Library prosa.results.fixed_priority.rta.limited_preemptive

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

Require Export prosa.results.fixed_priority.rta.bounded_nps.
Require Export prosa.analysis.facts.preemption.rtc_threshold.limited.
Require Export prosa.analysis.facts.readiness.sequential.
Require Export prosa.model.task.preemption.limited_preemptive.

RTA for FP-schedulers with Fixed Preemption Points

In this module we prove the RTA theorem for FP-schedulers with fixed preemption points.

Setup and Assumptions


Section RTAforFixedPreemptionPointsModelwithArrivalCurves.

We assume ideal uni-processor schedules.
  #[local] Existing Instance ideal.processor_state.

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

Consider an arbitrary task set ts, ...
  Variable ts : list Task.

... assume that all jobs come from the 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.

First, 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.
  Context `{JobPreemptionPoints Job}
          `{TaskPreemptionPoints Task}.
  Hypothesis H_valid_model_with_fixed_preemption_points:
    valid_fixed_preemption_points_model arr_seq ts.

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.

Recall that we assume sequential readiness.
  #[local] Instance sequential_readiness : JobReady _ _ :=
    sequential_ready_instance arr_seq.

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_respects_preemption_model:
    schedule_respects_preemption_model arr_seq sched.

Consider an FP policy that indicates a higher-or-equal priority relation, and assume that the relation is reflexive and transitive.
  Context {FP : FP_policy Task}.
  Hypothesis H_priority_is_reflexive : reflexive_priorities.
  Hypothesis H_priority_is_transitive : transitive_priorities.

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_FP_policy_at_preemption_point arr_seq sched FP.

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 with higher priority ...
  Let total_hep_rbf := total_hep_request_bound_function_FP ts tsk.

... and the request bound function of all tasks with higher priority other than task tsk.
  Let total_ohep_rbf := total_ohep_request_bound_function_FP ts tsk.

Next, we define a bound for the priority inversion caused by tasks of lower priority.
  Let blocking_bound :=
    \max_(tsk_other <- ts | ~~ hep_task tsk_other tsk)
     (task_max_nonpreemptive_segment tsk_other - ε).

Let L be any positive fixed point of the busy interval recurrence, determined by the sum of blocking and higher-or-equal-priority workload.
  Variable L : duration.
  Hypothesis H_L_positive : L > 0.
  Hypothesis H_fixed_point : L = blocking_bound + total_hep_rbf L.

Response-Time Bound

To reduce the time complexity of the analysis, recall the notion of search space.
  Let is_in_search_space := is_in_search_space tsk L.

Next, consider any value R, and assume that for any given arrival A from 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 : duration),
      is_in_search_space A →
      ∃ (F : duration),
        A + F ≥ blocking_bound
                + (task_rbf (A + ε) - (task_last_nonpr_segment tsk - ε))
                + total_ohep_rbf (A + F) ∧
        R ≥ F + (task_last_nonpr_segment tsk - ε).

Now, we can reuse 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_fp_with_fixed_preemption_points:
    response_time_bounded_by tsk R.
  Proof.
    move: (H_valid_model_with_fixed_preemption_points) ⇒ [MLP [BEG [END [INCR [HYP1 [HYP2 HYP3]]]]]].
    move: (MLP) ⇒ [BEGj [ENDj _]].
    edestruct (posnP (task_cost tsk)) as [ZERO|POSt].
    { intros j ARR TSK.
      move: (H_valid_job_cost _ ARR) ⇒ POSt.
      move: TSK ⇒ /eqP TSK; move: POSt; rewrite /valid_job_cost TSK ZERO leqn0; move ⇒ /eqP Z.
      by rewrite /job_response_time_bound /completed_by Z.
    }
    try ( eapply uniprocessor_response_time_bound_fp_with_bounded_nonpreemptive_segments
      with (L0 := L) ) ||
    eapply uniprocessor_response_time_bound_fp_with_bounded_nonpreemptive_segments
      with (L := L).
    all: rt_eauto.
    - by apply sequential_readiness_implies_work_bearing_readiness; rt_auto.
    - by apply sequential_readiness_implies_sequential_tasks; rt_auto.
    - intros A SP.
      destruct (H_R_is_maximum _ SP) as[FF [EQ1 EQ2]].
      ∃ FF; 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.
  Qed.

End RTAforFixedPreemptionPointsModelwithArrivalCurves.