Library prosa.analysis.abstract.abstract_rta

Abstract Response-Time Analysis

In this module, we propose the general framework for response-time analysis (RTA) of uni-processor scheduling of real-time tasks with arbitrary arrival models. We prove that the maximum (with respect to the set of offsets) among the solutions of the response-time bound recurrence is a response time bound for tsk. Note that in this section we do not rely on any hypotheses about job sequentiality.
Section Abstract_RTA.

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 {JA : JobArrival Job}.
  Context {JC : JobCost Job}.
  Context `{JobPreemptable Job}.

Consider any kind of uni-service ideal processor state model.
Consider any arrival sequence with consistent, non-duplicate arrivals...
Next, consider any schedule of this arrival sequence...
... where jobs do not execute before their arrival nor after completion.
Assume that the job costs are no larger than the task costs.
Consider a task set ts...
  Variable ts : list Task.

... and a task tsk of 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.
Assume we are provided with abstract functions for interference and interfering workload.
We assume that the scheduler is work-conserving.
For simplicity, let's define some local names.
Let L be a constant which bounds any busy interval of task tsk.
Next, assume that interference_bound_function is a bound on the interference incurred by jobs of task tsk.
For simplicity, let's define a local name for the search space.
Consider any value R 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 tsk - task_rtct tsk).
  Variable R: nat.
  Hypothesis H_R_is_maximum:
     A,
      is_in_search_space A
       F,
        A + F task_rtct tsk
                + interference_bound_function tsk A (A + F)
        R F + (task_cost tsk - task_rtct tsk).

In this section we show a detailed proof of the main theorem that establishes that R is a response-time bound of task tsk.
  Section ProofOfTheorem.

Consider any job j of tsk with positive cost.
    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.

Assume we have a busy interval [t1, t2) of job j that is bounded by L.
    Variable t1 t2: instant.
    Hypothesis H_busy_interval: busy_interval 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.

In order to prove that R is a response-time bound of job j, we use hypothesis H_R_is_maximum. Note that the relative arrival time (A) is not necessarily from the search space. However, earlier we have proven that for any A there exists another A_sp from the search space that shares the same IBF value. Moreover, we've also shown that there exists an F_sp such that F_sp is a solution of the response time recurrence for parameter A_sp. Thus, despite the fact that the relative arrival time may not lie in the search space, we can still use the assumption H_R_is_maximum.
More formally, consider any A_sp and F_sp such that:..
    Variable A_sp F_sp : duration.

(a) A_sp is less than or equal to A...
    Hypothesis H_A_gt_Asp : A_sp A.

(b) interference_bound_function(A, x) is equal to interference_bound_function(A_sp, x) for all x less than L...
    Hypothesis H_equivalent :
      are_equivalent_at_values_less_than
        (interference_bound_function tsk A)
        (interference_bound_function tsk A_sp) L.

(c) A_sp is in the search space, ...
(d) A_sp + F_sp is a solution of the response time recurrence...
(e) and finally, F_sp + (task_last - ε) is no greater than R.
    Hypothesis H_R_gt_Fsp : R F_sp + (task_cost tsk - task_rtct tsk).

In this section, we consider the case where the solution is so large that the value of t1 + A_sp + F_sp goes beyond the busy interval. Although this case may be impossible in some scenarios, it can be easily proven, since any job that completes by the end of the busy interval remains completed.
    Section FixpointOutsideBusyInterval.

By assumption, suppose that t2 is less than or equal to t1 + A_sp + F_sp.
      Hypothesis H_big_fixpoint_solution : t2 t1 + (A_sp + F_sp).

Then we prove that job_arrival j + R is no less than t2.
      Lemma t2_le_arrival_plus_R:
        t2 job_arrival j + R.
      Proof.
        move: H_busy_interval ⇒ [[/andP [GT LT] [QT1 NTQ]] QT2].
        apply leq_trans with (t1 + (A_sp + F_sp)); first by done.
        apply leq_trans with (t1 + A + F_sp).
        { by rewrite !addnA leq_add2r leq_add2l. }
        rewrite /A subnKC; last by done.
        rewrite leq_add2l.
          by apply leq_trans with (F_sp + (task_cost tsk - task_rtct tsk));
            first rewrite leq_addr.
      Qed.

But since we know that the job is completed by the end of its busy interval, we can show that job j is completed by job arrival j + R.
      Lemma job_completed_by_arrival_plus_R_1:
        completed_by sched j (job_arrival j + R).
      Proof.
        move: H_busy_interval ⇒ [[/andP [GT LT] [QT1 NTQ]] QT2].
        apply completion_monotonic with t2; try done.
        apply t2_le_arrival_plus_R.
        eapply job_completes_within_busy_interval; eauto 2.
      Qed.

    End FixpointOutsideBusyInterval.

In this section, we consider the complementary case where t1 + A_sp + F_sp lies inside the busy interval.
    Section FixpointInsideBusyInterval.

So, assume that t1 + A_sp + F_sp is less than t2.
      Hypothesis H_small_fixpoint_solution : t1 + (A_sp + F_sp) < t2.

Next, let's consider two other cases: CASE 1: the value of the fix-point is no less than the relative arrival time of job j.
      Section FixpointIsNoLessThanArrival.

Suppose that A_sp + F_sp is no less than relative arrival of job j.
In this section, we prove that the fact that job j is not completed by time job_arrival j + R leads to a contradiction. Which in turn implies that the opposite is true -- job j completes by time job_arrival j + R.
        Section ProofByContradiction.

Recall that by lemma "solution_for_A_exists" there is a solution F of the response-time recurrence for the given relative arrival time A (which is not necessarily from the search space).
Thus, consider a constant F such that:..
          Variable F : duration.
(a) the sum of A_sp and F_sp is equal to the sum of A and F...
          Hypothesis H_Asp_Fsp_eq_A_F : A_sp + F_sp = A + F.
(b) F is at most F_sp...
          Hypothesis H_F_le_Fsp : F F_sp.
(c) and A + F is a solution for the response-time recurrence for A.
          Hypothesis H_A_F_fixpoint:
            A + F task_rtct tsk + interference_bound_function tsk A (A + F).

Next, we assume that job j is not completed by time job_arrival j + R.
          Hypothesis H_j_not_completed : ~~ completed_by sched j (job_arrival j + R).

Some additional reasoning is required since the term task_cost tsk - task_rtct tsk does not necessarily bound the term job_cost j - job_rtct j. That is, a job can have a small run-to-completion threshold, thereby becoming non-preemptive much earlier than guaranteed according to task run-to-completion threshold, while simultaneously executing the last non-preemptive segment that is longer than task_cost tsk - task_rtct tsk (e.g., this is possible in the case of floating non-preemptive sections).
In this case we cannot directly apply lemma "j_receives_at_least_run_to_completion_threshold". Therefore we introduce two temporal notions of the last non-preemptive region of job j and an execution optimism. We use these notions inside this proof, so we define them only locally.
Let the last non-preemptive region of job j (last) be the difference between the cost of the job and the j's run-to-completion threshold (i.e. job_cost j - job_rtct j). We know that after j has reached its run-to-completion threshold, it will additionally be executed job_last j units of time.
          Let job_last := job_cost j - job_rtct j.

And let execution optimism (optimism) be the difference between the tsk's run-to-completion threshold and the j's run-to-completion threshold (i.e. task_rtct - job_rtct). Intuitively, optimism is how much earlier job j has received its run-to-completion threshold than it could at worst.
          Let optimism := task_rtct tsk - job_rtct j.

From lemma "j_receives_at_least_run_to_completion_threshold" with parameters progress_of_job := job_rtct j and delta := (A + F) - optimism) we know that service of j by time t1 + (A + F) - optimism is no less than job_rtct j. Hence, job j is completed by time t1 + (A + F) - optimism + last.
          Lemma j_is_completed_by_t1_A_F_optimist_last :
            completed_by sched j (t1 + (A + F - optimism) + job_last).
          Proof.
            move: H_busy_interval ⇒ [[/andP [GT LT] _] _].
            have ESERV :=
              @j_receives_at_least_run_to_completion_threshold
                _ _ _ PState _ _ arr_seq sched interference interfering_workload
                _ j _ _ t1 t2 _ (job_rtct j) _ ((A + F) - optimism).
            specialize (ESERV JA JC).
            feed_n 6 ESERV; eauto 2.
            specialize (ESERV JC _).
            feed_n 2 ESERV.
            { eapply job_run_to_completion_threshold_le_job_cost; eauto. }
            { rewrite -{2}(leqRW H_A_F_fixpoint).
              rewrite /definitions.cumul_interference.
              rewrite -[in X in _ X]addnBAC; last by rewrite leq_subr.
              rewrite {2}/optimism.
              rewrite subKn; last by apply H_valid_run_to_completion_threshold.
              rewrite leq_add2l.
              apply leq_trans with (cumul_interference j t1 (t1 + (A + F))).
              { rewrite /cumul_interference /definitions.cumul_interference
                   [in X in _ X](@big_cat_nat _ _ _ (t1 + (A + F - optimism))) //=.
                all: by lia. }
              { apply H_job_interference_is_bounded with t2; try done.
                - by rewrite -H_Asp_Fsp_eq_A_F.
                - apply/negP; intros CONTR.
                  move: H_j_not_completed ⇒ /negP C; apply: C.
                  apply completion_monotonic with (t1 + (A + F)); try done.
                  rewrite addnA subnKC // leq_add2l.
                  apply leq_trans with F_sp; first by done.
                  by lia. } }
            apply: job_completes_after_reaching_run_to_completion_threshold; eauto 2.
          Qed.

However, t1 + (A + F) - optimism + last job_arrival j + R! To prove this fact we need a few auxiliary inequalities that are needed because we use the truncated subtraction in our development. So, for example a + (b - c) = a + b - c only if b c.
          Section AuxiliaryInequalities.

Recall that we consider a busy interval of a job j, and j has arrived A time units after the beginning the busy interval. From basic properties of a busy interval it follows that job j incurs interference at any time instant t ∈ [t1, t1 + A). Therefore interference_bound_function(tsk, A, A + F) is at least A.
            Lemma relative_arrival_le_interference_bound:
              A interference_bound_function tsk A (A + F).
            Proof.
              move: H_j_not_completed; clear H_j_not_completed; move ⇒ /negP CONTRc.
              move: (H_busy_interval) ⇒ [[/andP [GT LT] _] _].
              apply leq_trans with (cumul_interference j t1 (t1 + (A+F))).
              { rewrite /cumul_interference.
                apply leq_trans with
                  (\sum_(t1 t < t1 + A) interference j t);last by
                  rewrite /definitions.cumul_interference [in X in _ X](@big_cat_nat _ _ _ (t1 + A)) //=; try by lia.
                { rewrite -{1}[A](sum_of_ones t1).
                  rewrite [in X in X _]big_nat_cond [in X in _ X]big_nat_cond.
                  rewrite leq_sum //.
                  movet /andP [/andP [NEQ1 NEQ2] _].
                  rewrite lt0b.
                  move: (H_work_conserving j t1 t2 t) ⇒ CONS.
                  feed_n 4 CONS; try done.
                  { apply/andP; split; first by done.
                    by apply leq_trans with (t1 + A); [done | lia]. }
                  move: CONS ⇒ [CONS1 _].
                  apply/negP; intros CONTR.
                  move: (CONS1 CONTR) ⇒ SCHED; clear CONS1 CONTR.
                  apply H_jobs_must_arrive_to_execute in SCHED.
                  move: NEQ2; rewrite ltnNge; move ⇒ /negP NEQ2; apply: NEQ2.
                    by rewrite subnKC. } }
              { apply H_job_interference_is_bounded with t2; try done.
                - by rewrite -H_Asp_Fsp_eq_A_F.
                - apply /negP; moveCONTR.
                  apply: CONTRc.
                  by apply completion_monotonic with (t1 + (A + F)); [lia | done]. }
            Qed.

As two trivial corollaries, we show that tsk's run-to-completion threshold is at most F_sp...
            Corollary tsk_run_to_completion_threshold_le_Fsp :
              task_rtct tsk F_sp.
            Proof.
              move: H_A_F_fixpointEQ.
              have L1 := relative_arrival_le_interference_bound.
              by lia.
            Qed.

... and optimism is at most F.
            Corollary optimism_le_F :
              optimism F.
            Proof.
              move: H_A_F_fixpointEQ.
              have L1 := relative_arrival_le_interference_bound.
              by lia.
            Qed.

          End AuxiliaryInequalities.

Next we show that t1 + (A + F) - optimism + last is at most job_arrival j + R, which is easy to see from the following sequence of inequalities:
t1 + (A + F) - optimism + last job_arrival j + (F - optimism) + job_last job_arrival j + (F_sp - optimism) + job_last job_arrival j + F_sp + (job_last - optimism) job_arrival j + F_sp + job_cost j - task_rtct tsk job_arrival j + F_sp + task_cost tsk - task_rtct tsk job_arrival j + R.
          Lemma t1_A_F_optimist_last_le_arrival_R :
            t1 + (A + F - optimism) + job_last job_arrival j + R.
          Proof.
            move: (H_busy_interval) ⇒ [[/andP [GT LT] _] _].
            have L1 := tsk_run_to_completion_threshold_le_Fsp.
            have L2 := optimism_le_F.
            apply leq_trans with (job_arrival j + (F - optimism) + job_last).
            { rewrite leq_add2r addnBA.
              - by rewrite /A !addnA subnKC // addnBA.
              - by apply leq_trans with F; last rewrite leq_addl.
            }
            { move: H_valid_run_to_completion_threshold ⇒ [PRT1 PRT2].
              rewrite -addnA leq_add2l.
              apply leq_trans with (F_sp - optimism + job_last ); first by rewrite leq_add2r leq_sub2r.
              apply leq_trans with (F_sp + (task_cost tsk - task_rtct tsk)); last by done.
              rewrite /optimism subnBA; last by apply PRT2.
              rewrite -addnBAC // /job_last.
              rewrite addnBA; last by eapply job_run_to_completion_threshold_le_job_cost; eauto 2.
              rewrite -addnBAC; last by lia.
              rewrite -addnBA // subnn addn0.
              rewrite addnBA; last by apply PRT1.
              rewrite addnBAC; last by done.
              rewrite leq_sub2r // leq_add2l.
              by move: H_job_of_tsk ⇒ /eqP <-; apply H_valid_job_cost.
            }
          Qed.

... which contradicts the initial assumption about j is not completed by time job_arrival j + R.
          Lemma j_is_completed_earlier_contradiction : False.
          Proof.
            move: H_j_not_completed ⇒ /negP C; apply: C.
            apply completion_monotonic with (t1 + ((A + F) - optimism) + job_last);
              auto using j_is_completed_by_t1_A_F_optimist_last, t1_A_F_optimist_last_le_arrival_R.
          Qed.

        End ProofByContradiction.

Putting everything together, we conclude that j is completed by job_arrival j + R.
        Lemma job_completed_by_arrival_plus_R_2:
          completed_by sched j (job_arrival j + R).
        Proof.
          move: H_busy_interval ⇒ [[/andP [GT LT] _] _].
          have L1 := solution_for_A_exists
                       tsk L (fun tsk A Rtask_rtct tsk
                                          + interference_bound_function tsk A R) A_sp F_sp.
          specialize (L1 _).
          feed_n 2 L1; try done.
          { move: (H_busy_interval_exists j H_j_arrives H_job_of_tsk H_job_cost_positive)
                ⇒ [t1' [t2' [BOUND BUSY]]].
            have EQ:= busy_interval_is_unique _ _ _ _ _ _ _ _ H_busy_interval BUSY.
            move : EQ ⇒ [EQ1 EQ2].
            subst t1' t2'; clear BUSY.
            by rewrite -(ltn_add2l t1); apply leq_trans with t2.
          }
          specialize (L1 A); feed_n 2 L1; first by apply/andP; split.
          + by intros x LTG; apply/eqP; rewrite eqn_add2l H_equivalent.
            move: L1 ⇒ [F [EQSUM [F2LEF1 FIX2]]].
            apply/negP; intros CONTRc; move: CONTRc ⇒ /negP CONTRc.
            by eapply j_is_completed_earlier_contradiction in CONTRc; eauto 2.
        Qed.

      End FixpointIsNoLessThanArrival.

CASE 2: the value of the fix-point is less than the relative arrival time of job j (which turns out to be impossible, i.e. the solution of the response-time recurrence is always equal to or greater than the relative arrival time).
      Section FixpointCannotBeSmallerThanArrival.

Assume that A_sp + F_sp is less than A.
Note that the relative arrival time of job j is less than L.
        Lemma relative_arrival_is_bounded: A < L.
        Proof.
          rewrite /A.
          move: (H_busy_interval_exists j H_j_arrives H_job_of_tsk H_job_cost_positive) ⇒ [t1' [t2' [BOUND BUSY]]].
          have EQ:= busy_interval_is_unique _ _ _ _ _ _ _ _ H_busy_interval BUSY. destruct EQ as [EQ1 EQ2].
          subst t1' t2'; clear BUSY.
          apply leq_trans with (t2 - t1); last by rewrite leq_subLR.
          move: (H_busy_interval)=> [[/andP [T1 T3] [_ _]] _].
          by apply ltn_sub2r; first apply leq_ltn_trans with (job_arrival j).
        Qed.

We can use j_receives_at_least_run_to_completion_threshold to prove that the service received by j by time t1 + (A_sp + F_sp) is no less than run-to-completion threshold.
        Lemma service_of_job_ge_run_to_completion_threshold:
          service sched j (t1 + (A_sp + F_sp)) job_rtct j.
        Proof.
          move: (H_busy_interval) ⇒ [[NEQ [QT1 NTQ]] QT2].
          move: (NEQ) ⇒ /andP [GT LT].
          move: (H_job_interference_is_bounded t1 t2 (A_sp + F_sp) j) ⇒ IB.
          feed_n 5 IB; try done.
          { apply/negPCOMPL.
            apply completion_monotonic with (t' := t1 + A) in COMPL; try done; last first.
            { by rewrite leq_add2l; apply ltnW. }
            { rewrite /A subnKC in COMPL; last by done.
              move: COMPL; rewrite /completed_by leqNgt; move ⇒ /negP COMPL; apply: COMPL.
              rewrite /service -(service_during_cat _ _ _ (job_arrival j)); last by apply/andP; split.
              rewrite (cumulative_service_before_job_arrival_zero) // add0n.
                by rewrite /service_during big_geq //.
            }
          }
          rewrite -/A in IB.
          have ALTT := relative_arrival_is_bounded.
          simpl in IB; rewrite H_equivalent in IB; last by apply ltn_trans with A.
          have ESERV :=
              @j_receives_at_least_run_to_completion_threshold
                _ _ _ PState _ _ arr_seq sched interference interfering_workload
                _ j _ _ t1 t2 _ (job_rtct j) _ (A_sp + F_sp).
          specialize (ESERV JA JC).
          feed_n 6 ESERV; eauto 2.
          specialize (ESERV JC _).
          feed_n 2 ESERV; eauto using job_run_to_completion_threshold_le_job_cost.
          by rewrite -{2}(leqRW H_Asp_Fsp_fixpoint) leq_add //; apply H_valid_run_to_completion_threshold.
          Qed.

However, this is a contradiction. Since job j has not yet arrived, its service is equal to 0. However, run-to-completion threshold is always positive.
        Lemma relative_arrival_time_is_no_less_than_fixpoint:
          False.
        Proof.
          move: (H_busy_interval) ⇒ [[NEQ [QT1 NTQ]] QT2].
          move: (NEQ) ⇒ /andP [GT LT].
          have ESERV := service_of_job_ge_run_to_completion_threshold.
          move: ESERV; rewrite leqNgt; move ⇒ /negP ESERV; apply: ESERV.
          rewrite /service cumulative_service_before_job_arrival_zero;
            eauto 5 using job_run_to_completion_threshold_positive.
          rewrite -[X in _ X](@subnKC t1) //.
            by rewrite -/A leq_add2l ltnW.
        Qed.

      End FixpointCannotBeSmallerThanArrival.

    End FixpointInsideBusyInterval.

  End ProofOfTheorem.

Using the lemmas above, we prove that R is a response-time bound.
  Theorem uniprocessor_response_time_bound:
    response_time_bounded_by tsk R.
  Proof.
    intros j ARR JOBtsk. unfold job_response_time_bound.
    move: (posnP (@job_cost _ JC j)) ⇒ [ZERO|POS].
    { by rewrite /completed_by ZERO. }
    move: (H_busy_interval_exists j ARR JOBtsk POS) ⇒ [t1 [t2 [T2 BUSY]]].
    move: (BUSY) ⇒ [[/andP [GE LT] _] QTt2].
    have A2LTL := relative_arrival_is_bounded _ ARR JOBtsk POS _ _ BUSY.
    set (A2 := job_arrival j - t1) in ×.
    move: (representative_exists tsk _ interference_bound_function _ A2LTL) ⇒ [A1 [ALEA2 [EQΦ INSP]]].
    move: (H_R_is_maximum _ INSP) ⇒ [F1 [FIX1 LE1]].
    destruct (t1 + (A1 + F1) t2) eqn:BIG.
    - eapply job_completed_by_arrival_plus_R_1; eauto 2.
    - apply negbT in BIG; rewrite -ltnNge in BIG.
      destruct (A2 A1 + F1) eqn:BOUND.
      + eapply job_completed_by_arrival_plus_R_2; eauto 2.
      + apply negbT in BOUND; rewrite -ltnNge in BOUND.
        exfalso; apply relative_arrival_time_is_no_less_than_fixpoint
                   with (j := j) (t1 := t1) (t2 := t2) (A_sp := A1) (F_sp := F1); auto.
  Qed.

End Abstract_RTA.