Library rt.restructuring.analysis.edf.rta.nonpr_reg.concrete_models.limited

Next, we assume we have the model with fixed preemption points. I.e., each task is divided into a number of nonpreemptive segments by inserting staticaly predefined preemption points.

  Context `{JobPreemptionPoints Job}.
  Context `{TaskPreemptionPoints Task}.
  Hypothesis H_valid_model_with_fixed_preemption_points:
    valid_fixed_preemption_points_model arr_seq ts.

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.

  Context `{MaxArrivals Task}.
  Hypothesis H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals.
  Hypothesis H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts.

Let tsk be any task in ts that is to be analyzed.

Variable tsk : Task.
Hypothesis H_tsk_in_ts : tsk \in ts.

Next, consider any ideal uniprocessor schedule with limited preemptionsof this arrival sequence ...

  Variable sched : schedule (ideal.processor_state Job).
  Hypothesis H_jobs_come_from_arrival_sequence:
    jobs_come_from_arrival_sequence sched arr_seq.
  Hypothesis H_schedule_with_limited_preemptions:
    valid_schedule_with_limited_preemptions arr_seq sched.

... where jobs do not execute before their arrival or after completion.

Hypothesis H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched.
Hypothesis H_completed_jobs_dont_execute : completed_jobs_dont_execute sched.

Assume we have sequential tasks, i.e, jobs from the same task execute in the order of their arrival.

Hypothesis H_sequential_tasks : sequential_tasks sched.

Next, we assume that the schedule is a work-conserving schedule...

Hypothesis H_work_conserving : work_conserving arr_seq sched.

... and the schedule respects the policy defined by the job_preemptable function (i.e., jobs have bounded nonpreemptive segments).

Hypothesis H_respects_policy : respects_policy_at_preemption_point arr_seq sched.

Let's define some local names for clarity.

  Let response_time_bounded_by := task_response_time_bound arr_seq sched.
  Let task_rbf_changes_at A := task_rbf_changes_at tsk A.
  Let bound_on_total_hep_workload_changes_at :=
    bound_on_total_hep_workload_changes_at ts tsk.

We introduce the abbreviation "rbf" for the task request bound function, which is defined as task_cost(T) × max_arrivals(T,Δ) for a task T.

Let rbf := task_request_bound_function.

Next, we introduce task_rbf as an abbreviation for 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).

Let total_rbf := total_request_bound_function ts.

We define a bound for the priority inversion caused by jobs with lower priority.

  Let blocking_bound :=
    \max_(tsk_other <- ts | (tsk_other != tsk ) && (D tsk_other > D tsk ))
     (task_max_nonpreemptive_segment tsk_other - ε ).

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 :=
    \sum_(tsk_o <- ts | tsk_o != tsk )
     rbf tsk_o (minn ((A + ε ) + D tsk - D tsk_o) Δ).

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.

To reduce the time complexity of the analysis, recall the notion of search space.

Let is_in_search_space A :=
(A < L ) && (task_rbf_changes_at A || bound_on_total_hep_workload_changes_at A ).

Consider any value R, and assume that for any given arrival offset A in the search space, there is a solution of the response-time bound recurrence which 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
                + (task_rbf (A + ε) - (task_last_nonpr_segment tsk - ε ))
                + bound_on_total_hep_workload A (A + F) ∧
        F + (task_last_nonpr_segment tsk - ε ) ≤ R.

Now, we can leverage the results for the abstract model with bounded nonpreemptive segments to establish a response-time bound for the more concrete model of fixed preemption points.

  Theorem uniprocessor_response_time_bound_edf_with_fixed_preemption_points:
    response_time_bounded_by tsk R.
  Proof.
    move: (H_valid_model_with_fixed_preemption_points) ⇒ [MLP [BEG [END [INCR [HYP1 [HYP2 HYP3]]]]]].
    move: (MLP) ⇒ [BEGj [ENDj _]].
    case: (posnP (task_cost tsk)) ⇒ [ZERO|POSt].
    { intros j ARR TSK.
      move: (H_job_cost_le_task_cost _ ARR) ⇒ POSt.
      move: POSt; rewrite /job_cost_le_task_cost TSK ZERO leqn0; move ⇒ /eqP Z.
        by rewrite /job_response_time_bound /completed_by Z.
    }
    eapply uniprocessor_response_time_bound_edf_with_bounded_nonpreemptive_segments with (L0 := L).
    all: eauto 2 with basic_facts.
    { rewrite subKn; first by done.
      rewrite /task_last_nonpr_segment -(leq_add2r 1) subn1 !addn1 prednK; last first.
      { rewrite /last0 -nth_last.
        apply HYP3; try by done.
        rewrite -(ltn_add2r 1) !addn1 prednK //.
        move: (number_of_preemption_points_in_task_at_least_two
                 _ _ H_valid_model_with_fixed_preemption_points _ H_tsk_in_ts POSt) ⇒ Fact2.
        move: (Fact2) ⇒ Fact3.
          by rewrite size_of_seq_of_distances // addn1 ltnS // in Fact2.
      }
      { apply leq_trans with (task_max_nonpreemptive_segment tsk).
        - by apply last_of_seq_le_max_of_seq.
        - rewrite -END; last by done.
          apply ltnW; rewrite ltnS; try done.
            by apply max_distance_in_seq_le_last_element_of_seq; eauto 2.
      }
    }
  Qed.

End RTAforFixedPreemptionPointsModelwithArrivalCurves.