Library rt.analysis.uni.jitter.workload_bound_fp

Require Import rt.util.all.
Require Import rt.model.arrival.basic.task rt.model.priority.
Require Import rt.model.arrival.jitter.job
               rt.model.arrival.jitter.task_arrival
               rt.model.arrival.jitter.arrival_bounds.
Require Import rt.model.schedule.uni.workload.
Require Import rt.model.schedule.uni.jitter.schedule.

(* In this file, we define the workload bound for jitter-aware FP scheduling. *)
Module WorkloadBoundFP.

  Import JobWithJitter SporadicTaskset UniprocessorScheduleWithJitter Priority Workload
         TaskArrivalWithJitter ArrivalBounds.

  (* First, we define a bound for the workload of a single task. *)
  Section SingleTask.

    Context {Task: eqType}.
    Variable task_cost: Task time.
    Variable task_period: Task time.
    Variable task_jitter: Task time.

    (* Consider any task tsk to be scheduled in a given interval of length delta. *)
    Variable tsk: Task.
    Variable delta: time.

    (* Based on the maximum number of jobs of tsk that can execute in the interval, ... *)
    Definition max_jobs := div_ceil (delta + task_jitter tsk) (task_period tsk).

    (* ... we define the following workload bound for the task. *)
    Definition task_workload_bound_FP := max_jobs × task_cost tsk.

  End SingleTask.

  (* In this section, we define a bound for the workload all tasks with
     higher or equal priority. *)

  Section AllTasks.

    Context {Task: eqType}.
    Variable task_cost: Task time.
    Variable task_period: Task time.
    Variable task_jitter: Task time.

    (* Assume any FP policy. *)
    Variable higher_eq_priority: FP_policy Task.

    (* Consider a task set ts... *)
    Variable ts: list Task.

    (* ...and let tsk be the task to be analyzed. *)
    Variable tsk: Task.

    (* Let delta be the length of the interval of interest. *)
    Variable delta: time.

    (* Recall the definition of higher-or-equal-priority task and
       the per-task workload bound for FP scheduling (defined above). *)

    Let is_hep_task tsk_other := higher_eq_priority tsk_other tsk.
    Let W tsk_other :=
      task_workload_bound_FP task_cost task_period task_jitter tsk_other delta.

    (* Using the sum of individual workload bounds, we define the following bound
       for the total workload of tasks of higher-or-equal priority (with respect
       to tsk) in any interval of length delta. *)

    Definition total_workload_bound_fp :=
      \sum_(tsk_other <- ts | is_hep_task tsk_other) W tsk_other.

  End AllTasks.

  (* In this section, we prove some basic lemmas about the workload bound. *)
  Section BasicLemmas.

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

    (* Assume any FP policy. *)
    Variable higher_eq_priority: FP_policy Task.

    (* Consider a task set ts... *)
    Variable ts: list Task.

    (* ...and let tsk be any task in ts. *)
    Variable tsk: Task.
    Hypothesis H_tsk_in_ts: tsk \in ts.

    (* Recall the workload bound for uniprocessor FP scheduling. *)
    Let workload_bound :=
      total_workload_bound_fp task_cost task_period task_jitter higher_eq_priority ts tsk.

    (* In this section we prove that the workload bound in a time window of
       length (task_cost tsk) is as large as (task_cost tsk) time units.
       (This is an important initial condition for the fixed-point iteration.) *)

    Section NoSmallerThanCost.

      (* Assume that the priority order is reflexive... *)
      Hypothesis H_priority_is_reflexive: FP_is_reflexive higher_eq_priority.

      (* ...and that the cost and period of task tsk are positive. *)
      Hypothesis H_cost_positive: task_cost tsk > 0.
      Hypothesis H_period_positive: task_period tsk > 0.

      (* Then, we prove that the workload bound in an interval of length (task_cost tsk)
         cannot be smaller than (task_cost tsk). *)

      Lemma total_workload_bound_fp_ge_cost:
        workload_bound (task_cost tsk) task_cost tsk.

    End NoSmallerThanCost.

    (* In this section, we prove that the workload bound is monotonically non-decreasing. *)
    Section NonDecreasing.

      (* Assume that the period of every task in the task set is positive. *)
      Hypothesis H_period_positive:
         tsk,
          tsk \in ts
          task_period tsk > 0.

      (* Then, the workload bound is a monotonically non-decreasing function.
         (This is an important property for the fixed-point iteration.) *)

      Lemma total_workload_bound_fp_non_decreasing:
         delta1 delta2,
          delta1 delta2
          workload_bound delta1 workload_bound delta2.

    End NonDecreasing.

  End BasicLemmas.

  (* In this section, we prove that any fixed point R = workload_bound R
     is indeed a workload bound for any interval of length R. *)

  Section ProofWorkloadBound.

    Context {Task: eqType}.
    Variable task_cost: Task time.
    Variable task_period: Task time.
    Variable task_jitter: Task time.

    Context {Job: eqType}.
    Variable job_arrival: Job time.
    Variable job_cost: Job time.
    Variable job_jitter: Job time.
    Variable job_task: Job Task.

    (* Let ts be any task set... *)
    Variable ts: seq Task.

    (* ...with positive task periods. *)
    Hypothesis H_positive_periods:
       tsk, tsk \in ts task_period tsk > 0.

    (* Consider any job arrival sequence with consistent, duplicate-free arrivals. *)
    Variable arr_seq: arrival_sequence Job.
    Hypothesis H_arrival_times_are_consistent: arrival_times_are_consistent job_arrival arr_seq.
    Hypothesis H_arrival_sequence_is_a_set: arrival_sequence_is_a_set arr_seq.

    (* First, let's define some local names for clarity. *)
    Let actual_arrivals := actual_arrivals_between job_arrival job_jitter arr_seq.

    (* Next, assume that all jobs come from the task set, ...*)
    Hypothesis H_all_jobs_from_taskset:
       j, arrives_in arr_seq j job_task j \in ts.

    (* ...the cost of each job is bounded by the cost of its task, ... *)
    Hypothesis H_job_cost_le_task_cost:
       j,
        arrives_in arr_seq j
        job_cost j task_cost (job_task j).

    (* ...and the jitter of each job is bounded by the jitter of its task. *)
    Hypothesis H_job_jitter_le_task_jitter:
       j,
        arrives_in arr_seq j
        job_jitter j task_jitter (job_task j).

    (* Assume that jobs arrived sporadically. *)
    Hypothesis H_sporadic_arrivals:
      sporadic_task_model task_period job_arrival job_task arr_seq.

    (* Then, let tsk be any task in ts. *)
    Variable tsk: Task.
    Hypothesis H_tsk_in_ts: tsk \in ts.

    (* For a given fixed-priority policy, ... *)
    Variable higher_eq_priority: FP_policy Task.

    (* ...we recall the definitions of higher-or-equal-priority workload and the workload bound. *)
    Let actual_hp_workload t1 t2 :=
      workload_of_higher_or_equal_priority_tasks job_cost job_task (actual_arrivals t1 t2)
                                                 higher_eq_priority tsk.
    Let workload_bound :=
      total_workload_bound_fp task_cost task_period task_jitter higher_eq_priority ts tsk.

    (* Next, consider any R that is a fixed point of the following equation,
       i.e., the claimed workload bound is equal to the interval length. *)

    Variable R: time.
    Hypothesis H_fixed_point: R = workload_bound R.

    (* Then, we prove that R bounds the workload of jobs of higher-or-equal-priority tasks
       with actual arrival time (including jitter) in [t, t + R). *)

    Lemma fp_workload_bound_holds:
       t,
        actual_hp_workload t (t + R) R.

  End ProofWorkloadBound.

End WorkloadBoundFP.