Library prosa.results.fixed_priority.rta.fully_nonpreemptive

RTA for Fully Non-Preemptive FP Model

In this module we prove the RTA theorem for the fully non-preemptive FP model.

Setup and Assumptions

We assume ideal uni-processor schedules.
  #[local] Existing Instance ideal.processor_state.

Consider any type of tasks ...
  Context {Task : TaskType}.
  Context {tc : TaskCost Task}.

... and any type of jobs associated with these tasks.
  Context {Job : JobType}.
  Context `{JobTask Job Task}.
  Context `{JobArrival Job}.
  Context `{JobCost Job}.

We assume that jobs and tasks are fully nonpreemptive.
  #[local] Existing Instance fully_nonpreemptive_job_model.
  #[local] Existing Instance fully_nonpreemptive_task_model.
  #[local] Existing Instance fully_nonpreemptive_rtc_threshold.

Consider any arrival sequence with consistent, non-duplicate arrivals.
Consider an arbitrary task set ts, ...
  Variable ts : list Task.

... assume that all jobs come from the task set, ...
... and the cost of a job cannot be larger than the task cost.
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.
Let tsk be any task in ts that is to be analyzed.
  Variable tsk : Task.
  Hypothesis H_tsk_in_ts : tsk \in ts.

Recall that we assume sequential readiness.
  #[local] Instance sequential_readiness : JobReady _ _ :=
    sequential_ready_instance arr_seq.

Next, consider any ideal non-preemptive uniprocessor schedule of this arrival sequence ...
Consider an FP policy that indicates a higher-or-equal priority relation, and assume that the relation is reflexive and transitive.
Next, we assume that the schedule is a work-conserving schedule ...
... and the schedule respects the scheduling policy.

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.
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 ...
... and the request bound function of all tasks with higher priority other than task tsk.
Next, we define a bound for the priority inversion caused by tasks of lower priority.
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.
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)
        R F + (task_cost tsk - ε).

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.
    move: (posnP (@task_cost _ tc 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 in NEQ.
        move: TSK ⇒ /eqPin NEQ.
        rewrite 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
        (L := L) ⇒ //.
    - exact: sequential_readiness_implies_work_bearing_readiness.
    - exact: sequential_readiness_implies_sequential_tasks.

End RTAforFullyNonPreemptiveFPModelwithArrivalCurves.