Library prosa.results.edf.rta.bounded_pi

Abstract RTA for EDF-schedulers with Bounded Priority Inversion

In this module we instantiate the Abstract Response-Time analysis (aRTA) to EDF-schedulers for ideal uni-processor model of real-time tasks with arbitrary arrival models.
Given EDF 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 `{TaskDeadline 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}.
  Context `{JobPreemptable Job}.

We assume the classic (i.e., Liu & Layland) model of readiness without jitter or self-suspensions, wherein pending jobs are always ready.
  #[local] Existing Instance basic_ready_instance.

For clarity, let's denote the relative deadline of a task as D.
  Let D tsk := task_deadline tsk.

Consider the EDF policy that indicates a higher-or-equal priority relation. Note that we do not relate the EDF policy with the scheduler. However, we define functions for Interference and Interfering Workload that actively use the concept of priorities.
  Let EDF := EDF Job.

Consider any arrival sequence with consistent, non-duplicate arrivals.
Next, consider any valid ideal uni-processor schedule of this arrival sequence that follows the scheduling policy.
To use the theorem uniprocessor_response_time_bound_seq from the Abstract RTA module, we need to specify functions of interference, interfering workload, and IBF_other. Next, we define interference and interfering workload; we return to IBF_other later.

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 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.
We introduce rbf as an abbreviation of the task request bound function, which is defined as task_cost(T) × max_arrivals(T,Δ) for some task T.
Next, we introduce task_rbf as an abbreviation of the task request bound function of task tsk.
  Let task_rbf := rbf tsk.

Using the sum of individual request bound functions, we define the request bound function of all tasks (total request bound function).
Assume that there exists a bound on the length of any priority inversion experienced by any job of task 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 = total_rbf L.

Next, we define an upper bound on interfering workload received from jobs of other tasks with higher-than-or-equal priority.
  Let bound_on_total_hep_workload (A Δ : duration) :=
    \sum_(tsk_o <- ts | tsk_o != tsk)
     rbf tsk_o (minn ((A + ε) + D tsk - D tsk_o) Δ).

To reduce the time complexity of the analysis, we introduce the notion of search space for EDF. Intuitively, this corresponds to all "interesting" arrival offsets that the job under analysis might have with regard to the beginning of its busy-window.
In the case of the search space for EDF, we consider three conditions. First, we ask whether task_rbf A task_rbf (A + ε).
  Definition task_rbf_changes_at (A : duration) := task_rbf A != task_rbf (A + ε).

Second, we ask whether there exists a task tsko from ts such that tsko tsk and rbf(tsko, A + D tsk - D tsko) rbf(tsko, A + ε + D tsk - D tsko). Note that we use a slightly uncommon notation has (λ tsko P tskₒ) ts, which can be interpreted as follows: the task set ts contains a task tsko such that a predicate P holds for tsko.
  Definition bound_on_total_hep_workload_changes_at A :=
    has (fun tsko
           (tsk != tsko)
             && (rbf tsko (A + D tsk - D tsko)
                     != rbf tsko ((A + ε) + D tsk - D tsko))) ts.

Third, we ask whether priority_inversion_bound (A - ε) priority_inversion_bound A.
The final search space for EDF is a set of offsets that are less than L and where priority_inversion_bound, task_rbf, or bound_on_total_hep_workload changes in value.
  Definition is_in_search_space (A : duration) :=
    (A < L) && (priority_inversion_changes_at A
                || task_rbf_changes_at A
                || bound_on_total_hep_workload_changes_at A).

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 A
                + (task_rbf (A + ε) - (task_cost tsk - task_rtct tsk))
                + bound_on_total_hep_workload A (A + F)
        R F + (task_cost tsk - task_rtct tsk).

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

Filling Out Hypothesis Of Abstract RTA Theorem

In this section we prove that all hypotheses necessary to use the abstract theorem are satisfied.
First, we prove that IBF_other is indeed an interference bound.
    Section TaskInterferenceIsBoundedByIBF_other.

      Section HepWorkloadBound.

Consider an arbitrary job j of tsk.
        Variable j : Job.
        Hypothesis H_j_arrives : arrives_in arr_seq j.
        Hypothesis H_job_of_tsk : job_of_task tsk j.
        Hypothesis H_job_cost_positive: job_cost_positive j.

Consider any busy interval [t1, t2) of job j.
        Variable t1 t2 : duration.
        Hypothesis H_busy_interval : definitions.busy_interval sched j t1 t2.

Let's define A as a relative arrival time of job j (with respect to time t1).
        Let A := job_arrival j - t1.

Consider an arbitrary shift Δ inside the busy interval ...
        Variable Δ : duration.
        Hypothesis H_Δ_in_busy : t1 + Δ < t2.

... and the set of all arrivals between t1 and t1 + Δ.
        Let jobs := arrivals_between arr_seq t1 (t1 + Δ).

We define a predicate EDF_from tsk. Predicate EDF_from tsk holds true for any job jo of task tsk such that job_deadline jo job_deadline j.
        Let EDF_from (tsk : Task) := fun (jo : Job) ⇒ EDF jo j && (job_task jo == tsk).

Now, consider the case where A + ε + D tsk - D tsk_o Δ.
        Section ShortenRange.

Consider an arbitrary task tsk_o tsk from ts.
          Variable tsk_o : Task.
          Hypothesis H_tsko_in_ts: tsk_o \in ts.
          Hypothesis H_neq: tsk_o != tsk.

And assume that A + ε + D tsk - D tsk_o Δ.
          Hypothesis H_Δ_ge : A + ε + D tsk - D tsk_o Δ.

Then we prove that the total workload of jobs with higher-or-equal priority from task tsk_o over time interval t1, t1 + Δ is bounded by workload over time interval t1, t1 + A + ε + D tsk - D tsk_o. The intuition behind this inequality is that jobs which arrive after time instant t1 + A + ε + D tsk - D tsk_o have lower priority than job j due to the term D tsk - D tsk_o.
          Lemma total_workload_shorten_range:
            workload_of_jobs (EDF_from tsk_o) (arrivals_between arr_seq t1 (t1 + Δ))
             workload_of_jobs (EDF_from tsk_o) (arrivals_between arr_seq t1 (t1 + (A + ε + D tsk - D tsk_o))).
            have BOUNDED: t1 + (A + ε + D tsk - D tsk_o) t1 + Δ by lia.
            rewrite (workload_of_jobs_nil_tail _ _ BOUNDED) // ⇒ j' IN'.
            rewrite /EDF_fromARR'.
            case: (eqVneq (job_task j') tsk_o) ⇒ TSK';
              last by rewrite andbF.
            rewrite andbT; apply: contraT ⇒ /negPn.
            rewrite /EDF/edf.EDF/job_deadline/job_deadline_from_task_deadline.
            move: H_job_of_tsk; rewrite TSK' /job_of_task ⇒ /eqPHEP.
            have LATEST: job_arrival j' t1 + A + D tsk - D tsk_o by rewrite /D/A; lia.
            have EARLIEST: t1 job_arrival j' by apply: job_arrival_between_ge.
            by case: (leqP (A + 1 + D tsk) (D tsk_o)); [rewrite /D/A|]; lia.

        End ShortenRange.

Using the above lemma, we prove that the total workload of the tasks is at most bound_on_total_hep_workload(A, Δ).
        Corollary sum_of_workloads_is_at_most_bound_on_total_hep_workload :
          \sum_(tsk_o <- ts | tsk_o != tsk) workload_of_jobs (EDF_from tsk_o) jobs
           bound_on_total_hep_workload A Δ.
          apply leq_sum_seqtsko INtsko NEQT.
          edestruct (leqP Δ (A + ε + D tsk - D tsko)) as [NEQ|NEQ]; [ | apply ltnW in NEQ].
          - exact: (workload_le_rbf' arr_seq tsko).
          - eapply leq_trans; first by eapply total_workload_shorten_range; eauto 2.
            exact: workload_le_rbf'.

      End HepWorkloadBound.

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.
However, in this module we analyze only one task -- tsk, therefore it is “hard-coded” inside the interference bound function IBF_other. Therefore, in order for the IBF_other signature to match the required signature in module abstract_seq_RTA, we wrap the IBF_other function in a function that accepts, but simply ignores the task.
      Corollary instantiated_task_interference_is_bounded :
          arr_seq sched tsk (fun tsk A RIBF_other A R).
        movet1 t2 R2 j ARR TSK BUSY LT NCOMPL A OFF.
        move: (OFF _ _ BUSY) ⇒ EQA; subst A.
        move: (posnP (@job_cost _ Cost j)) ⇒ [ZERO|POS].
        - exfalso; move: NCOMPL ⇒ /negP COMPL; apply: COMPL.
          by rewrite /completed_by /completed_by ZERO.
        - rewrite -/(cumul_task_interference _ _ _ _ _).
          rewrite (leqRW (cumulative_task_interference_split _ _ _ _ _ _ _ _ _ _ _ _ _ )) //=.
          rewrite /I leq_add //; first exact: cumulative_priority_inversion_is_bounded.
          eapply leq_trans; first exact: cumulative_interference_is_bounded_by_total_service.
          eapply leq_trans; first exact: service_of_jobs_le_workload.
          eapply leq_trans.
          + eapply reorder_summation.
            movej' IN _.
            apply H_all_jobs_from_taskset. eapply in_arrivals_implies_arrived. exact IN.
          + move : TSK ⇒ /eqP TSK.
            rewrite TSK.
            apply: sum_of_workloads_is_at_most_bound_on_total_hep_workload ⇒ //.
            by apply /eqP.

    End TaskInterferenceIsBoundedByIBF_other.

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_of_task tsk j.
      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_other(A, Δ).
      Let total_interference_bound tsk (A Δ : duration) :=
        task_rbf (A + ε) - task_cost tsk + IBF_other A Δ.

Next, consider any A from the search space (in 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 ⇒ [-> | [/andP [POSA LTL] [x [LTx INSP2]]]];
                                                  apply/andP; split ⇒ //.
        { apply/orP; left; apply/orP; right.
          rewrite /task_rbf_changes_at /task_rbf /rbf task_rbf_0_zero //; eauto 2.
          apply contraT ⇒ /negPn /eqP ZERO.
          rewrite -(ltnn 0) {2}ZERO add0n.
          apply: (@leq_trans (task_cost tsk)); last by exact: task_rbf_1_ge_task_cost.
          apply: (@leq_trans (job_cost j)) ⇒ //.
          move: (H_job_of_tsk) ⇒ /eqP <-.
          by apply: (H_valid_job_cost _ H_j_arrives). }
        { apply contraT; rewrite !negb_or ⇒ /andP [/andP [/negPn/eqP PI /negPn/eqP RBF] WL].
          exfalso; apply INSP2.
          rewrite /total_interference_bound subnK // RBF.
          apply /eqP; rewrite eqn_add2l /IBF_other PI eqn_add2l.
          rewrite /bound_on_total_hep_workload subnK //.
          apply /eqP; rewrite big_seq_cond [RHS]big_seq_cond.
          apply eq_big ⇒ // tsk_i /andP [TS OTHER].
          move: WL; rewrite /bound_on_total_hep_workload_changes_at ⇒ /hasPn WL.
          move: {WL} (WL tsk_i TS) ⇒ /nandP [/negPn/eqP EQ|/negPn/eqP WL];
                                       first by move: OTHER; rewrite EQ ⇒ /neqP.
          case: (ltngtP (A + ε + D tsk - D tsk_i) x) ⇒ [ltn_x|gtn_x|eq_x];
                                                        rewrite /minn.
          { by rewrite ifT //; lia. }
          { rewrite ifF //.
            by move: gtn_x; rewrite leq_eqVlt ⇒ /orP [/eqP EQ|LEQ]; lia. }
          { case: (A + D tsk - D tsk_i < x).
            - by rewrite WL.
            - by rewrite eq_x. } }

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

      End SolutionOfResponseTimeReccurenceExists.

  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_edf:
    task_response_time_bound arr_seq sched tsk R.
    movejs 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 with
      (task_IBF := fun tsk A RIBF_other A R) (L := L) ⇒ //.
    - exact: instantiated_i_and_w_are_coherent_with_schedule.
    - exact: EDF_implies_sequential_tasks.
    - 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 AbstractRTAforEDFwithArrivalCurves.