Library rt.restructuring.analysis.fixed_priority.rta.nonpr_reg.concrete_models.limited

Require Export rt.restructuring.analysis.fixed_priority.rta.nonpr_reg.response_time_bound.
Require Export rt.restructuring.analysis.basic_facts.preemption.rtc_threshold.limited.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq path fintype bigop.

Throughout this file, we assume ideal uniprocessor schedules.

Require Import rt.restructuring.model.processor.ideal.

Throughout this file, we assume the basic (i.e., Liu & Layland) readiness model.

Require Import rt.restructuring.model.readiness.basic.

Throughout this file, we assume the task model with fixed preemption points.

Require Import rt.restructuring.model.preemption.limited_preemptive.
Require Import rt.restructuring.model.task.preemption.limited_preemptive.

RTA for FP-schedulers with Fixed Premption Points

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

Section RTAforFixedPreemptionPointsModelwithArrivalCurves.

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}.

Consider any arrival sequence with consistent, non-duplicate arrivals.

  Variable arr_seq : arrival_sequence Job.
  Hypothesis H_arrival_times_are_consistent : consistent_arrival_times arr_seq.
  Hypothesis H_arr_seq_is_a_set : arrival_sequence_uniq 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_job_cost_le_task_cost:
cost_of_jobs_from_arrival_sequence_le_task_cost arr_seq.

First, we assume we have the model with fixed preemption points. I.e., each task is divided into a number of nonpreemptive segments by inserting staticaly 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.

Next, consider any ideal uniprocessor schedule with limited preemptionsof this arrival sequence ...

  Variable sched : schedule (ideal.processor_state Job).
  Hypothesis H_jobs_come_from_arrival_sequence:
    jobs_come_from_arrival_sequence sched arr_seq.
  Hypothesis H_valid_schedule_with_limited_preemptions:
    valid_schedule_with_limited_preemptions arr_seq sched.

... where jobs do not execute before their arrival or after completion.

Hypothesis H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched.
Hypothesis H_completed_jobs_dont_execute : completed_jobs_dont_execute sched.

Consider an FP policy that indicates a higher-or-equal priority relation, and assume that the relation is reflexive and transitive.

  Variable higher_eq_priority : FP_policy Task.
  Hypothesis H_priority_is_reflexive : reflexive_priorities.
  Hypothesis H_priority_is_transitive : transitive_priorities.

Assume we have sequential tasks, i.e, jobs from the same task execute in the order of their arrival.

Hypothesis H_sequential_tasks : sequential_tasks 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 policy defined by thejob_preemptable function (i.e., jobs have bounded nonpreemptive segments).

Hypothesis H_respects_policy : respects_policy_at_preemption_point arr_seq sched.

Let's define some local names for clarity.

  Let task_rbf := task_request_bound_function tsk.
  Let total_hep_rbf := total_hep_request_bound_function_FP _ ts tsk.
  Let total_ohep_rbf := total_ohep_request_bound_function_FP _ ts tsk.
  Let response_time_bounded_by := task_response_time_bound arr_seq sched.

Next, we define a bound for the priority inversion caused by tasks of lower priority.

  Let blocking_bound :=
    \max_(tsk_other <- ts | ~~ higher_eq_priority 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.

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

Let is_in_search_space (A : duration) := (A < L ) && (task_rbf A != task_rbf (A + ε)).

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) ∧
        F + (task_last_nonpr_segment tsk - ε ) ≤ R.

Now, we can reuse the results for the abstract model with bounded nonpreemptive segments to establish a response-time bound for the more concrete model of fixed preemption points.

  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_job_cost_le_task_cost _ ARR) ⇒ POSt.
      move: POSt; rewrite /job_cost_le_task_cost TSK ZERO leqn0; move ⇒ /eqP Z.
        by rewrite /job_response_time_bound /completed_by Z.
    }
    eapply uniprocessor_response_time_bound_fp_with_bounded_nonpreemptive_segments
      with (L0 := L).
    all: eauto 2 with basic_facts.
    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.