Library prosa.results.edf.rta.limited_preemptive

RTA for EDF with Fixed Preemption Points

In this module we prove the RTA theorem for EDF-schedulers with fixed preemption points.
Throughout this file, we assume the EDF priority policy, ideal uni-processor schedules, and the basic (i.e., Liu & Layland) readiness model.
Require Import prosa.model.priority.edf.
Require Import prosa.model.processor.ideal.
Require Import prosa.model.readiness.basic.

Furthermore, we assume the task model with fixed preemption points.

Setup and Assumptions

Consider any type of tasks ...
  Context {Task : TaskType}.
  Context `{TaskCost Task}.
  Context `{TaskDeadline Task}.

... and any type of jobs associated with these tasks.
  Context {Job : JobType}.
  Context `{JobTask Job Task}.
  Context `{JobArrival Job}.
  Context `{JobCost Job}.

Consider any arrival sequence with consistent, non-duplicate arrivals.
Consider an arbitrary task set ts, ...
  Variable ts : list Task.

... assume that all jobs come from this task set, ...
... and the cost of a job cannot be larger than the task cost.
Next, we assume we have the model with fixed preemption points. I.e., each task is divided into a number of non-preemptive segments by inserting statically predefined preemption points.
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.

Next, consider any ideal uni-processor schedule with limited preemptions of this arrival sequence ...
... where jobs do not execute before their arrival or after completion.
Assume we have sequential tasks, i.e, jobs from the same task execute in the order of their arrival.
Next, we assume that the schedule is a work-conserving schedule...
... and the schedule respects the policy defined by the job_preemptable function (i.e., jobs have bounded non-preemptive segments).

Total Workload and Length of Busy Interval

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.
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).
We define a bound for the priority inversion caused by jobs with lower priority.
Next, we define an upper bound on interfering workload received from jobs of other tasks with higher-than-or-equal priority.
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.

Response-Time Bound

To reduce the time complexity of the analysis, recall the notion of search space.
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 non-preemptive segments to establish a response-time bound for the more concrete model of fixed preemption points.

  Let response_time_bounded_by := task_response_time_bound arr_seq sched.

  Theorem uniprocessor_response_time_bound_edf_with_fixed_preemption_points:
    response_time_bounded_by tsk R.
    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_valid_job_cost _ ARR) ⇒ POSt.
      move: POSt; rewrite /valid_job_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.

End RTAforFixedPreemptionPointsModelwithArrivalCurves.