Built with Alectryon, running Coq+SerAPI v8.15.0+0.15.0. Bubbles () indicate interactive fragments: hover for details, tap to reveal contents. Use Ctrl+↑ Ctrl+↓ to navigate, Ctrl+🖱️ to focus. On Mac, use ⌘ instead of Ctrl.
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "_ + _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ - _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ <= _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ < _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ >= _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ > _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ <= _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ < _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ <= _ < _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ < _ < _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ * _" was already used in scope nat_scope.
[notation-overridden,parsing]

Require Export BinInt.


(** * GEL Priority Policy * *)

(** We define the class of "Global-EDF-like" (GEL) priority policies,
    as first introduced by Leontyev et al. ("Multiprocessor Extensions
    to Real-Time Calculus", Real-Time Systems, 2011).

    GEL scheduling is a (large) subset of the class of JLFP policies,
    which structurally resembles EDF scheduling, hence the name. Under
    GEL scheduling, each task has a constant "priority point offset."
    Each job's priority point is given by its arrival time plus its
    task's offset, with the interpretation that an earlier priority
    point implies higher priority.

    EDF and FIFO are both special cases of GEL with the offset
    respectively being the relative deadline and zero. *)

(** To begin, we define the offset type. Note that an offset may be negative. *)
Definition offset := Z.

(** We define a task-model parameter to express each task's relative priority point. *)
Class PriorityPoint (Task : TaskType) := task_priority_point : Task -> offset.

(** Based on the task-level relative priority-point parameter, we
    define a job's absolute priority point in a straightforward manner.
    To this end, we need to introduce some context first. *)
Section AbsolutePriorityPoint.
  (** For any type of tasks with relative priority points, ... *)
  Context {Job : JobType} {Task : TaskType} `{JobTask Job Task} `{PriorityPoint Task} `{JobArrival Job}.

  (** ... a job's absolute priority point is given by its arrival time
          plus its task's relative priority point. *)
  Definition job_priority_point (j : Job) := (Z.of_nat (job_arrival j) + task_priority_point (job_task j))%Z.
End AbsolutePriorityPoint.

(** The resulting definition of GEL is straightforward: a job [j1]'s
    priority is higher than or equal to a job [j2]'s priority iff
    [j1]'s absolute priority point is no later than [j2]'s absolute
    priority point.  *)
#[export] Instance GEL_priority (Job : JobType) (Task : TaskType)
          `{PriorityPoint Task} `{JobArrival Job} `{JobTask Job Task} : JLFP_policy Job :=
{
  hep_job (j1 j2 : Job) := (job_priority_point j1 <=? job_priority_point j2)%Z
}.

(** In this section, we note three basic properties of the GEL policy:
    the priority relation is reflexive, transitive, and total.  *)
Section PropertiesOfGEL.

  (** Consider any type of tasks with relative priority points...*)
  Context {Task : TaskType} `{PriorityPoint Task}.

  (** ...and jobs of these tasks. *)
  Context {Job : JobType} `{JobArrival Job} `{JobTask Job Task}.

  (** GEL is reflexive. *)
  
Task: TaskType
H: PriorityPoint Task
Job: JobType
H0: JobArrival Job
H1: JobTask Job Task
reflexive_priorities

  
Task: TaskType
H: PriorityPoint Task
Job: JobType
H0: JobArrival Job
H1: JobTask Job Task
reflexive_priorities
 by move=> t j; apply Z.leb_refl. Qed.

  (** GEL is transitive. *)
  
Task: TaskType
H: PriorityPoint Task
Job: JobType
H0: JobArrival Job
H1: JobTask Job Task
transitive_priorities

  
Task: TaskType
H: PriorityPoint Task
Job: JobType
H0: JobArrival Job
H1: JobTask Job Task
transitive_priorities
 by move=> t y x z; apply Zbool.Zle_bool_trans. Qed.

  (** GEL is total. *)
  
Task: TaskType
H: PriorityPoint Task
Job: JobType
H0: JobArrival Job
H1: JobTask Job Task
total_priorities

  
Task: TaskType
H: PriorityPoint Task
Job: JobType
H0: JobArrival Job
H1: JobTask Job Task
total_priorities

    
Task: TaskType
H: PriorityPoint Task
Job: JobType
H0: JobArrival Job
H1: JobTask Job Task
t: instant
j1, j2: Job
hep_job_at t j1 j2 \/ hep_job_at t j2 j1

    
Task: TaskType
H: PriorityPoint Task
Job: JobType
H0: JobArrival Job
H1: JobTask Job Task
t: instant
j1, j2: Job
REL: ~~
(job_priority_point j1 <=? job_priority_point j2)%Z
hep_job_at t j2 j1

    
Task: TaskType
H: PriorityPoint Task
Job: JobType
H0: JobArrival Job
H1: JobTask Job Task
t: instant
j1, j2: Job
REL: (job_priority_point j2 < job_priority_point j1)%Z
hep_job_at t j2 j1

    by move: (Z.lt_le_incl _ _  REL) => /Z.leb_spec0.
  Qed.

End PropertiesOfGEL.

(** We add the above facts into a "Hint Database" basic_rt_facts, so Coq
    will be able to apply them automatically where needed. *)
Global Hint Resolve
     GEL_is_reflexive
     GEL_is_transitive
     GEL_is_total
  : basic_rt_facts.