RTA for FP-schedulers with bounded nonpreemptive segments

In this module we prove a general RTA theorem for FP-schedulers
Module RTAforFPwithBoundedNonpreemptiveSegmentsWithArrivalCurves.

  Import Job ArrivalCurves TaskArrival Priority UniprocessorSchedule Workload Service
         ResponseTime MaxArrivalsWorkloadBound LimitedPreemptionPlatform RBF
         AbstractRTAforFPwithArrivalCurves BusyIntervalJLFP PriorityInversionIsBounded.

  Section Analysis.

    Context {Task: eqType}.
    Variable task_max_nps task_cost: Task time.

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

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

    (* Next, consider any uniprocessor schedule of this arrival sequence...*)
    Variable sched: schedule Job.
    Hypothesis H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence sched arr_seq.

    (* ... where jobs do not execute before their arrival nor after completion. *)
    Hypothesis H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute job_arrival sched.
    Hypothesis H_completed_jobs_dont_execute: completed_jobs_dont_execute job_cost sched.

    (* Assume we have sequential jobs, i.e, jobs from the same 
       task execute in the order of their arrival. *)

    Hypothesis H_sequential_jobs: sequential_jobs job_arrival job_cost sched job_task.

    (* Assume that a job cost cannot be larger than a task cost. *)
    Hypothesis H_job_cost_le_task_cost:
        task_cost job_cost job_task arr_seq.

    (* Consider an FP policy that indicates a higher-or-equal priority relation,
       and assume that the relation is reflexive and transitive. *)

    Variable higher_eq_priority: FP_policy Task.
    Hypothesis H_priority_is_reflexive: FP_is_reflexive higher_eq_priority.
    Hypothesis H_priority_is_transitive: FP_is_transitive higher_eq_priority.

    (* We consider an arbitrary function can_be_preempted which defines 
       a preemption model with bounded nonpreemptive segments. *)

    Variable can_be_preempted: Job time bool.
    Let preemption_time := preemption_time sched can_be_preempted.
    Hypothesis H_correct_preemption_model:
      correct_preemption_model arr_seq sched can_be_preempted.
    Hypothesis H_model_with_bounded_nonpreemptive_segments:
        job_cost job_task arr_seq can_be_preempted job_max_nps task_max_nps.

    (* Next, we assume that the schedule is a work-conserving schedule... *)
    Hypothesis H_work_conserving: work_conserving job_arrival job_cost arr_seq sched.

    (* ... and the schedule respects the policy defined by the 
       can_be_preempted function (i.e., bounded nonpreemptive segments). *)

    Hypothesis H_respects_policy:
        job_arrival job_cost job_task arr_seq sched can_be_preempted higher_eq_priority.

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

    (* ..and 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.

    (* Let tsk be any task in ts that is to be analyzed. *)
    Variable tsk: Task.
    Hypothesis H_tsk_in_ts: tsk \in ts.

    (* Let max_arrivals be a family of proper 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. *)

    Variable max_arrivals: Task time nat.
    Hypothesis H_family_of_proper_arrival_curves:
      family_of_proper_arrival_curves job_task arr_seq max_arrivals ts.

    (* Consider a proper job lock-in service and task lock-in functions, i.e... *)
    Variable job_lock_in_service: Job time.
    Variable task_lock_in_service: Task time.

    (* ...we assume that for all jobs in the arrival sequence the lock-in service is 
       (1) positive, (2) no bigger than costs of the corresponding jobs, and (3) a job
       becomes nonpreemptive after it reaches its lock-in service... *)

    Hypothesis H_proper_job_lock_in_service:
      proper_job_lock_in_service job_cost arr_seq sched job_lock_in_service.

    (* ...and that task_lock_in_service tsk is (1) no bigger than tsk's cost, (2) for any
       job of task tsk job_lock_in_service is bounded by task_lock_in_service. *)

    Hypothesis H_proper_task_lock_in_service:
        task_cost job_task arr_seq job_lock_in_service task_lock_in_service tsk.

    (* We also lift the FP priority relation to a corresponding JLFP priority relation. *)
    Let jlfp_higher_eq_priority := FP_to_JLFP job_task higher_eq_priority.

    (* Let's define some local names for clarity. *)
    Let job_pending_at := pending job_arrival job_cost sched.
    Let job_scheduled_at := scheduled_at sched.
    Let job_completed_by := completed_by job_cost sched.
    Let job_backlogged_at := backlogged job_arrival job_cost sched.
    Let arrivals_between := jobs_arrived_between arr_seq.
    Let max_length_of_priority_inversion :=
      max_length_of_priority_inversion job_max_nps arr_seq jlfp_higher_eq_priority.
    Let response_time_bounded_by :=
      is_response_time_bound_of_task job_arrival job_cost job_task arr_seq sched.
    Let task_rbf := task_request_bound_function task_cost max_arrivals tsk.
    Let total_hep_rbf :=
      total_hep_request_bound_function_FP task_cost higher_eq_priority max_arrivals ts tsk.
    Let total_ohep_rbf :=
      total_ohep_request_bound_function_FP task_cost higher_eq_priority max_arrivals ts tsk.

    (* We also define a bound for the priority inversion caused by jobs with lower priority. *)
    Definition blocking_bound :=
      \max_(tsk_other <- ts | ~~ higher_eq_priority tsk_other tsk)
       (task_max_nps tsk_other - ε).

Priority inversion is bounded

In this section, we prove that a priority inversion for task tsk is bounded by the maximum length of nonpreemtive segments among the tasks with lower priority.
    Section PriorityInversionIsBounded.

      (* First, we prove that the maximum length of a priority inversion of a job j is 
         bounded by the maximum length of a nonpreemptive section of a task with 
         lower-priority task (i.e., the blocking term). *)

      Lemma priority_inversion_is_bounded_by_blocking:
         j t,
          arrives_in arr_seq j
          job_task j = tsk
          max_length_of_priority_inversion j t blocking_bound.

      (* Using the above lemma, we prove that the priority inversion of the task is bounded by blocking_bound. *)
      Lemma priority_inversion_is_bounded:
          job_arrival job_cost job_task arr_seq sched jlfp_higher_eq_priority tsk blocking_bound.

    End PriorityInversionIsBounded.

Response-Time Bound

In this section, we prove that the maximum among the solutions of the response-time bound recurrence is a response-time bound for tsk.
    Section ResponseTimeBound.

      (* Let L be any positive fixed point of the busy interval recurrence. *)
      Variable L: time.
      Hypothesis H_L_positive: L > 0.
      Hypothesis H_fixed_point: L = blocking_bound + total_hep_rbf L.

      (* To reduce the time complexity of the analysis, recall the notion of search space. *)
      Let is_in_search_space A := (A < L) && (task_rbf A != task_rbf (A + ε)).

      (* Next, consider any value R, and assume that for any given arrival offset A from the search 
         space there is a solution of the response-time bound recurrence that is bounded by R. *)

      Variable R: nat.
      Hypothesis H_R_is_maximum:
          is_in_search_space A
            A + F = blocking_bound
                    + (task_rbf (A + ε) - (task_cost tsk - task_lock_in_service tsk))
                    + total_ohep_rbf (A + F)
            F + (task_cost tsk - task_lock_in_service tsk) R.

      (* Then, using the results for the general RTA for FP-schedulers, we establish a 
         response-time bound for the more concrete model of bounded nonpreemptive segments. 
         Note that in case of the general RTA for FP-schedulers, we just _assume_ that 
         the priority inversion is bounded. In this module we provide the preemption model
         with bounded nonpreemptive segments and _prove_ that the priority inversion is 
         bounded. *)

      Theorem uniprocessor_response_time_bound_fp_with_bounded_nonpreemptive_segments:
        response_time_bounded_by tsk R.

    End ResponseTimeBound.

  End Analysis.

End RTAforFPwithBoundedNonpreemptiveSegmentsWithArrivalCurves.