_bounded_by_blocking:
       j t1 t2,
        arrives_in arr_seq j
        job_of_task tsk j
        busy_interval_prefix arr_seq sched j t1 t2
        max_lp_nonpreemptive_segment arr_seq j t1 blocking_bound ts tsk.
    Proof.
      movej t1 t2 ARR TSK BUSY; move: TSK ⇒ /eqP TSK.
      rewrite /blocking_bound /max_lp_nonpreemptive_segment.
      apply: leq_trans; first exact: max_np_job_segment_bounded_by_max_np_task_segment.
      apply: (@leq_trans (\max_(j_lp <- arrivals_between arr_seq 0 t1
                | (~~ hep_task (job_task j_lp) tsk) && (0 < job_cost j_lp))
                            (task_max_nonpreemptive_segment (job_task j_lp) - ε))).
      { rewrite /hep_job /FP_to_JLFP TSK.
        apply: leq_big_maxj' JINB NOTHEP.
        rewrite leq_sub2r //. }
      { apply /bigmax_leq_seqPj' JINB /andP[NOTHEP POS].
        apply leq_bigmax_cond_seq with
            (x := (job_task j')) (F := fun tsktask_max_nonpreemptive_segment tsk - 1);
          last by done.
        apply: H_all_jobs_from_taskset.
        by apply: in_arrivals_implies_arrived (JINB). }
    Qed.

Using the above lemma, we prove that the priority inversion of the task is bounded by the blocking_bound.
    Lemma priority_inversion_is_bounded:
      priority_inversion_is_bounded_by
        arr_seq sched tsk (constant (blocking_bound ts tsk)).
    Proof.
      have PIB: priority_inversion_is_bounded_by arr_seq sched tsk (fun blocking_bound ts tsk); last by done.
      apply: priority_inversion_is_bounded ⇒ //.
      by exact: priority_inversion_is_bounded_by_blocking.
    Qed.

  End PriorityInversionIsBounded.

Response-Time Bound

In this section, we prove that the maximum among the solutions of the response-time bound recurrence is a response-time bound for tsk.
  Section ResponseTimeBound.

Let L be any positive fixed point of the busy interval recurrence.
    Variable L : duration.
    Hypothesis H_L_positive : L > 0.
    Hypothesis H_fixed_point : L = blocking_bound ts tsk + total_hep_rbf L.

To reduce the time complexity of the analysis, recall the notion of search space.
Next, consider any value R, and assume that for any given arrival offset A from the search space there is a solution of the response-time bound recurrence that is bounded by R.
    Variable R : duration.
    Hypothesis H_R_is_maximum:
       (A : duration),
        is_in_search_space A
         (F : duration),
          A + F blocking_bound ts tsk
                  + (task_rbf (A + ε) - (task_cost tsk - task_rtct tsk))
                  + total_ohep_rbf (A + F)
          F + (task_cost tsk - task_rtct tsk) R.

Then, using the results for the general RTA for FP-schedulers, we establish a response-time bound for the more concrete model of bounded nonpreemptive segments. Note that in case of the general RTA for FP-schedulers, we just assume that the priority inversion is bounded. In this module we provide the preemption model with bounded nonpreemptive segments and prove that the priority inversion is bounded.