Library prosa.results.edf.rta.fully_preemptive

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

Require Import prosa.model.readiness.basic.
Require Import prosa.results.edf.rta.bounded_nps.
Require Export prosa.analysis.facts.preemption.task.preemptive.
Require Export prosa.analysis.facts.preemption.rtc_threshold.preemptive.
Require Export prosa.analysis.facts.readiness.basic.
Require Import prosa.model.task.preemption.fully_preemptive.
Require Import prosa.model.priority.edf.

RTA for Fully Preemptive EDF

In this section we prove the RTA theorem for the fully preemptive EDF model

Setup and Assumptions

Section RTAforFullyPreemptiveEDFModelwithArrivalCurves.

We assume that jobs and tasks are fully preemptive.

  #[local] Existing Instance fully_preemptive_job_model.
  #[local] Existing Instance fully_preemptive_task_model.
  #[local] Existing Instance fully_preemptive_rtc_threshold.

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.

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.

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 uniprocessor schedule of the arrival sequence ...

  Variable sched : schedule (ideal.processor_state Job).
  Hypothesis H_sched_valid: valid_schedule sched arr_seq.
  Hypothesis H_jobs_come_from_arrival_sequence:
    jobs_come_from_arrival_sequence sched arr_seq.

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.

If jobs are fully preemptive, lower priority jobs do not cause priority inversion. Hence, the blocking bound is always 0 for any A.

Let blocking_bound (A : duration) := 0.

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 ≥ 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 non-preemptive segments to establish a response-time bound for the more concrete model of fully preemptive scheduling.

  Let response_time_bounded_by := task_response_time_bound arr_seq sched.

  Theorem uniprocessor_response_time_bound_fully_preemptive_edf:
    response_time_bounded_by tsk R.
  Proof.
    apply: uniprocessor_response_time_bound_edf_with_bounded_nonpreemptive_segments; rt_eauto.
    move ⇒ A /andP [LT CHANGE].
    have BLOCK: ∀ A', bounded_nps.blocking_bound ts tsk A' = blocking_bound A'.
    { by move⇒ A'; rewrite /bounded_nps.blocking_bound /parameters.task_max_nonpreemptive_segment
         /fully_preemptive_task_model subnn big1_eq. }
    specialize (H_R_is_maximum A); feed H_R_is_maximum; first by apply/andP; split; done.
    move: H_R_is_maximum ⇒ [F [FIX BOUND]].
    ∃ F; split.
    + by rewrite BLOCK add0n subnn subn0.
    + by rewrite subnn addn0.
  Qed.

End RTAforFullyPreemptiveEDFModelwithArrivalCurves.