# 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.
Consider any type of tasks with a given cost ...

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

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.

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

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.
Section Facts.

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

Consider an arbitrary task tsk ...

... 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:
valid_arrival_curve (arrivals tsk)
Proof.
moveARR [ZERO MONO].
split.
- by rewrite /task_max_rbf ZERO muln0.
- movex y LEQ.
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:
valid_arrival_curve (arrivals tsk)
Proof.
moveARR [ZERO MONO].
split.
- by rewrite /task_min_rbf ZERO muln0.
- movex y LEQ.
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.
specialize (RESPECT t1 t2 LEQ).
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.
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.
specialize (RESPECT t1 t2 LEQ).
apply leq_trans with (n := task_min_cost tsk × number_of_task_arrivals arr_seq tsk t1 t2) ⇒ //.
- rewrite /min_arrivals /number_of_task_arrivals -sum1_size big_distrr //= muln1 leq_sum_seq // ⇒ j.
rewrite mem_filter ⇒ /andP [/eqP TSK _] _.
rewrite -TSK.
Qed.

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

Let ts be an arbitrary task set...

... 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 ...
Proof.
moveVALID tsk IN.
specialize (VALID tsk IN).
by apply valid_arrival_curve_to_max_rbf.
Qed.

... and in the lower-bounding case.
Proof.
moveVALID 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...
jobs_have_valid_job_costs
Proof.
by apply respects_arrival_curve_to_max_rbf, SET_RESPECTS.
Qed.

...as well as in the lower-bounding case.
jobs_have_valid_min_job_costs
Proof.
by apply respects_arrival_curve_to_min_rbf, SET_RESPECTS.
Qed.