Library prosa.results.fixed_priority.rta.fully_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.task.preemptive.
Require Export prosa.analysis.facts.preemption.rtc_threshold.preemptive.
Require Export prosa.analysis.facts.readiness.sequential.
Require Export prosa.model.task.preemption.fully_preemptive.

RTA for Fully Preemptive FP Model

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

Setup and Assumptions

Section RTAforFullyPreemptiveFPModelwithArrivalCurves.

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 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 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.

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 ideal uniprocessor schedule of this 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.

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.

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 = 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 : duration.
  Hypothesis H_R_is_maximum:
    ∀ (A : duration),
      is_in_search_space A →
      ∃ (F : duration ),
        A + F ≥ task_rbf (A + ε) + total_ohep_rbf (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_fp:
    response_time_bounded_by tsk R.
  Proof.
    have BLOCK: blocking_bound ts tsk = 0.
    { by rewrite /blocking_bound /parameters.task_max_nonpreemptive_segment
               /fully_preemptive_task_model subnn big1_eq. }
    eapply uniprocessor_response_time_bound_fp_with_bounded_nonpreemptive_segments.
    all: rt_eauto.
    rewrite /work_bearing_readiness.
    - by apply sequential_readiness_implies_work_bearing_readiness; rt_auto.
    - by apply sequential_readiness_implies_sequential_tasks ⇒ //; rt_auto.
    - by rewrite BLOCK add0n.
    - move ⇒ A /andP [LT NEQ].
      edestruct H_R_is_maximum as [F [FIX BOUND]].
      { by apply/andP; split; eauto 2. }
      ∃ F; split.
      + by rewrite BLOCK add0n subnn subn0.
      + by rewrite subnn addn0.
  Qed.

End RTAforFullyPreemptiveFPModelwithArrivalCurves.