Library prosa.results.fixed_priority.rta.fully_nonpreemptive

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.

Next, consider any ideal non-preemptive uniprocessor 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.
  Hypothesis H_nonpreemptive_sched : nonpreemptive_schedule sched.

... 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.

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

  Context `{FP_policy Task}.
  Hypothesis H_priority_is_reflexive : reflexive_priorities.
  Hypothesis H_priority_is_transitive : transitive_priorities.

Assume we have sequential tasks, i.e, tasks 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 nonpreemptive segments).

Hypothesis H_respects_policy : respects_policy_at_preemption_point arr_seq sched.

Total Workload and Length of Busy Interval

We introduce the 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 with higher priority ...

Let total_hep_rbf := total_hep_request_bound_function_FP ts tsk.

... and the request bound function of all tasks with higher priority other than task tsk.

Let total_ohep_rbf := total_ohep_request_bound_function_FP ts tsk.

Next, we define a bound for the priority inversion caused by tasks of lower priority.

Let blocking_bound :=
\max_(tsk_other <- ts | ~~ hep_task tsk_other tsk ) (task_cost tsk_other - ε ).

Let L be any positive fixed point of the busy interval recurrence, determined by the sum of blocking and higher-or-equal-priority workload.

  Variable L : duration.
  Hypothesis H_L_positive : L > 0.
  Hypothesis H_fixed_point : L = blocking_bound + total_hep_rbf L.

Response-Time Bound

To reduce the time complexity of the analysis, recall the notion of search space.

Let is_in_search_space (A : duration) := (A < L ) && (task_rbf A != task_rbf (A + ε)).

Next, consider any value R, and assume that for any given arrival A from 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 - ε ))
                + total_ohep_rbf (A + F) ∧
        F + (task_cost tsk - ε ) ≤ R.

Now, we can leverage the results for the abstract model with bounded nonpreemptive segments to establish a response-time bound for the more concrete model of fully nonpreemptive scheduling.

  Let response_time_bounded_by := task_response_time_bound arr_seq sched.

  Theorem uniprocessor_response_time_bound_fully_nonpreemptive_fp:
    response_time_bounded_by tsk R.
  Proof.
    move: (posnP (@task_cost _ H tsk)) ⇒ [ZERO|POS].
    { intros j ARR TSK.
      have ZEROj: job_cost j = 0.
      { move: (H_valid_job_cost j ARR) ⇒ NEQ.
        rewrite /valid_job_cost TSK ZERO in NEQ.
          by apply/eqP; rewrite -leqn0.
      }
        by rewrite /job_response_time_bound /completed_by ZEROj.
    }
    eapply uniprocessor_response_time_bound_fp_with_bounded_nonpreemptive_segments with
        (L0 := L).
    all: eauto 2 with basic_facts.
  Qed.

End RTAforFullyNonPreemptiveFPModelwithArrivalCurves.