Library prosa.results.rta.ovh.fp.fully_preemptive

RTA for Fully Preemptive FP Scheduling on Uniprocessors with Overheads

In the following, we derive a response-time analysis for FP schedulers, assuming a workload of fully preemptive sporadic real-time tasks, characterized by arbitrary arrival curves, executing upon a uniprocessor subject to scheduling overheads. To this end, we instantiate the sequential variant of abstract Restricted-Supply 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:

the processor model,
tasks, jobs, and their parameters,
the task set and the task under analysis,
the sequence of job arrivals,
the absence of self-suspensions,
an arbitrary schedule of the task set, and finally,
an upper bound on overhead-induced delays.

Processor Model

Consider a unit-speed uniprocessor subject to scheduling overheads.

#[local] Existing Instance overheads.processor_state.

Tasks and Jobs

Consider tasks characterized by a WCET task_cost and an arrival curve max_arrivals, ...

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

... and their associated jobs, where each job has a corresponding task job_task, an execution time job_cost, and an arrival time job_arrival.

Context {Job : JobType} `{JobTask Job Task} `{JobCost Job} `{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 Task Set and the Task Under Analysis

Consider an arbitrary task set ts, and ...

Variable ts : seq Task.

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

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

The Job Arrival Sequence

Allow for any possible arrival sequence arr_seq consistent with the parameters of the task set ts. That is, arr_seq is a valid arrival sequence in which each job's cost is upper-bounded by its task's WCET, every job stems from a task in ts, and the number of arrivals respects each task's max_arrivals bound.

Variable arr_seq : arrival_sequence Job.
Hypothesis H_valid_task_arrival_sequence : valid_task_arrival_sequence ts arr_seq.

Additionally, we assume that all jobs in arr_seq have positive execution costs. This requirement is not fundamental to the analysis approach itself but reflects an artifact of the current proof structure specific to upper bounds on the total duration of overheads.

Hypothesis H_arrivals_have_positive_job_costs :
arrivals_have_positive_job_costs arr_seq.

Absence of Self-Suspensions

We assume the classic (i.e., Liu & Layland) model of readiness without jitter or self-suspensions, wherein pending jobs are always ready.

#[local] Existing Instance basic_ready_instance.

The Schedule

Consider an arbitrary fixed-priority policy FP that indicates a higher-or-equal priority relation among the tasks in ts, 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.

Consider a work-conserving, valid uniprocessor schedule with explicit overheads that corresponds to the given arrival sequence arr_seq (and hence the given task set ts).

  Variable sched : schedule (overheads.processor_state Job).
  Hypothesis H_valid_schedule : valid_schedule sched arr_seq.
  Hypothesis H_work_conserving : work_conserving arr_seq sched.

We assume that the schedule respects the given FP scheduling policy.

Hypothesis H_respects_policy :
respects_FP_policy_at_preemption_point arr_seq sched FP.

Furthermore, we require that the schedule ensures two additional properties: jobs of the same task are executed in the order of their arrival, ...

Hypothesis H_sequential_tasks : sequential_tasks arr_seq sched.

... and preemptions occur only when strictly required by the scheduling policy (specifically, a job is never preempted by another job of equal priority).

Hypothesis H_no_superfluous_preemptions : no_superfluous_preemptions sched.

Bounding the Total Overhead Duration

We assume that the scheduling overheads encountered in the schedule sched are bounded by the following upper bounds:

the maximum dispatch overhead is bounded by DB,
the maximum context-switch overhead is bounded by CSB, and
the maximum cache-related preemption delay is bounded by CRPDB.

Variable DB CSB CRPDB : duration.
Hypothesis H_valid_overheads_model : overhead_resource_model sched DB CSB CRPDB.

To conservatively account for the maximum cumulative delay that task tsk may experience due to scheduling overheads, we introduce an overhead bound. This term upper-bounds the maximum cumulative duration during which processor cycles are "lost" to dispatch, context-switch, and preemption-related delays in a given interval.

Under FP scheduling, the bound in an interval of length Δ is determined by the arrivals of tasks with higher-or-equal priority relative to tsk. Up to n such arrivals can lead to at most 1 + 2n segments without a schedule change, each potentially incurring dispatch, context-switch, and preemption-related overhead.

We denote this bound by overhead_bound for the task under analysis tsk.

Let overhead_bound Δ :=
(DB + CSB + CRPDB ) × (1 + 2 × \sum_(tsk_o <- ts | hep_task tsk_o tsk ) max_arrivals tsk_o Δ ).

Maximum Length of a Busy Interval

In order to apply aRSA, we require a bound on the maximum busy-window length. To this end, let L be any positive solution of the busy-interval "recurrence" (i.e., inequality) overhead_bound L + total_hep_rbf L ≤ L, as defined below.

As the lemma busy_intervals_are_bounded_rs_fp shows, under FP scheduling, this condition is sufficient to guarantee that the maximum busy-window length is at most L, i.e., the length of any busy interval is bounded by L.

  Definition busy_window_recurrence_solution (L : duration) :=
    L > 0
    ∧ L ≥ overhead_bound L + total_hep_request_bound_function_FP ts tsk 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 for task tsk if, for any given offset A in the search space (w.r.t. the busy-window bound L), the response-time bound "recurrence" (i.e., inequality) has a solution F not exceeding A + R.

  Definition rta_recurrence_solution L R :=
    ∀ (A : duration),
      is_in_search_space tsk L A →
      ∃ (F : duration ),
        F ≥ overhead_bound F
            + task_request_bound_function tsk (A + ε)
            + total_ohep_request_bound_function_FP ts tsk F
        ∧ A + R ≥ F.

Finally, using the sequential variant of abstract restricted-supply analysis, we establish that, given a bound on the maximum busy-window length L, any such R is indeed a sound response-time bound for task tsk under fully-preemptive fixed-priority scheduling on a unit-speed uniprocessor subject to scheduling overheads.

  Theorem uniprocessor_response_time_bound_fully_preemptive_fp :
    ∀ (L : duration),
      busy_window_recurrence_solution L →
      ∀ (R : duration),
        rta_recurrence_solution L R →
        task_response_time_bound arr_seq sched tsk R.
  Proof.
    set (sSBF := fp_ovh_sbf_slow ts DB CSB CRPDB tsk).
    move⇒ L [BW_POS BW_SOL] 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 VBSBF : valid_busy_sbf arr_seq sched tsk sSBF by apply overheads_sbf_busy_valid ⇒ //=.
    have USBF : unit_supply_bound_function sSBF by apply overheads_sbf_unit ⇒ //=.
    have POStsk: 0 < task_cost tsk
      by move: TSKs ⇒ /eqP <-; apply: leq_trans; [apply POS | apply H_valid_task_arrival_sequence].
    eapply uniprocessor_response_time_bound_restricted_supply_seq with (L := L) (SBF := sSBF) ⇒ //=.
    - exact: instantiated_i_and_w_are_coherent_with_schedule.
    - apply instantiated_interference_and_workload_consistent_with_sequential_tasks ⇒ //.
    - eapply busy_intervals_are_bounded_rs_fp with (SBF := sSBF) ⇒ //=.
      + exact: instantiated_i_and_w_are_coherent_with_schedule.
      + rewrite BLOCK add0n; apply bound_preserved_under_slowed.
        unfold fp_blackout_bound, overhead_bound in *; lia.
    - apply: valid_pred_sbf_switch_predicate; last (eapply overheads_sbf_busy_valid) ⇒ //=.
      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 ⇒ //=.
        { by apply: non_pathological_max_arrivals =>//; apply H_valid_task_arrival_sequence. }
        apply: search_space_switch_IBF; last by exact: SP.
        by move⇒ A1 Δ1; rewrite //= BLOCK.
      + move ⇒ F [FIX LE]; ∃ F; split ⇒ //.
        rewrite /task_intra_IBF /task_rtct /fully_preemptive_rtc_threshold.
        rewrite BLOCK subnn //= add0n addn0 subn0; split.
        × apply bound_preserved_under_slowed.
          move: FIX.
          by rewrite /sSBF /fp_blackout_bound /overhead_bound; lia.
        × exact: overheads_sbf_monotone.
  Qed.

End RTAforFullyPreemptiveFPModelwithArrivalCurves.