Library prosa.results.edf.rta.fully_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.task.nonpreemptive.
Require Export prosa.analysis.facts.preemption.rtc_threshold.nonpreemptive.
Require Export prosa.analysis.facts.readiness.basic.
Require Export prosa.model.task.preemption.fully_nonpreemptive.
Require Import prosa.model.priority.edf.

RTA for Fully Non-Preemptive EDF

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

Setup and Assumptions

Section RTAforFullyNonPreemptiveEDFModelwithArrivalCurves.

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 and tasks are fully nonpreemptive.

  #[local] Existing Instance fully_nonpreemptive_job_model.
  #[local] Existing Instance fully_nonpreemptive_task_model.
  #[local] Existing Instance fully_nonpreemptive_rtc_threshold.

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 non-preemptive uniprocessor schedule of this arrival sequence ...

  Variable sched : schedule (ideal.processor_state Job).
  Hypothesis H_sched_valid: valid_schedule sched arr_seq.
  Hypothesis H_nonpreemptive_sched : nonpreemptive_schedule 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 also define a bound for the priority inversion caused by jobs with lower priority.

  Let blocking_bound A :=
    \max_(tsk_o <- ts | (blocking_relevant tsk_o )
                         && (task_deadline tsk_o > task_deadline tsk + A ))
     (task_cost tsk_o - 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: nat.
  Hypothesis H_R_is_maximum:
    ∀ A,
      is_in_search_space A →
      ∃ F ,
        A + F ≥ blocking_bound A + (task_rbf (A + ε) - (task_cost tsk - ε ))
                + bound_on_total_hep_workload A (A + F) ∧
        R ≥ F + (task_cost tsk - ε ).

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 fully nonpreemptive scheduling.

  Let response_time_bounded_by := task_response_time_bound arr_seq sched.

  Theorem uniprocessor_response_time_bound_fully_nonpreemptive_edf:
    response_time_bounded_by tsk R.
  Proof.
    case: (posnP (task_cost tsk)) ⇒ [ZERO|POS].
    { intros j ARR TSK.
      have ZEROj: job_cost j = 0.
      { move: (H_valid_job_cost j ARR) ⇒ NEQ.
        rewrite /valid_job_cost in NEQ.
        move: TSK ⇒ /eqP → in NEQ.
        rewrite ZERO in NEQ.
        by apply/eqP; rewrite -leqn0.
      }
      by rewrite /job_response_time_bound /completed_by ZEROj.
    }
    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.
  Qed.

End RTAforFullyNonPreemptiveEDFModelwithArrivalCurves.