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# Library prosa.model.task.arrival.curve_as_rbf

Require Export prosa.util.all.
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 cost ...
Context {Task : TaskType} `{TaskCost 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 conversion to a request-bound function as the product of the task cost and the number of arrivals during Δ. In the upper-bounding case, the cost of a task will represent the WCET of its jobs. Symmetrically, in the lower-bounding case, the cost of a task will represent the BCET of its jobs.
Definition task_max_rbf (arrivals : Task → duration → nat) task Δ := task_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 MaxArrivalsRBF : MaxRequestBound Task := task_max_rbf max_arrivals.
Global Program Instance MinArrivalsRBF : MinRequestBound Task := 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.
Theorem valid_arrival_curve_to_max_rbf :
∀ (arrivals : Task → duration → nat),
valid_arrival_curve (arrivals tsk) →
valid_request_bound_function ((task_max_rbf arrivals) tsk).
Proof.
move ⇒ ARR [ZERO MONO].
split.
- by rewrite /task_max_rbf ZERO muln0.
- move ⇒ x y LEQ.
rewrite /task_max_rbf.
destruct (task_cost tsk); first by rewrite mul0n.
by rewrite leq_pmul2l //; apply MONO.
Qed.
The same idea can be applied in the lower-bounding case.
Theorem valid_arrival_curve_to_min_rbf :
∀ (arrivals : Task → duration → nat),
valid_arrival_curve (arrivals tsk) →
valid_request_bound_function ((task_min_rbf arrivals) tsk).
Proof.
move ⇒ ARR [ZERO MONO].
split.
- by rewrite /task_min_rbf ZERO muln0.
- move ⇒ x y LEQ.
rewrite /task_min_rbf.
destruct (task_min_cost tsk); first by rewrite mul0n.
by rewrite leq_pmul2l //; apply MONO.
Qed.
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).
Theorem respects_arrival_curve_to_max_rbf :
jobs_have_valid_job_costs →
respects_max_arrivals arr_seq tsk (MaxArr tsk) →
respects_max_request_bound arr_seq tsk ((task_max_rbf MaxArr) tsk).
Proof.
move⇒ TASK_COST RESPECT t1 t2 LEQ.
specialize (RESPECT t1 t2).
apply leq_trans with (n := task_cost tsk × number_of_task_arrivals arr_seq tsk t1 t2) ⇒ //.
- rewrite /max_arrivals /number_of_task_arrivals -sum1_size big_distrr //= muln1 leq_sum_seq // ⇒ j.
rewrite mem_filter ⇒ /andP [/eqP TSK _] _.
rewrite -TSK.
by apply TASK_COST.
- by destruct (task_cost tsk) eqn:C; rewrite /task_max_rbf C // leq_pmul2l.
Qed.
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).
Theorem respects_arrival_curve_to_min_rbf :
jobs_have_valid_min_job_costs →
respects_min_arrivals arr_seq tsk (MinArr tsk) →
respects_min_request_bound arr_seq tsk ((task_min_rbf MinArr) tsk).
Proof.
move⇒ TASK_COST RESPECT t1 t2 LEQ.
specialize (RESPECT t1 t2 LEQ).
apply leq_trans with (n := task_min_cost tsk × number_of_task_arrivals arr_seq tsk t1 t2) ⇒ //.
- by destruct (task_min_cost tsk) eqn:C; rewrite /task_min_rbf C // leq_pmul2l.
- rewrite /min_arrivals /number_of_task_arrivals -sum1_size big_distrr //= muln1 leq_sum_seq // ⇒ j.
rewrite mem_filter ⇒ /andP [/eqP TSK _] _.
rewrite -TSK.
by apply TASK_COST.
Qed.
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 :
valid_taskset_arrival_curve ts MaxArr →
valid_taskset_request_bound_function ts MaxArrivalsRBF.
Proof.
move⇒ VALID tsk IN.
specialize (VALID tsk IN).
by apply valid_arrival_curve_to_max_rbf.
Qed.
... and in the lower-bounding case.
Corollary valid_taskset_arrival_curve_to_min_rbf :
valid_taskset_arrival_curve ts MinArr →
valid_taskset_request_bound_function ts MinArrivalsRBF.
Proof.
move⇒ VALID tsk IN.
specialize (VALID tsk IN).
by apply valid_arrival_curve_to_min_rbf.
Qed.
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 :
jobs_have_valid_job_costs →
taskset_respects_max_arrivals arr_seq ts →
taskset_respects_max_request_bound arr_seq ts.
Proof.
move⇒ TASK_COST SET_RESPECTS tsk IN.
by apply respects_arrival_curve_to_max_rbf, SET_RESPECTS.
Qed.
...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.
Proof.
move⇒ TASK_COST SET_RESPECTS tsk IN.
by apply respects_arrival_curve_to_min_rbf, SET_RESPECTS.
Qed.
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.