Library prosa.results.fixed_priority.rta.fully_preemptive

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.

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

RTA for Fully Preemptive FP Model

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

Throughout this file, we assume the FP priority policy, ideal uni-processor schedules, and the basic (i.e., Liu & Layland) readiness model.

Require Import prosa.model.processor.ideal.
Require Import prosa.model.readiness.basic.

Furthermore, we assume the fully preemptive task model.

Require Import prosa.model.task.preemption.fully_preemptive.

Setup and Assumptions

Section RTAforFullyPreemptiveFPModelwithArrivalCurves.

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_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 ideal uniprocessor schedule of 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.

... 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.

  Context `{FP_policy Task}.
  Hypothesis H_priority_is_reflexive : reflexive_priorities.
  Hypothesis H_priority_is_transitive : transitive_priorities.

Assume we have sequential tasks, i.e, tasks 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 the job_preemptable function (i.e., jobs have bounded non-preemptive segments).

Hypothesis H_respects_policy : respects_policy_at_preemption_point arr_seq sched.

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 (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 : 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) ∧
        F ≤ R.

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.fully_preemptive_model subnn big1_eq. }
    eapply uniprocessor_response_time_bound_fp_with_bounded_nonpreemptive_segments.
    all: eauto 2 with basic_facts.
    - 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.