Library prosa.results.edf.rta.bounded_nps

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.
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.
Next, consider any ideal uni-processor schedule of this arrival sequence ...
... where jobs do not execute before their arrival or after completion.
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.
Assume we have sequential tasks, i.e, jobs from the same task execute in the order of their arrival.
Next, we assume that the schedule is a work-conserving schedule...
... and the schedule respects the policy defined by the job_preemptable function (i.e., jobs have bounded non-preemptive segments).
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.

Consider a valid preemption model...
...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.
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.
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).
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.
We also define a bound for the priority inversion caused by jobs with lower priority.

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.
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.

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.
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.