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# Library prosa.model.task.arrivals

Require Export prosa.model.task.concept.
In this module, we provide basic definitions concerning the job arrivals of a given task.
Section TaskArrivals.
Consider any type of job associated with any type of tasks.
Context {Job : JobType}.
Context {Task : TaskType}.
Context `{JobTask Job Task}.
Context `{JobArrival Job}.
Context `{JobCost Job}.
Consider any job arrival sequence ...
Variable arr_seq : arrival_sequence Job.
... and any task.
Variable tsk : Task.
First, we construct the list of jobs of task tsk that arrive in a given half-open interval `[t1, t2)`.
Definition task_arrivals_between (t1 t2 : instant) :=
[seq j <- arrivals_between arr_seq t1 t2 | job_of_task tsk j].
Based on that, we define the list of jobs of task tsk that arrive up to and including time t, ...
Definition task_arrivals_up_to (t : instant) :=
task_arrivals_between 0 t.+1.
... the list of jobs of task tsk that arrive strictly before time t, ...
Definition task_arrivals_before (t : instant) :=
task_arrivals_between 0 t.
... the list of jobs of task tsk that arrive exactly at an instant t, ...
Definition task_arrivals_at (tsk : Task) (t : instant) :=
[seq j <- arr_seq t | job_of_task tsk j].
... and finally count the number of job arrivals.
Definition number_of_task_arrivals (t1 t2 : instant) :=
size (task_arrivals_between t1 t2).
... and also count the cost of job arrivals.
Definition cost_of_task_arrivals (t1 t2 : instant) :=
\sum_(j <- task_arrivals_between t1 t2) job_cost j.
In the following, suppose there is a deadline associated with each job.
Context `{JobDeadline Job}.
We define the list of jobs of task tsk that arrive in a given half-open interval `[t1, t2)` and that also have a deadline within the closed interval `[t1, t2]`, ...
Definition task_arrivals_with_deadline_within (t1 t2 : instant) :=
[seq j <- arrivals_between arr_seq t1 t2 | job_of_task tsk j & job_deadline j ≤ t2].
... and similarly define the count of such jobs.
Definition number_of_task_arrivals_with_deadline_within (t1 t2 : instant) :=
size (task_arrivals_with_deadline_within t1 t2).
End TaskArrivals.
In this section we introduce a few definitions regarding arrivals of a particular task prior to a job.
Section PriorArrivals.
Consider any type of jobs associated with any type of tasks.
Context {Job : JobType}.
Context {Task : TaskType}.
Context `{JobTask Job Task}.
Context `{JobArrival Job}.
Consider any arrival sequence of jobs ...
Variable arr_seq : arrival_sequence Job.
... and any job j.
Variable j : Job.
We first define a sequence of jobs of a task that arrive before or at job_arrival j.
Definition task_arrivals_up_to_job_arrival :=
task_arrivals_up_to arr_seq (job_task j) (job_arrival j).
Next, we define a sequence of jobs of a task that arrive strictly before job_arrival j.
Definition task_arrivals_before_job_arrival :=
task_arrivals_before arr_seq (job_task j) (job_arrival j).
Finally, we define a sequence of jobs of a task that arrive at job_arrival j.
Definition task_arrivals_at_job_arrival :=
task_arrivals_at arr_seq (job_task j) (job_arrival j).
End PriorArrivals.
In this section, we define the notion of a job's index.
Section JobIndex.
Consider any type of tasks, ...
Context {Task : TaskType}.
... any type of jobs associated with the tasks, ...
Context {Job : JobType}.
Context `{JobArrival Job}.
Context `{JobTask Job Task}.
... and any arrival sequence.
Variable arr_seq : arrival_sequence Job.
Given a job j, we define a concept of job index as the number of jobs of the same task as j that arrived before or at the same instant as j. Please note that job indices start from zero.
Definition job_index (j : Job) :=
index j (task_arrivals_up_to_job_arrival arr_seq j).
End JobIndex.
In this section we define the notion of a previous job for any job with a positive job_index.
Section PreviousJob.
Consider any type of jobs associated with any type of tasks.
Context {Job : JobType}.
Context {Task : TaskType}.
Context `{JobTask Job Task}.
Context `{JobArrival Job}.
Consider any job arrival sequence.
Variable arr_seq : arrival_sequence Job.
Let task_arrivals_up_to_job_arrival j := task_arrivals_up_to_job_arrival arr_seq j.
Let prev_index j := job_index arr_seq j - 1.
For any job j with a positive job_index we define the notion of a previous job.
Definition prev_job (j : Job) := nth j (task_arrivals_up_to_job_arrival j) (prev_index j).
End PreviousJob.