# Library prosa.model.task.arrival.request_bound_functions

Require Export prosa.util.rel.
Require Export prosa.model.task.arrivals.

# Request Bound Functions (RBF)

In the following, we define the notion of Request Bound Functions (RBF), which can be used to reason about the job cost arrivals. In contrast to arrival curves which constrain the number of arrivals per time interval, request bound functions bound the sum of costs of arriving jobs.

## Task Parameters for the Request Bound Functions

The request bound functions give an upper bound and, optionally, a lower bound on the cost of new job arrivals during any given interval.
We let max_request_bound tsk Δ denote a bound on the maximum cost of arrivals of jobs of task tsk in any interval of length Δ.
Class MaxRequestBound (Task : TaskType) :=
{ max_request_bound : Task → duration → work }.
Conversely, we let min_request_bound tsk Δ denote a bound on the minimum cost of arrivals of jobs of task tsk in any interval of length Δ.
Class MinRequestBound (Task : TaskType) :=
{ min_request_bound : Task → duration → work }.

## Parameter Semantics

In the following, we precisely define the semantics of the request bound functions.
Section RequestBoundFunctions.
Consider any type of tasks ...
Context {Task : TaskType}.
... and any type of jobs associated with these tasks.
Context {Job : JobType}.
Context `{JobTask Job Task}.
Context `{JobCost Job}.
Consider any job arrival sequence.
Variable arr_seq : arrival_sequence Job.

### Definition of Request Bound Functions

First, what constitutes a valid request bound function for a task?
We say that a given bound request_bound is a valid request bound function iff request_bound is a monotonic function that equals 0 for the empty interval delta = 0.
Definition valid_request_bound_function (request_bound : duration → work) :=
request_bound 0 = 0
∧ monotone leq request_bound.
We say that request_bound is an upper request bound for task tsk iff, for any interval `[t1, t2)`, request_bound (t2 - t1) bounds the sum of costs of jobs of tsk that arrive in that interval.
Definition respects_max_request_bound (tsk : Task) (max_request_bound : duration → work) :=
∀ (t1 t2 : instant),
t1 ≤ t2 →
cost_of_task_arrivals arr_seq tsk t1 t2 ≤ max_request_bound (t2 - t1).
We analogously define the lower request bound.
Definition respects_min_request_bound (tsk : Task) (min_request_bound : duration → work) :=
∀ (t1 t2 : instant),
t1 ≤ t2 →
min_request_bound (t2 - t1) ≤ cost_of_task_arrivals arr_seq tsk t1 t2.
End RequestBoundFunctions.

## Model Validity

Based on the just-established semantics, we define the properties of a valid request bound model.
Section RequestBoundFunctionsModel.
Consider any type of tasks ...
Context {Task : TaskType}.
... and any type of jobs associated with these tasks.
Context {Job : JobType}.
Context `{JobTask Job Task}.
Context `{JobCost Job}.
Consider any job arrival sequence...
Variable arr_seq : arrival_sequence Job.
...and all kinds of request bounds.
Context `{MaxRequestBound Task}
`{MinRequestBound Task}.
Let ts be an arbitrary task set.
Variable ts : TaskSet Task.
We say that request_bound is a valid arrival curve for a task set if it is valid for any task in the task set
Definition valid_taskset_request_bound_function (request_bound : Task → duration → work) :=
∀ (tsk : Task), tsk \in ts → valid_request_bound_function (request_bound tsk).
Finally, we lift the per-task semantics of the respective request bound functions to the entire task set for both the maximum ...
Definition taskset_respects_max_request_bound :=
∀ (tsk : Task), tsk \in ts → respects_max_request_bound arr_seq tsk (max_request_bound tsk).
... and minimum request-bound functions.
Definition taskset_respects_min_request_bound :=
∀ (tsk : Task), tsk \in ts → respects_min_request_bound arr_seq tsk (min_request_bound tsk).
End RequestBoundFunctionsModel.