Library prosa.results.rs.fp.fully_preemptive

RTA for Fully Preemptive FP Scheduling on Restricted-Supply Uniprocessors

In the following, we derive a response-time analysis for FP schedulers, assuming a workload of sporadic real-time tasks characterized by arbitrary arrival curves executing upon a uniprocessor with arbitrary supply restrictions. To this end, we instantiate the sequential variant of abstract Restricted-Supply Response-Time Analysis (aRSA) as provided in the prosa.analysis.abstract.restricted_supply module.

Section RTAforFullyPreemptiveFPModelwithArrivalCurves.

Defining the System Model

Before any formal claims can be stated, an initial setup is needed to define the system model under consideration. To this end, we next introduce and define the following notions using Prosa's standard definitions and behavioral semantics:

processor model,
tasks, jobs, and their parameters,
the sequence of job arrivals,
worst-case execution time (WCET) and the absence of self-suspensions,
the task under analysis,
an arbitrary schedule of the task set, and finally,
a supply-bound function.

Processor Model

Consider a restricted-supply uniprocessor model.

#[local] Existing Instance rs_processor_state.

Tasks and Jobs

Consider any type of tasks, each characterized by a WCET task_cost and an arrival curve max_arrivals, ...

  Context {Task : TaskType}.
  Context `{TaskCost Task}.
  Context `{MaxArrivals Task}.

... and any type of jobs associated with these tasks, where each job has a task job_task, a cost job_cost, and an arrival time job_arrival.

  Context {Job : JobType}.
  Context `{JobTask Job Task}.
  Context `{JobCost Job}.
  Context `{JobArrival Job}.

Furthermore, 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.

The Job Arrival Sequence

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

Variable arr_seq : arrival_sequence Job.
Hypothesis H_valid_arrival_sequence : valid_arrival_sequence arr_seq.

Absence of Self-Suspensions and WCET Compliance

We assume the sequential model of readiness without jitter or self-suspensions, wherein a pending job j is ready as soon as all prior jobs from the same task completed.

#[local] Instance sequential_readiness : JobReady _ _ :=
sequential_ready_instance arr_seq.

We further require that a job's cost cannot exceed its task's stated WCET.

Hypothesis H_valid_job_cost : arrivals_have_valid_job_costs arr_seq.

The Task Set

We consider an arbitrary task set ts ...

Variable ts : seq Task.

... and assume that all jobs stem from tasks in this task set.

Hypothesis H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts.

We assume that max_arrivals is a family of valid arrival curves that constrains the arrival sequence arr_seq, i.e., for any task tsk in ts, max_arrival tsk is (1) an arrival bound of tsk, and ...

Hypothesis H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts.

... (2) a monotonic function that equals 0 for the empty interval delta = 0.

Hypothesis H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals.

The Task Under Analysis

Let tsk be any task in ts that is to be analyzed.

Variable tsk : Task.
Hypothesis H_tsk_in_ts : tsk \in ts.

The Schedule

Consider any arbitrary, work-conserving, valid restricted-supply uni-processor schedule of the given arrival sequence arr_seq (and hence the given task set ts).

  Variable sched : schedule (rs_processor_state Job).
  Hypothesis H_valid_schedule : valid_schedule sched arr_seq.
  Hypothesis H_work_conserving : work_conserving 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_task_priorities FP.
  Hypothesis H_priority_is_transitive : transitive_task_priorities FP.

We assume that the schedule respects the FP scheduling policy.

Hypothesis H_respects_policy_at_preemption_point :
respects_FP_policy_at_preemption_point arr_seq sched FP.

Supply-Bound Function

Assume the minimum amount of supply that any job of task tsk receives is defined by a monotone unit-supply-bound function SBF.

  Context {SBF : SupplyBoundFunction}.
  Hypothesis H_SBF_monotone : sbf_is_monotone SBF.
  Hypothesis H_unit_SBF : unit_supply_bound_function SBF.

We assume that SBF properly characterizes all busy intervals (w.r.t. task tsk) in sched. That is, (1) SBF 0 = 0 and (2) for any duration Δ, at least SBF Δ supply is available in any busy-interval prefix of length Δ.

Hypothesis H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF.

Workload Abbreviation

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 total_hep_rbf as an abbreviation for the request-bound function of all tasks with higher-or-equal priority ...

Let total_hep_rbf := total_hep_request_bound_function_FP ts tsk.

... and total_ohep_rbf as an abbreviation for the request-bound function of all tasks with higher-or-equal priority other than task tsk.

Let total_ohep_rbf := total_ohep_request_bound_function_FP ts tsk.

Length of Busy Interval

The next step is to establish a bound on the maximum busy-window length, which aRSA requires to be given.

To this end, let L be any positive fixed point of the busy-interval recurrence. As the lemma busy_intervals_are_bounded_rs_fp shows, under FP scheduling, this is sufficient to guarantee that all busy intervals are bounded by L.

  Variable L : duration.
  Hypothesis H_L_positive : 0 < L.
  Hypothesis H_fixed_point : total_hep_rbf L ≤ SBF L.

Response-Time Bound

Having established all necessary preliminaries, it is finally time to state the claimed response-time bound R.

A value R is a response-time bound if, for any given offset A in the search space, the response-time bound recurrence has a solution F not exceeding R.

  Definition rta_recurrence_solution R :=
    ∀ (A : duration),
      is_in_search_space tsk L A →
      ∃ (F : duration ),
        A ≤ F ≤ A + R
        ∧ rbf tsk (A + ε) + total_ohep_rbf F ≤ SBF F.

Finally, using the sequential variant of abstract restricted-supply analysis, we establish that any such R is a sound response-time bound for the concrete model of fully-preemptive fixed-priority scheduling with arbitrary supply restrictions.

  Theorem uniprocessor_response_time_bound_fully_preemptive_fp :
    ∀ (R : duration),
      rta_recurrence_solution R →
      task_response_time_bound arr_seq sched tsk R.
  Proof.
    move⇒ R SOL js ARRs TSKs.
    have BLOCK: ∀ tsk , blocking_bound ts tsk = 0.
    { by move⇒ tsk2; rewrite /blocking_bound /parameters.task_max_nonpreemptive_segment
                 /fully_preemptive_task_model subnn big1_eq. }
    have [ZERO|POS] := posnP (job_cost js);
      first by rewrite /job_response_time_bound /completed_by ZERO.
    have READ : work_bearing_readiness arr_seq sched
      by exact: sequential_readiness_implies_work_bearing_readiness.
    eapply uniprocessor_response_time_bound_restricted_supply_seq with (L := L) ⇒ //.
    - exact: instantiated_i_and_w_are_coherent_with_schedule.
    - exact: sequential_readiness_implies_sequential_tasks.
    - exact: instantiated_interference_and_workload_consistent_with_sequential_tasks.
    - by apply: busy_intervals_are_bounded_rs_fp ⇒ //; rewrite BLOCK add0n.
    - apply: valid_pred_sbf_switch_predicate; last by exact: H_valid_SBF.
      move ⇒ ? ? ? ? [? ?]; split ⇒ //.
      by apply instantiated_busy_interval_prefix_equivalent_busy_interval_prefix.
    - apply: instantiated_task_intra_interference_is_bounded; eauto 1 ⇒ //; first last.
      + by apply athep_workload_le_total_ohep_rbf.
      + apply: service_inversion_is_bounded ⇒ // ⇒ jo t1 t2 ARRo TSKo BUSYo.
        by unshelve rewrite (leqRW (nonpreemptive_segments_bounded_by_blocking _ _ _ _ _ _ _ _ _)) ⇒ //.
    - move ⇒ A SP; move: (SOL A) ⇒ [].
      + apply: search_space_sub ⇒ //; apply: search_space_switch_IBF; last by exact: SP.
        by move⇒ A1 Δ1; rewrite //= BLOCK.
      + move ⇒ F [/andP [_ LE] FIX]; ∃ F; split ⇒ //.
        rewrite /task_intra_IBF /task_rtct /fully_preemptive_rtc_threshold.
        by rewrite BLOCK subnn //= add0n addn0 subn0.
  Qed.

End RTAforFullyPreemptiveFPModelwithArrivalCurves.