Library prosa.results.edf.rta.floating_nonpreemptive

RTA for EDF with Floating Non-Preemptive Regions

In this module we prove the RTA theorem for floating non-preemptive regions EDF model.

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}.

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.

We assume that jobs are limited-preemptive.
  #[local] Existing Instance limited_preemptive_job_model.

Consider any arrival sequence with consistent, non-duplicate arrivals.
Assume we have the model with floating non-preemptive regions. I.e., for each task only the length of the maximal non-preemptive segment is known and each job level is divided into a number of non-preemptive segments by inserting preemption points.
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.
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 valid ideal uni-processor schedule with limited preemptions of this arrival sequence ...
Next, we assume that the schedule is a work-conserving schedule...
... and the schedule respects the scheduling policy.

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).
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 ts tsk A + task_rbf (A + ε) + bound_on_athep_workload ts tsk A (A + F)
        R F.

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 with floating nonpreemptive regions.

  Let response_time_bounded_by := task_response_time_bound arr_seq sched.

  Theorem uniprocessor_response_time_bound_edf_with_floating_nonpreemptive_regions:
    response_time_bounded_by tsk R.
    move: (H_valid_task_model_with_floating_nonpreemptive_regions) ⇒ [LIMJ JMLETM].
    move: (LIMJ) ⇒ [BEG [END _]].
    eapply uniprocessor_response_time_bound_edf_with_bounded_nonpreemptive_segments with (L := L) ⇒ //.
    rewrite subnn.
    intros A SP.
    apply H_R_is_maximum in SP.
    move: SP ⇒ [F [EQ LE]].
    by rewrite subn0 addn0; split.

End RTAforModelWithFloatingNonpreemptiveRegionsWithArrivalCurves.