Library prosa.results.fixed_priority.rta.fully_nonpreemptive

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.nonpreemptive.
Require Export prosa.analysis.facts.preemption.rtc_threshold.nonpreemptive.
Require Export prosa.analysis.facts.readiness.sequential.
Require Export prosa.model.task.preemption.fully_nonpreemptive.

RTA for Fully Non-Preemptive FP Model

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

Setup and Assumptions


Section RTAforFullyNonPreemptiveFPModelwithArrivalCurves.

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 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 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 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.

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_cost 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 : duration.
  Hypothesis H_R_is_maximum:
    ∀ (A : duration),
      is_in_search_space A →
      ∃ (F : duration),
        A + F ≥ blocking_bound
                + (task_rbf (A + ε) - (task_cost tsk - ε))
                + total_ohep_rbf (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_fp:
    response_time_bounded_by tsk R.
  Proof.
    move: (posnP (@task_cost _ H 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_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.
  Qed.

End RTAforFullyNonPreemptiveFPModelwithArrivalCurves.