Library prosa.results.fixed_priority.rta.bounded_pi

Throughout this file, we assume ideal uni-processor schedules.
Require Import prosa.model.processor.ideal.

Throughout this file, we assume the basic (i.e., Liu & Layland) readiness model.
Require Import prosa.model.readiness.basic.

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 `{JobArrival Job}.
  Context `{JobCost Job}.
  Context `{JobPreemptable Job}.

Consider any arrival sequence with consistent, non-duplicate arrivals.
Next, consider any ideal uniprocessor schedule of this arrival sequence ...
... where jobs do not execute before their arrival or after completion.
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_run_to_completion_threshold tsk is (1) no bigger than tsk's cost, (2) for any job of task tsk job_run_to_completion_threshold is bounded by task_run_to_completion_threshold.
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.
  Context `{FP_policy Task}.
  Hypothesis H_priority_is_reflexive : reflexive_priorities.

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.
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 F + (task cost - task lock-in service) R.
  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_run_to_completion_threshold tsk))
                + total_ohep_rbf (A + F)
        F + (task_cost tsk - task_run_to_completion_threshold tsk) R.

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.
  Let interference (j : Job) (t : instant) :=
    ideal_jlfp_rta.interference sched j t.

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.
Finally, we define the interference bound function 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.
First, we prove that in the instantiation of interference and interfering workload, we really take into account everything that can interfere with tsk's jobs, and thus, the scheduler satisfies the abstract notion of work conserving schedule.
    Lemma instantiated_i_and_w_are_consistent_with_schedule:
      work_conserving_ab tsk interference interfering_workload.
      intros j t1 t2 t ARR TSK POS BUSY NEQ; split; intros HYP.
      - move: HYP ⇒ /negP.
        rewrite negb_or /is_priority_inversion /is_priority_inversion
        move ⇒ /andP [HYP1 HYP2].
        case SCHED: (sched t) ⇒ [s | ].
        + rewrite SCHED in HYP1, HYP2.
          move: HYP1 HYP2.
          rewrite !Bool.negb_involutive negb_and Bool.negb_involutive.
          moveHYP1 /orP [/negP HYP2| /eqP HYP2].
          × by exfalso.
          × by subst s; rewrite scheduled_at_def //; apply eqprop_to_eqbool.
        + exfalso; clear HYP1 HYP2.
          eapply instantiated_busy_interval_equivalent_edf_busy_interval in BUSY; eauto with basic_facts.
            by move: BUSY ⇒ [PREF _]; eapply not_quiet_implies_not_idle;
                              eauto 2 using eqprop_to_eqbool.
      - move: (HYP); rewrite scheduled_at_def; move ⇒ /eqP HYP2; apply/negP.
        rewrite /interference /ideal_jlfp_rta.interference /is_priority_inversion
                  /is_interference_from_another_hep_job HYP2 negb_or.
         apply/andP; split.
         + rewrite Bool.negb_involutive; eauto 2.
             by eapply H_priority_is_reflexive with (t := 0).
         + by rewrite negb_and Bool.negb_involutive; apply/orP; right.

Next, we prove that the interference and interfering workload functions are consistent with sequential tasks.
    Lemma instantiated_interference_and_workload_consistent_with_sequential_tasks:
        arr_seq sched tsk interference interfering_workload.
      intros j t1 t2 ARR TSK POS BUSY.
      eapply instantiated_busy_interval_equivalent_edf_busy_interval in BUSY; eauto with basic_facts.
      eapply all_jobs_have_completed_equiv_workload_eq_service; eauto 2; intros s ARRs TSKs.
      move: (BUSY) ⇒ [[_ [QT _]] _].
      apply QT.
      - by apply in_arrivals_implies_arrived in ARRs.
      - move: TSKs ⇒ /eqP TSKs.
        rewrite /hep_job /FP_to_JLFP TSK -TSKs; eauto 2.
          by eapply (H_priority_is_reflexive 0); eauto.
      - by eapply in_arrivals_implies_arrived_before; eauto 2.

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 interference interfering_workload L.
      intros j ARR TSK POS.
      edestruct (exists_busy_interval) with (delta := L) as [t1 [t2 [T1 [T2 GGG]]]]; eauto 2.
      { by intros; rewrite {2}H_fixed_point leq_add //; apply total_workload_le_total_rbf'. }
       t1, t2; split; first by done.
      - by done.
      - by eapply instantiated_busy_interval_equivalent_edf_busy_interval; eauto 2 with basic_facts.

Next, we prove that 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 t, the relative arrival time of a job j of task t, and the length of the interval to the maximum amount of interference (for more details see files limited.abstract_RTA.definitions and limited.abstract_RTA.abstract_seq_rta).
However, in this module we analyze only one task -- tsk, therefore it is “hard-coded” inside the interference bound function 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 IBF signature to match the required signature in module abstract_seq_RTA, we wrap the 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 interference interfering_workload (fun t A RIBF R).
      intros ? ? ? ? ARR TSK ? NCOMPL BUSY; simpl.
      move: (posnP (@job_cost _ H3 j)) ⇒ [ZERO|POS].
      { by exfalso; rewrite /completed_by ZERO in NCOMPL. }
      eapply instantiated_busy_interval_equivalent_edf_busy_interval in BUSY; eauto 2 with basic_facts.
      rewrite /interference; erewrite cumulative_task_interference_split; eauto 2 with basic_facts; last first.
      { move: BUSY ⇒ [[_ [_ [_ /andP [GE LT]]]] _].
          by eapply arrived_between_implies_in_arrivals; eauto 2. }
      unfold IBF, interference.
      rewrite leq_add; try done.
      { move: (H_priority_inversion_is_bounded j ARR TSK) ⇒ BOUND.
        apply leq_trans with (cumulative_priority_inversion sched j t1 (t1 + R0)); first by done.
        apply leq_trans with (cumulative_priority_inversion sched j t1 t2); last first.
        { by apply BOUND; move: BUSY ⇒ [PREF QT2]. }
        rewrite [X in _ X](@big_cat_nat _ _ _ (t1 + R0)) //=.
        - by rewrite leq_addr.
        - by rewrite leq_addr.
        - by rewrite ltnW.
      { erewrite instantiated_cumulative_interference_of_hep_tasks_equal_total_interference_of_hep_tasks;
          eauto 2; last by unfold quiet_time; move: BUSY ⇒ [[_ [T1 T2]] _].
        apply leq_trans with
               (fun jhp : Job ⇒ (FP_to_JLFP _ _) jhp j && (job_task jhp != job_task j))
               (arrivals_between arr_seq t1 (t1 + R0))
        { by apply service_of_jobs_le_workload; last apply ideal_proc_model_provides_unit_service. }
        { rewrite /workload_of_jobs /total_ohep_rbf /total_ohep_request_bound_function_FP.
            by rewrite -TSK; apply total_workload_le_total_rbf.

Finally, we show that there exists a solution for the response-time recurrence.
Consider any job j of tsk.
      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.

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 + IBF Δ.
      Let total_interference_bound tsk A Δ :=
        task_rbf (A + ε) - task_cost tsk + 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 tsk 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).
          + by apply leq_trans with (job_cost j); rewrite -?H_job_of_tsk; eauto 2.
          + eapply task_rbf_1_ge_task_cost; eauto 2.
        - apply/andP; split; first by done.
          apply/negP; intros EQ; move: EQ ⇒ /eqP EQ.
          apply INSP2.
          unfold total_interference_bound in ×.
          rewrite subn1 addn1 prednK; last by done.
            by rewrite -EQ.

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_run_to_completion_threshold tsk) + IBF (A + F)
          F + (task_cost tsk - task_run_to_completion_threshold tsk) R.
        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.
        apply/eqP; rewrite {1}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 _ H3 js)) ⇒ [ZERO|POS].
    { by rewrite /job_response_time_bound /completed_by ZERO. }
    eapply uniprocessor_response_time_bound_seq; eauto 3.
    - by apply instantiated_i_and_w_are_consistent_with_schedule.
    - by apply instantiated_interference_and_workload_consistent_with_sequential_tasks.
    - by apply instantiated_busy_intervals_are_bounded.
    - by apply instantiated_task_interference_is_bounded.
    - by eapply correct_search_space; eauto 2.

End AbstractRTAforFPwithArrivalCurves.