Library prosa.analysis.abstract.restricted_supply.fp_bounded_bi

 Require Export prosa.analysis.facts.blocking_bound_fp.
Require Export prosa.analysis.abstract.restricted_supply.abstract_rta.
Require Export prosa.analysis.abstract.restricted_supply.iw_instantiation.
Require Export prosa.analysis.definitions.sbf.busy.

Sufficient Condition for Bounded Busy Intervals for RS FP

In this section, we show that the existence of L such that B + total_hep_rbf L ≤ SBF L, where B is the blocking bound and SBF is a supply-bound function, is a sufficient condition for the existence of bounded busy intervals under FP scheduling with a restricted-supply processor model.
Section BoundedBusyIntervals.

Consider any type of tasks ...
  Context {Task : TaskType}.
  Context `{TaskCost Task}.
  Context `{TaskMaxNonpreemptiveSegment Task}.

... and any type of jobs associated with these tasks.
  Context {Job : JobType}.
  Context `{JobTask Job Task}.
  Context `{JobArrival Job}.
  Context `{JobCost Job}.

Consider any kind of fully supply-consuming unit-supply uniprocessor 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.

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.

Consider any valid arrival sequence.
  Variable arr_seq : arrival_sequence Job.
  Hypothesis H_valid_arrival_sequence : valid_arrival_sequence arr_seq.

Next, consider a schedule of this arrival sequence, ...
  Variable sched : schedule PState.

... allow for any work-bearing notion of job readiness, ...
  Context `{!JobReady Job PState}.
  Hypothesis H_job_ready : work_bearing_readiness arr_seq sched.

... and assume that the schedule is valid.
  Hypothesis H_sched_valid : valid_schedule sched arr_seq.

In addition, we assume the existence of a function mapping jobs to their preemption points ...
  Context `{JobPreemptable Job}.

... and assume that it defines a valid preemption model with bounded non-preemptive segments.
  Hypothesis H_valid_model_with_bounded_nonpreemptive_segments :
    valid_model_with_bounded_nonpreemptive_segments arr_seq sched.

Furthermore, we assume that the schedule is work-conserving ...
  Hypothesis H_work_conserving : work_conserving arr_seq sched.

... and that it respects the scheduling policy.
  Hypothesis H_respects_policy : respects_FP_policy_at_preemption_point arr_seq sched FP.

Recall that busy_intervals_are_bounded_by is an abstract notion. Hence, we need to introduce interference and interfering workload. We will use the restricted-supply instantiations.
We say that job j incurs interference at time t iff it cannot execute due to (1) the lack of supply at time t, (2) service inversion (i.e., a lower-priority job receiving service at t), or a higher-or-equal-priority job receiving service.
  #[local] Instance rs_jlfp_interference : Interference Job :=
    rs_jlfp_interference arr_seq sched.

The interfering workload, in turn, is defined as the sum of the blackout predicate, service inversion predicate, and the interfering workload of jobs with higher or equal priority.
  #[local] Instance rs_jlfp_interfering_workload : InterferingWorkload Job :=
    rs_jlfp_interfering_workload arr_seq sched.

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 that the cost of a job does not exceed its task's WCET.
  Hypothesis H_valid_job_cost : arrivals_have_valid_job_costs arr_seq.

Consider a valid, unit supply-bound function "busy" SBF. That is, (1) SBF 0 = 0, (2) for any duration Δ, the supply produced during a busy-interval prefix of length Δ is at least SBF Δ, and (3) SBF makes steps of at most one.
  Context {SBF : SupplyBoundFunction}.
  Hypothesis H_valid_SBF : valid_busy_sbf arr_seq sched SBF.
  Hypothesis H_unit_SBF : unit_supply_bound_function SBF.

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.

Consider a valid preemption model.
  Hypothesis H_valid_preemption_model :
    valid_preemption_model arr_seq sched.

Let L be any positive fixed point of the busy-interval recurrence.
  Variable L : duration.
  Hypothesis H_L_positive : 0 < L.
  Hypothesis H_fixed_point :
    blocking_bound ts tsk + total_hep_request_bound_function_FP ts tsk L ≤ SBF L.

Next, we provide a step-by-step proof of busy-interval boundedness.
  Section StepByStepProof.

Consider any job j of task tsk that has a positive job cost and is in the arrival sequence.
    Variable j : Job.
    Hypothesis H_j_arrives : arrives_in arr_seq j.
    Hypothesis H_job_of_tsk : job_of_task tsk j.
    Hypothesis H_job_cost_positive : job_cost_positive j.

First, we note that since job j is pending at time job_arrival j, there is a (potentially unbounded) busy interval that starts no later than with the arrival of j.
    Local Lemma busy_interval_prefix_exists :
      ∃ t1,
        t1 ≤ job_arrival j
        ∧ busy_interval_prefix arr_seq sched j t1 (job_arrival j).+1.

Next, consider two cases: (1) when the busy-interval prefix continues until time instant t1 + L and (2) when the busy interval prefix terminates earlier. In either case, we can show that the busy-interval prefix is bounded.
We start with the first case, where the busy-interval prefix continues until time instant t1 + L.
    Section Case1.

Consider a time instant t1 such that [t1, job_arrival j]>> and [t1, t1 + L) are both busy-interval prefixes of job j.
Note that at this point we do not necessarily know that job_arrival j ≤ L; hence, we assume the existence of both prefixes.
      Variable t1 : instant.
      Hypothesis H_busy_prefix_arr : busy_interval_prefix arr_seq sched j t1 (job_arrival j).+1.
      Hypothesis H_busy_prefix_L : busy_interval_prefix arr_seq sched j t1 (t1 + L).

The crucial point to note is that the sum of the job's cost (represented as workload_of_job) and the interfering workload in the interval [t1, t1 + L) is bounded by L due to the fixed point H_fixed_point.
      Local Lemma workload_is_bounded :
        workload_of_job arr_seq j t1 (t1 + L) + cumulative_interfering_workload j t1 (t1 + L) ≤ L.

It follows that t1 + L is a quiet time, which means that the busy prefix ends (i.e., it is bounded).
      Local Lemma busy_prefix_is_bounded_case1 :
        ∃ t2,
          job_arrival j < t2
          ∧ t2 ≤ t1 + L
          ∧ busy_interval arr_seq sched j t1 t2.

    End Case1.

Next, we consider the case when the interval [t1, t1 + L) is not a busy-interval prefix.
    Section Case2.

First, we prove a few helper lemmas.
      Section HelperLemmas.

Let <t1, t2)>> be a busy-interval prefix.
        Variable t1 t2 : instant.
        Hypothesis H_busy_prefix : busy_interval_prefix arr_seq sched j t1 t2.

Then, the service of higher-or-equal-priority jobs is less than the workload of higher-or-equal-priority jobs in any subinterval [t1, t) of the interval [t1, t2).
        Local Lemma service_lt_workload_in_busy :
          ∀ t,
            t1 < t < t2 →
            service_of_hep_jobs arr_seq sched j t1 t < workload_of_hep_jobs arr_seq j t1 t.

Consider a subinterval [t1, t1 + Δ) of the interval [t1, t2). We show that the sum of j's workload and the cumulative workload in [t1, t1 + Δ) is strictly larger than Δ.
        Local Lemma workload_exceeds_interval :
          ∀ Δ,
            0 < Δ →
            t1 + Δ < t2 →
            Δ < workload_of_job arr_seq j t1 (t1 + Δ) + cumulative_interfering_workload j t1 (t1 + Δ).

      End HelperLemmas.

Consider a time instant t1 such that [t1, job_arrival j]>> is a busy-interval prefix of j and [t1, t1 + L) is not.
      Variable t1 : instant.
      Hypothesis H_arrives : t1 ≤ job_arrival j.
      Hypothesis H_busy_prefix_arr : busy_interval_prefix arr_seq sched j t1 (job_arrival j).+1.
      Hypothesis H_no_busy_prefix_L : ¬ busy_interval_prefix arr_seq sched j t1 (t1 + L).

From the properties of busy intervals, one can conclude that j's arrival time is less than t1 + L.
      Lemma job_arrival_is_bounded :
        job_arrival j < t1 + L.
Lemma job_arrival_is_bounded implies that the busy-interval prefix starts at time t1, continues until job_arrival j + 1, and then terminates before t1 + L. Or, in other words, there is point in time t2 such that (1) j's arrival is bounded by t2, (2) t2 is bounded by t1 + L, and (3) [t1, t2) is busy interval of job j.
      Lemma busy_prefix_is_bounded_case2:
        ∃ t2, job_arrival j < t2 ∧ t2 ≤ t1 + L ∧ busy_interval arr_seq sched j t1 t2.
    End Case2.

  End StepByStepProof.

Combining the cases analyzed above, we conclude that busy intervals of jobs released by task tsk are bounded by L.
  Lemma busy_intervals_are_bounded_rs_fp :
    busy_intervals_are_bounded_by arr_seq sched tsk L.

End BoundedBusyIntervals.