Library prosa.results.ovh.fp.limited_preemptive

RTA for FP Scheduling with Fixed Preemption Points on Uniprocessors with Overheads

In the following, we derive a response-time analysis for FP schedulers, assuming a workload of sporadic real-time tasks with fixed preemption points, 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 RTAforLimitedPreemptiveFPModelwithArrivalCurves.

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,
a supply-bound function to account for 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, an arrival curve max_arrivals, and a list of preemption points task_preemption_points, ...

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

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

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

We assume that jobs are limited-preemptive.

#[local] Existing Instance limited_preemptive_job_model.

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.

We assume a model with fixed preemption points. I.e., each task is divided into a number of non-preemptive segments by inserting statically predefined preemption points.

Hypothesis H_valid_model_with_fixed_preemption_points :
valid_fixed_preemption_points_model arr_seq ts.

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 limited preemptions and 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.
  Hypothesis H_schedule_with_limited_preemptions :
    schedule_respects_preemption_model 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 ).

Workload Abbreviations

For brevity in the following definitions, we introduce a number of local abbreviations.

We let rbf denote 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.

Additionally, we let total_hep_rbf denote the cumulative request-bound function w.r.t. all tasks with higher-or-equal priority ...

Let total_hep_rbf := total_hep_request_bound_function_FP ts tsk.

... and use total_ohep_rbf as an abbreviation for the cumulative request-bound function w.r.t. all tasks with higher-or-equal priority other than task tsk itself.

Let total_ohep_rbf := total_ohep_request_bound_function_FP ts tsk.

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) blocking_bound ts tsk + total_hep_rbf L ≤ SBF tsk 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
          + blocking_bound ts tsk
          + total_hep_rbf 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 R.

  Definition rta_recurrence_solution L R :=
    ∀ (A : duration),
      is_in_search_space tsk L A →
      ∃ (F : duration ),
        A ≤ F ≤ A + R
        ∧ F ≥ overhead_bound F
              + blocking_bound ts tsk
              + (rbf tsk (A + ε) - (task_last_nonpr_segment tsk - ε ))
              + total_ohep_rbf F
        ∧ A + R ≥ F + (overhead_bound (A + R) - overhead_bound F )
                  + (task_last_nonpr_segment tsk - ε ).

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 FP scheduling with limited preemptions on a unit-speed uniprocessor subject to scheduling overheads.

  Theorem uniprocessor_response_time_bound_limited_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 VAL1 : valid_preemption_model arr_seq sched
      by apply valid_fixed_preemption_points_model_lemma, H_valid_model_with_fixed_preemption_points.
    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.
    - by exact: instantiated_interference_and_workload_consistent_with_sequential_tasks ⇒ //.
    - eapply busy_intervals_are_bounded_rs_fp with (SBF := sSBF); try done.
      + by eapply instantiated_i_and_w_are_coherent_with_schedule.
      + by apply bound_preserved_under_slowed; unfold fp_blackout_bound, overhead_bound, total_hep_rbf in *; lia.
    - apply: valid_pred_sbf_switch_predicate; last (eapply overheads_sbf_busy_valid) ⇒ //=.
      by move ⇒ ? ? ? ? [? ?]; split ⇒ //; 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.
        unshelve rewrite (leqRW (nonpreemptive_segments_bounded_by_blocking _ _ _ _ _ _ _ _ _)) ⇒ //.
        by instantiate (1 := fun _ ⇒ blocking_bound ts tsk).
    - move ⇒ A SP; move: (SOL A) ⇒ [].
      + apply: search_space_sub ⇒ //=.
        by apply: non_pathological_max_arrivals =>//; apply H_valid_task_arrival_sequence.
      + move ⇒ F [/andP [_ LE] [FIX1 FIX2]].
        have [δ [LEδ EQ]]:= slowed_subtraction_value_preservation
                              (fp_blackout_bound ts DB CSB CRPDB tsk) F (ltac:(apply fp_blackout_bound_monotone ⇒ //)).
        ∃ δ; split; [lia | split].
        × rewrite /sSBF /fp_ovh_sbf_slow -EQ.
          apply: leq_trans; last by apply leq_subRL_impl; rewrite -!addnA in FIX1; apply FIX1.
          have NEQ: total_ohep_request_bound_function_FP ts tsk δ ≤ total_ohep_rbf F by apply total_ohep_rbf_monotone ⇒ //.
          erewrite last_segment_eq_cost_minus_rtct; [ | eauto | eauto ].
          by move: FIX1; rewrite /task_intra_IBF; set (c := _ _ (A +1) - ( _ )); unfold total_ohep_rbf in *; lia.
        × rewrite /sSBF /fp_ovh_sbf_slow -EQ.
          apply bound_preserved_under_slowed, leq_subRL_impl; apply: leq_trans; last by apply FIX2.
          erewrite last_segment_eq_cost_minus_rtct; [ | eauto | eauto ].
          rewrite /task_rtct /constant /fp_blackout_bound /overhead_bound.
          unfold overhead_bound in *; lia.
  Qed.

End RTAforLimitedPreemptiveFPModelwithArrivalCurves.