Library prosa.results.rta.arm.edf.limited_preemptive

RTA for EDF Scheduling with Fixed Preemption Points on Uniprocessors under the Average Resource Model

In the following, we derive a response-time analysis for EDF schedulers, assuming a workload of sporadic real-time tasks with fixed preemption points, characterized by arbitrary arrival curves, executing upon a uniprocessor under the average resource model (inspired by the paper "Periodic Resource Model for Compositional Real-Time Guarantees" by Shin & Lee, RTSS 2003). 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 RTAforLimitedPreemptiveEDFModelwithArrivalCurves.

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:

tasks, jobs, and their parameters,
the task set and the task under analysis,
the processor model,
the sequence of job arrivals,
the absence of self-suspensions,
an arbitrary schedule of the task set, and finally,
the resource-supply model.

Tasks and Jobs

Consider tasks characterized by a WCET task_cost, relative deadline task_deadline, an arrival curve max_arrivals, and a list of preemption points task_preemption_points, ...

Context {Task : TaskType} `{TaskCost Task} `{TaskDeadline 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 job's 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.

Processor Model

Consider any kind of fully-supply-consuming, unit-supply processor state model.

  Context `{PState : ProcessorState Job}.
  Hypothesis H_uniprocessor_proc_model : uniprocessor_model PState.
  Hypothesis H_unit_supply_proc_model : unit_supply_proc_model PState.
  Hypothesis H_consumed_supply_proc_model : fully_consuming_proc_model PState.

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.

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

  Variable sched : schedule PState.
  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 EDF scheduling policy.

Hypothesis H_respects_policy :
respects_JLFP_policy_at_preemption_point arr_seq sched (EDF Job).

Average Resource Model

Assume that the processor supply follows the *average resource model*. Under this model, for any interval [t1, t2), and given a resource period Π, a resource allocation time Θ, and a supply delay ν, the processor provides at least (t2 - t1 - ν) ⋅ Θ / Π units of supply. Intuitively, this means that on _average, the processor delivers Θ units of output every Π units of time, while the distribution of supply is not ideal and is subject to fluctuations bounded by ν. Furthermore, let arm_sbf Π Θ ν denote the SBF, which, as proven in prosa.analysis.facts.model.sbf.average, is a valid SBF.

Variables Π Θ ν : duration.
Hypothesis H_average_resource_model : average_resource_model Π Θ ν sched.

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 "recurrences" (i.e., set of inequalities) arm_sbf Π Θ ν L ≥ total_request_bound_function ts L and arm_sbf Π Θ ν L ≥ longest_busy_interval_with_pi ts tsk, as defined below.

As the lemma busy_intervals_are_bounded_rs_edf shows, under EDF 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
    ∧ arm_sbf Π Θ ν L ≥ total_request_bound_function ts L
    ∧ arm_sbf Π Θ ν L ≥ longest_busy_interval_with_pi ts tsk.

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 ts tsk L A →
      ∃ (F : duration ),
        arm_sbf Π Θ ν F ≥ blocking_bound ts tsk A
                          + (task_request_bound_function tsk (A + ε) - (task_last_nonpr_segment tsk - ε ))
                          + bound_on_athep_workload ts tsk A F
        ∧ arm_sbf Π Θ ν (A + R) ≥ arm_sbf Π Θ ν F + (task_last_nonpr_segment tsk - ε )
        ∧ 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 EDF scheduling with limited preemptions on a unit-speed uniprocessor under the average resource model.

  Theorem uniprocessor_response_time_bound_limited_edf :
    ∀ (L : duration),
      busy_window_recurrence_solution L →
      ∀ (R : duration),
        rta_recurrence_solution L R →
        task_response_time_bound arr_seq sched tsk R.
  Proof.
    move⇒ L [BW_POS [BW_SOL1 BW_SOL2]] R SOL js ARRs TSKs.
    have [ZERO|POS] := posnP (job_cost js); first by rewrite /job_response_time_bound /completed_by ZERO.
    have VAL1 : valid_preemption_model arr_seq sched
      by apply valid_fixed_preemption_points_model_lemma, H_valid_model_with_fixed_preemption_points.
    have VBSBF : valid_busy_sbf arr_seq sched tsk (arm_sbf Π Θ ν).
    { by apply: valid_pred_sbf_switch_predicate; last apply arm_sbf_valid. }
    have US : unit_supply_bound_function (arm_sbf Π Θ ν) by exact: arm_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 := arm_sbf Π Θ ν) ⇒ //.
    - exact: instantiated_i_and_w_are_coherent_with_schedule.
    - exact: EDF_implies_sequential_tasks.
    - exact: instantiated_interference_and_workload_consistent_with_sequential_tasks.
    - apply: busy_intervals_are_bounded_rs_edf ⇒ //.
      by apply: instantiated_i_and_w_are_coherent_with_schedule.
    - apply: valid_pred_sbf_switch_predicate; last first.
      + by apply arm_sbf_valid.
      + by move ⇒ ? ? ? ? [? ?]; split ⇒ //.
    - apply: instantiated_task_intra_interference_is_bounded; eauto 1 ⇒ //; first last.
      + by (apply: bound_on_athep_workload_is_valid; try apply H_fixed_point) ⇒ //.
      + apply: service_inversion_is_bounded ⇒ // ⇒ jo t1 t2 ARRo TSKo BUSYo.
        by apply: 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.
      + move ⇒ F [FIX1 [FIX2 FIX3]]; ∃ F; split ⇒ //; split.
        × erewrite last_segment_eq_cost_minus_rtct; [ | eauto | eauto ].
          by rewrite /task_intra_IBF; lia.
        × by erewrite last_segment_eq_cost_minus_rtct; [ | eauto | eauto ]; lia.
  Qed.

End RTAforLimitedPreemptiveEDFModelwithArrivalCurves.