Library prosa.model.task.offset

Require Export prosa.model.task.concept.
Require Export prosa.behavior.all.
Require Export prosa.model.task.arrivals.
Require Export prosa.util.all.
Require Export prosa.analysis.facts.behavior.all.

Task Offset

We define a task-model parameter task_offset that maps each task to its corresponding offset, that is the instant when its first job arrives.
In the following section, we define two important properties that an offset of any task should satisfy.
Section ValidTaskOffset.

Consider any type of tasks with offsets, ...
  Context {Task : TaskType}.
  Context `{TaskOffset Task}.

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

... and any arrival sequence.
  Variable arr_seq : arrival_sequence Job.

No jobs of a task tsk arrive before task_offset tsk.
  Definition no_jobs_before_offset (tsk : Task) :=
      job_task j = tsk
      job_arrival j task_offset tsk.

Furthermore for a task tsk, there exists a job that arrives exactly at the offset.
  Definition job_released_at_offset (tsk : Task) :=
      arrives_in arr_seq j'
      job_task j' = tsk
      job_arrival j' = task_offset tsk.

An offset is valid iff it satisfies both of the above conditions.
In the context of a set of tasks ts, ...
  Variable ts : TaskSet Task.

... all tasks in the set must have valid offsets.
  Definition valid_offsets := tsk, tsk \in ts valid_offset tsk.

End ValidTaskOffset.

In this section we define the notion of a maximum task offset.
Section MaxTaskOffset.

Consider any type of tasks with offsets, ...
  Context {Task : TaskType}.
  Context `{TaskOffset Task}.

... and any task set.
  Variable ts : TaskSet Task.

We define the sequence of task offsets of all tasks in ts, ...
  Definition task_offsets := map task_offset ts.

... and the maximum among all the task offsets.