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Require Export prosa.model.aggregate.workload.

(** * Interference and Interfering Workload *)

(** In the following, we introduce definitions of interference,
    interfering workload and a function that bounds cumulative
    interference. *)
Section Definitions.
  (** Consider any type of tasks ... *)
  Context {Task : TaskType}.

  (** ... and any type of jobs associated with these tasks. *)
  Context {Job : JobType} {jt : JobTask Job Task} {jc : JobCost Job}.

  (** Consider any kind of processor state model. *)
  Context {PState : ProcessorState Job}.

  (** Consider any arrival sequence ... *)
  Variable arr_seq : arrival_sequence Job.

  (** ... and any schedule. *)
  Variable sched : schedule PState.

  (** Definitions of interference for FP policies. *)
  Section FPDefinitions.
    Context {FP : FP_policy Task}.

    (** We first define interference from higher-priority tasks. *)
    Definition hp_task_interference (j : Job) (t : instant) :=
      has (fun jhp => hp_task (job_task jhp) (job_task j)
                      && receives_service_at sched jhp t)
        (arrivals_up_to arr_seq t).

    Context {JLFP : JLFP_policy Job}.

    (** Then, to define interference from equal-priority tasks, we first
        define higher-or-equal-priority jobs from an equal-priority task... *)
    Definition ep_task_hep_job j1 j2 :=
      hep_job j1 j2 && ep_task (job_task j1) (job_task j2).

    (** ... and higher-or-equal-priority-jobs of equal-priority tasks that
        do not stem from the same task. *)
    Definition other_ep_task_hep_job j1 j2 :=
      ep_task_hep_job j1 j2 && (job_task j1 != job_task j2).

    (** This enables us to define interference from equal-priority tasks. *)
    Definition hep_job_from_other_ep_task_interference (j : Job) (t : instant) :=
      has (other_ep_task_hep_job^~ j) (served_jobs_at arr_seq sched t).

    (** Similarly, to define interference from strictly higher-priority tasks, we first
        define higher-or-equal-priority jobs from a strictly higher-priority task, which... *)
    Definition hp_task_hep_job :=
      fun j1 j2 => hep_job j1 j2 && (hp_task (job_task j1) (job_task j2)).

    (** ... enables us to define interference from strictly higher-priority tasks. *)
    Definition hep_job_from_hp_task_interference (j : Job) (t : instant) :=
      has (hp_task_hep_job^~ j) (served_jobs_at arr_seq sched t).

    (** Using the above definitions, we define the cumulative interference incurred in the interval
        <<[t1, t2)>> from (1) higher-or-equal-priority jobs from strictly higher-priority tasks... *)
    Definition cumulative_interference_from_hep_jobs_from_hp_tasks (j : Job) (t1 t2 : instant) :=
      \sum_(t1 <= t < t2) hep_job_from_hp_task_interference j t.

    (** ... and (2) higher-or-equal-priority jobs from equal-priority tasks. *)
    Definition cumulative_interference_from_hep_jobs_from_other_ep_tasks (j : Job) (t1 t2 : instant) :=
      \sum_(t1 <= t < t2) hep_job_from_other_ep_task_interference j t.

  End FPDefinitions.

  (** Definitions of interference for JLFP policies. *)
  Section JLFPDefinitions.

    (** Consider a JLFP-policy that indicates a higher-or-equal priority
        relation. *)
    Context {JLFP : JLFP_policy Job}.

    (** We say that job [j] is incurring interference from another
        job with higher-or-equal priority at time [t] if there exists a
        job [jhp] (different from [j]) with a higher-or-equal priority
        that executes at time [t]. *)
    Definition another_hep_job_interference (j : Job) (t : instant) :=
      has (another_hep_job^~ j) (served_jobs_at arr_seq sched t).

    (** Similarly, we say that job [j] is incurring interference from a
        job with higher-or-equal priority of another task at time [t]
        if there exists a job [jhp] (of a different task) with
        higher-or-equal priority that executes at time [t]. *)
    Definition another_task_hep_job_interference (j : Job) (t : instant) :=
      has (another_task_hep_job^~ j) (served_jobs_at arr_seq sched t).

    (** Similarly, we say that job [j] is incurring interference from a
        job with higher-or-equal priority of the same task at time [t]
        if there exists another job [jhp] (of the same task) with
        higher-or-equal priority that executes at time [t]. *)
    Definition another_hep_job_of_same_task_interference (j : Job) (t : instant) :=
      has (fun jhp => another_hep_job_of_same_task jhp j
                      && receives_service_at sched jhp t)
        (arrivals_up_to arr_seq t).

    (** Now, we define the notion of cumulative interfering workload,
        called [other_hep_jobs_interfering_workload], that says how many
        units of workload are generated by jobs with higher-or-equal
        priority released at time [t]. *)
    Definition other_hep_jobs_interfering_workload (j : Job) (t : instant) :=
      \sum_(jhp <- arrivals_at arr_seq t | another_hep_job jhp j) job_cost jhp.

    (** For each of the concepts defined above, we introduce a
        corresponding cumulative function: *)

    (** (a) cumulative interference from other jobs with higher-or-equal priority ... *)
    Definition cumulative_another_hep_job_interference (j : Job) (t1 t2 : instant) :=
      \sum_(t1 <= t < t2) another_hep_job_interference j t.

    (** ... (b) and cumulative interference from jobs with higher or
        equal priority from other tasks, ... *)
    Definition cumulative_another_task_hep_job_interference (j : Job) (t1 t2 : instant) :=
      \sum_(t1 <= t < t2) another_task_hep_job_interference j t.

    (** ... and (c) cumulative workload from jobs with higher or equal priority. *)
    Definition cumulative_other_hep_jobs_interfering_workload (j : Job) (t1 t2 : instant) :=
      \sum_(t1 <= t < t2) other_hep_jobs_interfering_workload j t.

  End JLFPDefinitions.

End Definitions.