Library prosa.model.task.arrival.curve_as_rbf

Require Export prosa.util.all.
Require Export prosa.analysis.facts.model.cumulative_cost.
Require Export prosa.model.task.arrival.request_bound_functions.
Require Export prosa.model.task.arrival.curves.

Converting an Arrival Curve + Worst-Case/Best-Case to a Request-Bound Function (RBF)

In the following, we show a way to convert a given arrival curve, paired with a worst-case/best-case execution time, to a request-bound function. Definitions and proofs will handle both lower-bounding and upper-bounding arrival curves.

Section ArrivalCurveToRBF.

Consider any type of tasks with a given cumulative cost and minimum scalar cost.

Context {Task : TaskType} `{TaskCumulativeCost Task} `{TaskMinCost Task}.

... and any type of jobs associated with these tasks.

Context {Job : JobType} `{JobTask Job Task} `{JobCost Job}.

Let MaxArr and MinArr represent two arrivals curves. MaxArr upper-bounds the possible number or arrivals for a given task, whereas MinArr lower-bounds it.

Context `{MaxArr : MaxArrivals Task} `{MinArr : MinArrivals Task}.

We define the upper-bounding conversion as the cumulative cost of the maximum number of arrivals during Δ. Symmetrically, in the lower-bounding case, the cumulative cost of the jobs is derived from the task's BCET via scalar multiplication.

  Definition task_max_rbf (arrivals : Task → duration → nat) task Δ :=
    task_cumulative_cost task (arrivals task Δ).
  Definition task_min_rbf (arrivals : Task → duration → nat) task Δ := task_min_cost task × arrivals task Δ.

Finally, we show that the newly defined functions are indeed request-bound functions.

  Global Program Instance max_arrivals_rbf : MaxRequestBound Task :=
  {
    max_request_bound := task_max_rbf max_arrivals
  }.
  Global Program Instance min_arrivals_rbf : MinRequestBound Task :=
  {
    min_request_bound := task_min_rbf min_arrivals
  }.

In the following section, we prove that the transformation yields a request-bound function that conserves correctness properties in both the upper-bounding and lower-bounding cases.

First, we establish the validity of the transformation for a single, given task.

Section SingleTask.

Consider an arbitrary task tsk ...

Variable tsk : Task.

... and any job arrival sequence.

Variable arr_seq : arrival_sequence Job.

First, note that any valid upper-bounding arrival curve, after being converted, is a valid request-bound function.

    Fact valid_arrival_curve_to_max_rbf :
      valid_task_cumulative_cost tsk →
      valid_arrival_curve (max_arrivals tsk) →
      valid_request_bound_function (max_request_bound tsk).

The same idea can be applied in the lower-bounding case.

    Fact valid_arrival_curve_to_min_rbf :
      valid_arrival_curve (min_arrivals tsk) →
      valid_request_bound_function (min_request_bound tsk).

Next, we prove that the task respects the request-bound function in the upper-bounding case. Note that, for this to work, we assume that the cost of tasks upper-bounds the cost of the jobs belonging to them (i.e., the task cost is the worst case).

    Fact respects_arrival_curve_to_max_rbf :
      valid_task_cumulative_cost tsk →
      respects_task_cumulative_cost arr_seq tsk →
      respects_max_arrivals arr_seq tsk (max_arrivals tsk) →
      respects_max_request_bound arr_seq tsk (max_request_bound tsk).

Finally, we prove that the task respects the request-bound function also in the lower-bounding case. This time, we assume that the cost of tasks lower-bounds the cost of the jobs belonging to them. (i.e., the task cost is the best-case).

    Fact respects_arrival_curve_to_min_rbf :
      jobs_have_valid_min_job_costs →
      respects_min_arrivals arr_seq tsk (min_arrivals tsk) →
      respects_min_request_bound arr_seq tsk (min_request_bound tsk).

  End SingleTask.

Next, we lift the results to the previous section to an arbitrary task set.

Section TaskSet.

Let ts be an arbitrary task set ...

Variable ts : TaskSet Task.

... and consider any job arrival sequence.

Variable arr_seq : arrival_sequence Job.

First, we generalize the validity of the transformation to a task set both in the upper-bounding case ...

    Corollary valid_taskset_arrival_curve_to_max_rbf :
      taskset_has_valid_cumulative_cost_bounds ts →
      valid_taskset_arrival_curve ts max_arrivals →
      valid_taskset_request_bound_function ts max_request_bound.

... and in the lower-bounding case.

    Corollary valid_taskset_arrival_curve_to_min_rbf :
      valid_taskset_arrival_curve ts min_arrivals →
      valid_taskset_request_bound_function ts min_request_bound.

Second, we show that a task set that respects a given arrival curve also respects the produced request-bound function, lifting the result obtained in the single-task case. The result is valid in the upper-bounding case...

    Corollary taskset_respects_arrival_curve_to_max_rbf :
      taskset_has_valid_cumulative_cost_bounds ts →
      taskset_respects_cumulative_cost_bounds arr_seq ts →
      taskset_respects_max_arrivals arr_seq ts →
      taskset_respects_max_request_bound arr_seq ts.

...as well as in the lower-bounding case.

    Corollary taskset_respects_arrival_curve_to_min_rbf :
      jobs_have_valid_min_job_costs →
      taskset_respects_min_arrivals arr_seq ts →
      taskset_respects_min_request_bound arr_seq ts.

  End TaskSet.

End ArrivalCurveToRBF.

We add the lemmas into the "Hint Database" basic_rt_facts so that they become available for proof automation.

Global Hint Resolve
  valid_arrival_curve_to_max_rbf
  respects_arrival_curve_to_max_rbf
  valid_taskset_arrival_curve_to_max_rbf
  taskset_respects_arrival_curve_to_max_rbf
  : basic_rt_facts.