Library prosa.results.edf.rta.bounded_nps

Require Import prosa.model.priority.edf.
Require Export prosa.analysis.facts.model.rbf.
Require Export prosa.analysis.facts.model.sequential.
Require Export prosa.results.edf.rta.bounded_pi.
Require Export prosa.analysis.facts.busy_interval.priority_inversion.

Throughout this file, we assume ideal uni-processor schedules.

Require Import prosa.model.processor.ideal.

Throughout this file, we assume the basic (i.e., Liu & Layland) readiness model.

Require Import prosa.model.readiness.basic.

RTA for EDF with Bounded Non-Preemptive Segments

In this section we instantiate the Abstract RTA for EDF-schedulers with Bounded Priority Inversion to EDF-schedulers for ideal uni-processor model of real-time tasks with arbitrary arrival models and bounded non-preemptive segments.

Recall that Abstract RTA for EDF-schedulers with Bounded Priority Inversion does not specify the cause of priority inversion. In this section, we prove that the priority inversion caused by execution of non-preemptive segments is bounded. Thus the Abstract RTA for EDF-schedulers is applicable to this instantiation.

Section RTAforEDFwithBoundedNonpreemptiveSegmentsWithArrivalCurves.

Consider any type of tasks ...

  Context {Task : TaskType}.
  Context `{TaskCost Task}.
  Context `{TaskDeadline Task}.
  Context `{TaskRunToCompletionThreshold 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}.

For clarity, let's denote the relative deadline of a task as D.

Let D tsk := task_deadline tsk.

Consider the EDF policy that indicates a higher-or-equal priority relation. Note that we do not relate the EDF policy with the scheduler. However, we define functions for Interference and Interfering Workload that actively use the concept of priorities.

Let EDF := EDF Job.

Consider any arrival sequence with consistent, non-duplicate arrivals.

  Variable arr_seq : arrival_sequence Job.
  Hypothesis H_arrival_times_are_consistent : consistent_arrival_times arr_seq.
  Hypothesis H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq.

Next, consider any ideal uni-processor schedule of this arrival sequence ...

  Variable sched : schedule (ideal.processor_state Job).
  Hypothesis H_jobs_come_from_arrival_sequence:
    jobs_come_from_arrival_sequence sched arr_seq.

... where jobs do not execute before their arrival or after completion.

Hypothesis H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched.
Hypothesis H_completed_jobs_dont_execute : completed_jobs_dont_execute sched.

In addition, we assume the existence of a function mapping jobs to theirs 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.

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

Hypothesis H_sequential_tasks : sequential_tasks sched.

Next, we assume that the schedule is a work-conserving schedule...

Hypothesis H_work_conserving : work_conserving arr_seq sched.

... and the schedule respects the policy defined by the job_preemptable function (i.e., jobs have bounded non-preemptive segments).

Hypothesis H_respects_policy : respects_policy_at_preemption_point 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 the cost of a job cannot be larger than the task cost.

Hypothesis H_valid_job_cost:
arrivals_have_valid_job_costs arr_seq.

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.

...and a valid task run-to-completion threshold function. That is, task_run_to_completion_threshold tsk is (1) no bigger than tsk's cost, (2) for any job of task tsk job_run_to_completion_threshold is bounded by task_run_to_completion_threshold.

Hypothesis H_valid_run_to_completion_threshold:
valid_task_run_to_completion_threshold arr_seq tsk.

We introduce as an abbreviation rbf for the task request bound function, which is defined as task_cost(T) × max_arrivals(T,Δ) for a task T.

Let rbf := task_request_bound_function.

Next, we introduce task_rbf as an abbreviation for the task request bound function of task tsk.

Let task_rbf := rbf tsk.

Using the sum of individual request bound functions, we define the request bound function of all tasks (total request bound function).

Let total_rbf := total_request_bound_function ts.

Next, we define an upper bound on interfering workload received from jobs of other tasks with higher-than-or-equal priority.

  Let bound_on_total_hep_workload A Δ :=
    \sum_(tsk_o <- ts | tsk_o != tsk )
     rbf tsk_o (minn ((A + ε ) + D tsk - D tsk_o) Δ).

Let's define some local names for clarity.

  Let max_length_of_priority_inversion :=
    max_length_of_priority_inversion arr_seq.
  Let task_rbf_changes_at A := task_rbf_changes_at tsk A.
  Let bound_on_total_hep_workload_changes_at :=
    bound_on_total_hep_workload_changes_at ts tsk.
  Let response_time_bounded_by := task_response_time_bound arr_seq sched.

We also define a bound for the priority inversion caused by jobs with lower priority.

  Definition blocking_bound :=
    \max_(tsk_o <- ts | (tsk_o != tsk ) && (D tsk < D tsk_o ))
     (task_max_nonpreemptive_segment tsk_o - ε ).

Priority inversion is bounded

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

Section PriorityInversionIsBounded.

First, we prove that the maximum length of a priority inversion of job j is bounded by the maximum length of a non-preemptive 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 →
        t ≤ job_arrival j →
        max_length_of_priority_inversion j t ≤ blocking_bound.

Using the lemma above, we prove that the priority inversion of the task is bounded by the maximum length of a nonpreemptive section of lower-priority tasks.

    Lemma priority_inversion_is_bounded:
      priority_inversion_is_bounded_by arr_seq sched 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 : duration.
    Hypothesis H_L_positive : L > 0.
    Hypothesis H_fixed_point : L = total_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_changes_at A || bound_on_total_hep_workload_changes_at A ).

Consider any value R, and assume that for any given arrival offset A in the search space, there is a solution of the response-time bound recurrence which is bounded by R.

    Variable R : duration.
    Hypothesis H_R_is_maximum:
      ∀ (A : duration),
        is_in_search_space A →
        ∃ (F : duration ),
          A + F = blocking_bound
                  + (task_rbf (A + ε) - (task_cost tsk - task_run_to_completion_threshold tsk ))
                  + bound_on_total_hep_workload A (A + F) ∧
          F + (task_cost tsk - task_run_to_completion_threshold tsk ) ≤ R.

Then, using the results for the general RTA for EDF-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 EDF-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_edf_with_bounded_nonpreemptive_segments:
      response_time_bounded_by tsk R.

  End ResponseTimeBound.

End RTAforEDFwithBoundedNonpreemptiveSegmentsWithArrivalCurves.