Library prosa.results.fixed_priority.rta.bounded_pi

Abstract RTA for FP-schedulers with Bounded Priority Inversion

In this module we instantiate the Abstract Response-Time analysis (aRTA) to FP-schedulers for ideal uni-processor model of real-time tasks with arbitrary arrival models.
Given FP priority policy and an ideal uni-processor scheduler model, we can explicitly specify interference, interfering_workload, and interference_bound_function. In this settings, we can define natural notions of service, workload, busy interval, etc. The important feature of this instantiation is that we can induce the meaningful notion of priority inversion. However, we do not specify the exact cause of priority inversion (as there may be different reasons for this, like execution of a non-preemptive segment or blocking due to resource locking). We only assume that that a priority inversion is bounded.
Consider any type of tasks ...
  Context {Task : TaskType}.
  Context `{TaskCost Task}.
  Context `{TaskRunToCompletionThreshold 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}.
  Context `{JobPreemptable Job}.

Consider an FP policy that indicates a higher-or-equal priority relation, and assume that the relation is reflexive. Note that we do not relate the FP policy with the scheduler. However, we define functions for Interference and Interfering Workload that actively use the concept of priorities. We require the FP policy to be reflexive, so a job cannot cause lower-priority interference (i.e. priority inversion) to itself.
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.

Instantiation of Interference

We say that job j incurs interference at time t iff it cannot execute due to a higher-or-equal-priority job being scheduled, or if it incurs a priority inversion.

Instantiation of Interfering Workload

The interfering workload, in turn, is defined as the sum of the priority inversion function and interfering workload of jobs with higher or equal priority.
Note that we differentiate between abstract and classical notions of work conserving schedule.
We assume that the schedule is a work-conserving schedule in the classical sense, and later prove that the hypothesis about abstract work-conservation also holds.
Assume we have sequential tasks, i.e, jobs from the same task execute in the order of their arrival.
Assume that a job cost cannot be larger than a task cost.
Consider an arbitrary task set ts.
  Variable ts : list Task.

Next, we assume that all jobs come from the task set.
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.
For clarity, let's define some local names.
We introduce task_rbf as an abbreviation of the task request bound function, which is defined as task_cost(tsk) × max_arrivals(tsk,Δ).
Using the sum of individual request bound functions, we define the request bound function of all tasks with higher-or-equal priority (with respect to tsk).
Similarly, we define the request bound function of all tasks other than tsk with higher-or-equal priority (with respect to tsk).
Assume that there exists a constant priority_inversion_bound that bounds the length of any priority inversion experienced by any job of tsk. Since we analyze only task tsk, we ignore the lengths of priority inversions incurred by any other tasks.
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 = priority_inversion_bound + total_hep_rbf L.

To reduce the time complexity of the analysis, recall the notion of search space. Intuitively, this corresponds to all "interesting" arrival offsets that the job under analysis might have with regard to the beginning of its busy-window.
  Definition is_in_search_space A := (A < L) && (task_rbf A != task_rbf (A + ε)).

Let R be a value that upper-bounds the solution of each response-time recurrence, i.e., for any relative arrival time A in the search space, there exists a corresponding solution F such that R F + (task cost - task lock-in service).
  Variable R : duration.
  Hypothesis H_R_is_maximum :
     (A : duration),
      is_in_search_space A
       (F : duration),
        A + F priority_inversion_bound
                + (task_rbf (A + ε) - (task_cost tsk - task_rtct tsk))
                + total_ohep_rbf (A + F)
        R F + (task_cost tsk - task_rtct tsk).

Finally, we define the interference bound function (task_IBF). task_IBF bounds the interference if tasks are sequential. Since tasks are sequential, we exclude interference from other jobs of the same task. For FP, we define task_IBF as the sum of the priority interference bound and the higher-or-equal-priority workload.

Filling Out Hypotheses Of Abstract RTA Theorem

In this section we prove that all preconditions necessary to use the abstract theorem are satisfied.
Recall that L is assumed to be a fixed point of the busy interval recurrence. Thanks to this fact, we can prove that every busy interval (according to the concrete definition) is bounded. In addition, we know that the conventional concept of busy interval and the one obtained from the abstract definition (with the interference and interfering workload) coincide. Thus, it follows that any busy interval (in the abstract sense) is bounded.
    Lemma instantiated_busy_intervals_are_bounded :
      busy_intervals_are_bounded_by arr_seq sched tsk L.
      movej ARR TSK POS.
      edestruct (exists_busy_interval) with (delta := L) (priority_inversion_bound := (fun (d : duration) ⇒ priority_inversion_bound))
        as [t1 [t2 [T1 [T2 BI]]]] ⇒ //; last first.
      { t1, t2; split⇒ [//|]; split⇒ [//|].
        by eapply instantiated_busy_interval_equivalent_busy_interval. }
      { intros; rewrite {2}H_fixed_point leq_add //.
        rewrite /workload_of_higher_or_equal_priority_jobs /total_hep_rbf
          /workload_of_jobs /hep_job /FP_to_JLFP.
        move: (TSK) ⇒ /eqP →.
        exact: sum_of_jobs_le_sum_rbf. }

Next, we prove that task_IBF is indeed an interference bound.
Recall that in module abstract_seq_RTA hypothesis task_interference_is_bounded_by expects to receive a function that maps some task tsk, the relative arrival time of a job j of task tsk, and the length of the interval to the maximum amount of interference.
However, in this module we analyze only one task -- tsk, therefore it is “hard-coded” inside the interference bound function task_IBF. Moreover, in case of a model with fixed priorities, interference that some job j incurs from higher-or-equal priority jobs does not depend on the relative arrival time of job j. Therefore, in order for the task_IBF signature to match the required signature in module abstract_seq_RTA, we wrap the task_IBF function in a function that accepts, but simply ignores, the task and the relative arrival time.
    Lemma instantiated_task_interference_is_bounded :
        arr_seq sched tsk (fun A Rtask_IBF R).
      movet1 t2 Δ j ARR TSK BUSY LT NCOMPL A OFF.
      move: (posnP (@job_cost _ Cost j)) ⇒ [ZERO|POS].
      { by exfalso; rewrite /completed_by ZERO in NCOMPL. }
      rewrite -/(cumul_task_interference _ _ _ _ _).
      rewrite (leqRW (cumulative_task_interference_split _ _ _ _ _ _ _ _ _ _ _ _ _)) //=.
      rewrite /task_IBF leq_add//.
      { apply leq_trans with (cumulative_priority_inversion arr_seq sched j t1 (t1 + Δ)); first by done.
        apply leq_trans with (cumulative_priority_inversion arr_seq sched j t1 t2);
          last by apply: H_priority_inversion_is_bounded ⇒ //; eauto 6 with basic_rt_facts.
        by rewrite [X in _ X](@big_cat_nat _ _ _ (t1 + Δ)) //= leq_addr. }
      { erewrite cumulative_i_thep_eq_service_of_othep ⇒ //;
          last by eauto 6 with basic_rt_facts.
        apply: leq_trans.
        { apply service_of_jobs_le_workload; first apply ideal_proc_model_provides_unit_service.
          by apply (valid_schedule_implies_completed_jobs_dont_execute sched arr_seq). }
        { rewrite /workload_of_jobs /total_ohep_rbf /total_ohep_request_bound_function_FP.
          rewrite /another_task_hep_job /hep_job /FP_to_JLFP.
          set (pred_task tsk_other := hep_task tsk_other tsk && (tsk_other != tsk)).
          rewrite (eq_big (fun jpred_task (job_task j)) job_cost) //;
            last by movej'; rewrite /pred_task; move: TSK ⇒ /eqP →.
          erewrite (eq_big pred_task); [|by done|by movetsk'; eauto].
          by apply: sum_of_jobs_le_sum_rbf; eauto. } }

Finally, we show that there exists a solution for the response-time recurrence.
To rule out pathological cases with the concrete search space, we assume that the task cost is positive and the arrival curve is non-pathological.
      Hypothesis H_task_cost_pos : 0 < task_cost tsk.
      Hypothesis H_arrival_curve_pos : 0 < max_arrivals tsk ε.

Given any job j of task tsk that arrives exactly A units after the beginning of the busy interval, the bound of the total interference incurred by j within an interval of length Δ is equal to task_rbf (A + ε) - task_cost tsk + task_IBF Δ.
      Let total_interference_bound A Δ :=
        task_rbf (A + ε) - task_cost tsk + task_IBF Δ.

Next, consider any A from the search space (in the abstract sense).
      Variable A : duration.
      Hypothesis H_A_is_in_abstract_search_space :
        search_space.is_in_search_space L total_interference_bound A.

We prove that A is also in the concrete search space.
      Lemma A_is_in_concrete_search_space :
        is_in_search_space A.
        move: H_A_is_in_abstract_search_space ⇒ [INSP | [/andP [POSA LTL] [x [LTx INSP2]]]].
        - rewrite INSP.
          apply/andP; split; first by done.
          rewrite neq_ltn; apply/orP; left.
          rewrite {1}/task_rbf; erewrite task_rbf_0_zero; eauto 2; try done.
          rewrite add0n /task_rbf; apply leq_trans with (task_cost tsk) ⇒ //.
          exact: task_rbf_1_ge_task_cost.
        - apply/andP; split; first by done.
          apply/negP; intros EQ; move: EQ ⇒ /eqP EQ.
          by apply INSP2; rewrite /total_interference_bound subn1 addn1 prednK //.

Then, there exists a solution for the response-time recurrence (in the abstract sense).
      Corollary correct_search_space :
         (F : duration),
          A + F task_rbf (A + ε) - (task_cost tsk - task_rtct tsk) + task_IBF (A + F)
          R F + (task_cost tsk - task_rtct tsk).
        move: (H_R_is_maximum A) ⇒ FIX.
        feed FIX; first by apply A_is_in_concrete_search_space.
        move: FIX ⇒ [F [FIX NEQ]].
         F; split; last by done.
        rewrite -{2}(leqRW FIX).
          by rewrite addnA [_ + priority_inversion_bound]addnC -!addnA.

    End SolutionOfResponseTimeRecurrenceExists.

  End FillingOutHypothesesOfAbstractRTATheorem.

Final Theorem

Based on the properties established above, we apply the abstract analysis framework to infer that R is a response-time bound for tsk.
  Theorem uniprocessor_response_time_bound_fp :
    response_time_bounded_by tsk R.
    intros js ARRs TSKs.
    move: (posnP (@job_cost _ Cost js)) ⇒ [ZERO|POS].
    { by rewrite /job_response_time_bound /completed_by ZERO. }
    eapply uniprocessor_response_time_bound_seq ⇒ //.
    - exact: instantiated_i_and_w_are_coherent_with_schedule.
    - exact: instantiated_interference_and_workload_consistent_with_sequential_tasks.
    - exact: instantiated_busy_intervals_are_bounded.
    - exact: instantiated_task_interference_is_bounded.
    - exact: correct_search_space.

End AbstractRTAforFPwithArrivalCurves.