Library prosa.results.edf.rta.floating_nonpreemptive

From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq path fintype bigop.

Require Import prosa.model.readiness.basic.
Require Export prosa.results.edf.rta.bounded_nps.
Require Export prosa.analysis.facts.preemption.rtc_threshold.floating.
Require Export prosa.analysis.facts.readiness.sequential.
Require Import prosa.model.priority.edf.

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

Section RTAforModelWithFloatingNonpreemptiveRegionsWithArrivalCurves.

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.

Variable arr_seq : arrival_sequence Job.
Hypothesis H_valid_arrival_sequence : valid_arrival_sequence arr_seq.

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.

  Context `{JobPreemptionPoints Job}
          `{TaskMaxNonpreemptiveSegment Task}.
  Hypothesis H_valid_task_model_with_floating_nonpreemptive_regions:
    valid_model_with_floating_nonpreemptive_regions arr_seq.

Consider an arbitrary task set ts, ...

Variable ts : list Task.

... assume that all jobs come from this task set, ...

Hypothesis H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts.

... and the cost of a job cannot be larger than the task cost.

Hypothesis H_valid_job_cost:
arrivals_have_valid_job_costs arr_seq.

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 valid ideal uni-processor schedule with limited preemptions of this arrival sequence ...

  Variable sched : schedule (ideal.processor_state Job).
  Hypothesis H_sched_valid: valid_schedule sched arr_seq.
  Hypothesis H_schedule_with_limited_preemptions:
    schedule_respects_preemption_model arr_seq 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 scheduling policy.

Hypothesis H_respects_policy : respects_JLFP_policy_at_preemption_point arr_seq sched (EDF Job).

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.

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.

  Definition blocking_bound A :=
    \max_(tsk_other <- ts | (blocking_relevant tsk_other )
                             && (task_deadline tsk_other > task_deadline tsk + A ))
     (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 ) + task_deadline tsk - task_deadline 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.

Response-Time Bound

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

Let is_in_search_space := bounded_nps.is_in_search_space ts tsk L.

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 A + task_rbf (A + ε) + bound_on_total_hep_workload 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.
  Proof.
    move: (H_valid_task_model_with_floating_nonpreemptive_regions) ⇒ [LIMJ JMLETM].
    move: (LIMJ) ⇒ [BEG [END _]].
    try ( eapply uniprocessor_response_time_bound_edf_with_bounded_nonpreemptive_segments with (L0 := L) ) ||
    eapply uniprocessor_response_time_bound_edf_with_bounded_nonpreemptive_segments with (L := L).
    all: rt_eauto.
    rewrite subnn.
    intros A SP.
    apply H_R_is_maximum in SP.
    move: SP ⇒ [F [EQ LE]].
    ∃ F.
    by rewrite subn0 addn0; split.
  Qed.

End RTAforModelWithFloatingNonpreemptiveRegionsWithArrivalCurves.