Library rt.analysis.jitter.bertogna_edf_theory

Require Import rt.util.all.
Require Import rt.model.jitter.job rt.model.jitter.task rt.model.jitter.task_arrival
               rt.model.jitter.schedule rt.model.jitter.platform rt.model.jitter.interference
               rt.model.jitter.workload rt.model.jitter.schedulability
               rt.model.jitter.priority rt.model.jitter.constrained_deadlines
Require Import rt.analysis.jitter.workload_bound

Module ResponseTimeAnalysisEDFJitter.

  Export JobWithJitter SporadicTaskset ScheduleOfSporadicTaskWithJitter Workload
         Schedulability ResponseTime Priority SporadicTaskArrival WorkloadBoundJitter
         InterferenceBoundEDFJitter Platform Interference ConstrainedDeadlines.

  (* In this section, we prove that any fixed point in Bertogna and
     Cirinei's RTA for EDF scheduling modified to account for jitter
     yields a safe response-time bound. This is an extension of the
     analysis found in Chapter 17.1.2 of Baruah et al.'s book
     Multiprocessor Scheduling for Real-time Systems. *)

  Section ResponseTimeBound.

    Context {sporadic_task: eqType}.
    Variable task_cost: sporadic_task time.
    Variable task_period: sporadic_task time.
    Variable task_deadline: sporadic_task time.
    Variable task_jitter: sporadic_task time.

    Context {Job: eqType}.
    Variable job_cost: Job time.
    Variable job_deadline: Job time.
    Variable job_task: Job sporadic_task.
    Variable job_jitter: Job time.

    (* Assume any job arrival sequence... *)
    Context {arr_seq: arrival_sequence Job}.

    (* ... in which jobs arrive sporadically and have valid parameters.
       Note: the jitter of a valid job is bounded by the jitter of its task. *)

    Hypothesis H_sporadic_tasks:
      sporadic_task_model task_period arr_seq job_task.
    Hypothesis H_valid_job_parameters:
       (j: JobIn arr_seq),
        valid_sporadic_job_with_jitter task_cost task_deadline task_jitter job_cost
                                                 job_deadline job_task job_jitter j.

    (* Assume that we have a task set where all tasks have valid
       parameters and constrained deadlines, ... *)

    Variable ts: taskset_of sporadic_task.
    Hypothesis H_valid_task_parameters:
      valid_sporadic_taskset task_cost task_period task_deadline ts.
    Hypothesis H_constrained_deadlines:
       tsk, tsk ts task_deadline tsk task_period tsk.

    (* ... and that all jobs in the arrival sequence come from the task set. *)
    Hypothesis H_all_jobs_from_taskset:
       (j: JobIn arr_seq), job_task j ts.

    (* Next, consider any schedule such that...*)
    Variable num_cpus: nat.
    Variable sched: schedule num_cpus arr_seq.

    (* are sequential, do not execute before the
       jitter has passed and nor longer than their execution costs. *)

    Hypothesis H_sequential_jobs: sequential_jobs sched.
    Hypothesis H_execute_after_jitter:
      jobs_execute_after_jitter job_jitter sched.
    Hypothesis H_completed_jobs_dont_execute:
      completed_jobs_dont_execute job_cost sched.

    (* Assume that there exists at least one processor. *)
    Hypothesis H_at_least_one_cpu: num_cpus > 0.

    (* Assume that the schedule is a work-conserving EDF schedule. *)
    Hypothesis H_work_conserving: work_conserving job_cost job_jitter sched.
    Hypothesis H_edf_policy: enforces_JLDP_policy job_cost job_jitter sched (EDF job_deadline).

    (* Let's define some local names to avoid passing many parameters. *)
    Let no_deadline_is_missed_by_tsk (tsk: sporadic_task) :=
      task_misses_no_deadline job_cost job_deadline job_task sched tsk.
    Let response_time_bounded_by (tsk: sporadic_task) :=
      is_response_time_bound_of_task job_cost job_task tsk sched.

    (* Next we consider the response-time recurrence.
       Assume that a response-time bound R is known...  *)

    Let task_with_response_time := (sporadic_task × time)%type.
    Variable rt_bounds: seq task_with_response_time.

    (* ...for any task in the task set, ... *)
    Hypothesis H_rt_bounds_contains_all_tasks: unzip1 rt_bounds = ts.

    (* Also, assume that R is a fixed-point of the response-time recurrence, ... *)
    Let I (tsk: sporadic_task) (delta: time) :=
      total_interference_bound_edf task_cost task_period task_deadline task_jitter tsk rt_bounds delta.
    Hypothesis H_response_time_is_fixed_point :
       tsk R,
        (tsk, R) rt_bounds
        R = task_cost tsk + div_floor (I tsk R) num_cpus.

    (* ..., and R is no larger than the deadline. *)
    Hypothesis H_tasks_miss_no_deadlines:
       tsk R,
        (tsk, R) rt_bounds
        task_jitter tsk + R task_deadline tsk.

    (* In order to prove that R is a response-time bound, we first provide some lemmas. *)
    Section Lemmas.

      (* Let (tsk, R) be any task to be analyzed, with its response-time bound R. *)
      Variable tsk: sporadic_task.
      Variable R: time.
      Hypothesis H_tsk_R_in_rt_bounds: (tsk, R) rt_bounds.

      (* Consider any job j of tsk. *)
      Variable j: JobIn arr_seq.
      Hypothesis H_job_of_tsk: job_task j = tsk.

      (* Let t1 be the first point in time where j can actually be scheduled. *)
      Let t1 := job_arrival j + job_jitter j.

      (* Assume that job j did not complete on time, ... *)
      Hypothesis H_j_not_completed: ¬ completed job_cost sched j (t1 + R).

      (* ...and that it is the first job not to satisfy its response-time bound. *)
      Hypothesis H_all_previous_jobs_completed_on_time :
         (j_other: JobIn arr_seq) tsk_other R_other,
          job_task j_other = tsk_other
          (tsk_other, R_other) rt_bounds
          job_arrival j_other + task_jitter tsk_other + R_other < job_arrival j + task_jitter tsk + R
          completed job_cost sched j_other (job_arrival j_other + task_jitter tsk_other + R_other).

      (* Let's call x the interference incurred by job j due to tsk_other, ...*)
      Let x (tsk_other: sporadic_task) :=
        task_interference job_cost job_task job_jitter sched j tsk_other t1 (t1 + R).

      (* and X the total interference incurred by job j due to any task. *)
      Let X := total_interference job_cost job_jitter sched j t1 (t1 + R).

      (* Recall Bertogna and Cirinei's workload bound ... *)
      Let workload_bound (tsk_other: sporadic_task) (R_other: time) :=
        W_jitter task_cost task_period task_jitter tsk_other R_other R.

      (*... and the EDF-specific bound, ... *)
      Let edf_specific_bound (tsk_other: sporadic_task) (R_other: time) :=
        edf_specific_interference_bound task_cost task_period task_deadline task_jitter tsk tsk_other R_other.

      (* ... which combined form the interference bound. *)
      Let interference_bound (tsk_other: sporadic_task) (R_other: time) :=
        interference_bound_edf task_cost task_period task_deadline task_jitter tsk R (tsk_other, R_other).

      (* Based on the definition of a different task, ... *)
      Let other_task := different_task tsk.

      (* ...let other_tasks denote the set of tasks that are different from tsk. *)
      Let other_tasks :=
        [seq tsk_other <- ts | other_task tsk_other].

      (* Now we establish results the interfering tasks. *)
      Section LemmasAboutInterferingTasks.

        (* Let (tsk_other, R_other) be any pair of higher-priority task and
           response-time bound computed in previous iterations. *)

        Variable tsk_other: sporadic_task.
        Variable R_other: time.
        Hypothesis H_response_time_of_tsk_other: (tsk_other, R_other) rt_bounds.

        (* Note that tsk_other is in task set ts ...*)
        Lemma bertogna_edf_tsk_other_in_ts: tsk_other ts.

        (* Also, R_other is larger than the cost of tsk_other. *)
        Lemma bertogna_edf_R_other_ge_cost :
          R_other task_cost tsk_other.

        (* Since tsk_other cannot interfere more than it executes, we show that
           the interference caused by tsk_other is no larger than workload bound W. *)

        Lemma bertogna_edf_workload_bounds_interference :
          x tsk_other workload_bound tsk_other R_other.

        (* Recall that the edf-specific interference bound also holds. *)
        Lemma bertogna_edf_specific_bound_holds :
          x tsk_other edf_specific_bound tsk_other R_other.

      End LemmasAboutInterferingTasks.

      (* Next we prove some lemmas that help to derive a contradiction.*)
      Section DerivingContradiction.

        (* 0) Since job j did not complete by its response time bound, it follows that
              the total interference X >= R - e_k + 1. *)

        Lemma bertogna_edf_too_much_interference : X R - task_cost tsk + 1.

      (* 1) Next, we prove that during the scheduling window of j, any job that is
            scheduled while j is backlogged comes from a different task.
            This follows from the fact that j is the first job not to complete
            by its response-time bound, so previous jobs of j's task must have
            completed by their periods and cannot be pending. *)

        Lemma bertogna_edf_interference_by_different_tasks :
           t j_other,
            t1 t < t1 + R
            backlogged job_cost job_jitter sched j t
            scheduled sched j_other t
            job_task j_other tsk.

      (* 2) In order to use the lemmas in constrained_deadlines.v, we show that
            all jobs released before the end of the interval complete by their
            periods. This follows trivially from the hypothesis that all jobs
            before (t1 + R) complete by their response-time bounds. 
            With this lemma, we can conclude that during job j's scheduling
            window there cannot be multiple pending jobs of each task.*)

        Lemma bertogna_edf_all_previous_jobs_complete_by_their_period:
           t (j0: JobIn arr_seq),
            t < t1 + R
            job_arrival j0 + task_period (job_task j0) t
            completed job_cost sched j0
               (job_arrival j0 + task_period (job_task j0)).

        (* Let's define a predicate to identify the other tasks that are scheduled. *)
        Let other_scheduled_task (t: time) (tsk_other: sporadic_task) :=
          task_is_scheduled job_task sched tsk_other t
          other_task tsk_other.

        (* 3) Now we prove that, at all times that j is backlogged, the number
              of tasks other than tsk that are scheduled is exactly the number
              of processors in the system. This is required to prove lemma (4). *)

        Lemma bertogna_edf_all_cpus_are_busy:
            t1 t < t1 + R
            backlogged job_cost job_jitter sched j t
            count (other_scheduled_task t) ts = num_cpus.

      (* 4) Next, we prove that the sum of the interference of each task is equal
          to the total interference multiplied by the number of processors. This
          holds because interference only occurs when all processors are busy.
          With this lemma we can relate per-task interference with the total
          interference incurred by j (backlogged time). *)

        Lemma bertogna_edf_interference_on_all_cpus :
          \sum_(tsk_k <- other_tasks) x tsk_k = X × num_cpus.

        (* Before stating the next lemma, let (num_tasks_exceeding delta) be the
           number of interfering tasks whose interference x is larger than delta. *)

        Let num_tasks_exceeding delta := count (fun ix i delta) (other_tasks).

        (* 5) Now we prove that, for any delta, if (num_task_exceeding delta > 0), then the
              cumulative interference caused by the complementary set of interfering tasks fills
              the remaining, not-completely-full (num_cpus - num_tasks_exceeding delta)
              processors. *)

        Lemma bertogna_edf_interference_in_non_full_processors :
            0 < num_tasks_exceeding delta < num_cpus
            \sum_(i <- other_tasks | x i < delta) x i delta × (num_cpus - num_tasks_exceeding delta).

        (* 6) Based on lemma (5), we prove that, for any interval delta, if the sum of per-task
              interference exceeds (delta * num_cpus), the same applies for the
              sum of the minimum of the interference and delta. *)

        Lemma bertogna_edf_minimum_exceeds_interference :
            \sum_(tsk_k <- other_tasks) x tsk_k delta × num_cpus
               \sum_(tsk_k <- other_tasks) minn (x tsk_k) delta
               delta × num_cpus.

        (* 7) Next, using lemmas (0), (4) and (6) we prove that the reduction-based
              interference bound is not enough to cover the sum of the minima over all tasks
              (artifact of the proof by contradiction). *)

        Lemma bertogna_edf_sum_exceeds_total_interference:
          \sum_((tsk_other, R_other) <- rt_bounds | other_task tsk_other)
            minn (x tsk_other) (R - task_cost tsk + 1) > I tsk R.

        (* 8) After concluding that the sum of the minima exceeds (R - e_i + 1),
              we prove that there exists a tuple (tsk_k, R_k) that satisfies
              min (x_k, R - e_i + 1) > min (W_k, I_edf, R - e_i + 1).
              This implies that either x_k > W_k or x_k > I_edf, which is a contradiction,
              since both W_k and I_edf are valid task interference bounds. *)

        Lemma bertogna_edf_exists_task_that_exceeds_bound :
           tsk_other R_other,
            (tsk_other, R_other) rt_bounds
            (minn (x tsk_other) (R - task_cost tsk + 1) > interference_bound tsk_other R_other).

      End DerivingContradiction.

    End Lemmas.

    Section MainProof.

      (* Let (tsk, R) be any task to be analyzed, with its response-time bound R. *)
      Variable tsk: sporadic_task.
      Variable R: time.
      Hypothesis H_tsk_R_in_rt_bounds: (tsk, R) rt_bounds.

      (* Using the lemmas above, we prove that R bounds the response time of task tsk. *)
      Theorem bertogna_cirinei_response_time_bound_edf :
        response_time_bounded_by tsk (task_jitter tsk + R).

    End MainProof.

  End ResponseTimeBound.

End ResponseTimeAnalysisEDFJitter.