Library prosa.analysis.definitions.request_bound_function

Require Export prosa.model.task.arrival.curves.
Require Export prosa.model.priority.classes.
Require Export prosa.analysis.facts.priority.classes.
Require Export prosa.util.sum.

Request Bound Function (RBF)

We define the notion of a task's request-bound function (RBF), as well as the total request bound function of a set of tasks.

Section TaskWorkloadBoundedByArrivalCurves.

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

... and any type of jobs associated with these tasks, where each task has a cost.
  Context {Job : JobType}.
  Context `{JobTask Job Task}.
  Context `{JobCost Job}.

Also assume an FP policy that indicates a higher-or-equal priority relation.
  Context `{FP_policy Task}.

Let MaxArrivals denote any function that takes a task and an interval length and returns the associated number of job arrivals of the task.
  Context `{MaxArrivals Task}.

RBF of a Single Task

In this section, we define a bound for the workload of a single task under uni-processor FP scheduling.
  Section SingleTask.

Consider any task tsk that is to be scheduled in an interval of length delta.
    Variable tsk : Task.
    Variable delta : duration.

We define the following workload bound for the task.
    Definition task_request_bound_function :=
      task_cost tsk × max_arrivals tsk delta.

  End SingleTask.

Total RBF of Multiple Tasks

In this section, we define a bound for the workload of multiple tasks.
  Section AllTasks.

Consider a task set ts...
    Variable ts : list Task.

...and let tsk be any task in task set.
    Variable tsk : Task.

Let delta be the length of the interval of interest.
    Variable delta : duration.

Recall the definition of higher-or-equal-priority task for FP scheduling.
    Let is_hep_task tsk_other := hep_task tsk_other tsk.
    Let is_other_hep_task tsk_other := hep_task tsk_other tsk && (tsk_other != tsk).
    Let is_hp_task tsk_other := hp_task tsk_other tsk.
    Let is_ep_task tsk_other := ep_task tsk_other tsk.

Using the sum of individual workload bounds, we define the following bound for the total workload of tasks in any interval of length delta.
    Definition total_request_bound_function :=
      \sum_(tsk <- ts) task_request_bound_function tsk delta.

Similarly, we define the following bound for the total workload of tasks of higher-or-equal priority (with respect to tsk) in any interval of length delta.
    Definition total_hep_request_bound_function_FP :=
      \sum_(tsk_other <- ts | is_hep_task tsk_other)
       task_request_bound_function tsk_other delta.

We also define a bound for the total workload of higher-or-equal-priority tasks other than tsk in any interval of length delta.
    Definition total_ohep_request_bound_function_FP :=
      \sum_(tsk_other <- ts | is_other_hep_task tsk_other)
        task_request_bound_function tsk_other delta.

We also define a bound for the total workload of higher-or-equal-priority tasks other than tsk in any interval of length delta.
    Definition total_ep_request_bound_function_FP :=
      \sum_(tsk_other <- ts | is_ep_task tsk_other)
        task_request_bound_function tsk_other delta.

Finally, we define a bound for the total workload of higher-priority tasks in any interval of length delta.
    Definition total_hp_request_bound_function_FP :=
      \sum_(tsk_other <- ts | is_hp_task tsk_other)
       task_request_bound_function tsk_other delta.

  End AllTasks.

End TaskWorkloadBoundedByArrivalCurves.