Library prosa.classic.implementation.uni.jitter.schedule
Require Import prosa.classic.util.all.
Require Import prosa.classic.model.arrival.basic.job prosa.classic.model.arrival.basic.arrival_sequence prosa.classic.model.priority.
Require Import prosa.classic.model.schedule.uni.jitter.schedule
prosa.classic.model.schedule.uni.jitter.platform.
Require Import prosa.classic.model.schedule.uni.transformation.construction.
From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype bigop seq path.
Module ConcreteScheduler.
Import Job ArrivalSequence UniprocessorScheduleWithJitter Platform Priority
ScheduleConstruction.
Section Implementation.
Context {Job: eqType}.
Variable job_arrival: Job → time.
Variable job_cost: Job → time.
Variable job_jitter: Job → time.
Variable arr_seq: arrival_sequence Job.
Variable higher_eq_priority: JLDP_policy Job.
Section ScheduleConstruction.
Variable sched_prefix: schedule Job.
Variable t: time.
Let is_pending := pending job_arrival job_cost job_jitter sched_prefix.
Definition pending_jobs :=
[seq j <- actual_arrivals_up_to job_arrival job_jitter arr_seq t | is_pending j t].
Definition highest_priority_job :=
seq_min (higher_eq_priority t) pending_jobs.
End ScheduleConstruction.
Let empty_schedule : schedule Job := fun t ⇒ None.
Definition scheduler :=
build_schedule_from_prefixes highest_priority_job empty_schedule.
Lemma scheduler_depends_only_on_prefix:
∀ sched1 sched2 t,
(∀ t0, t0 < t → sched1 t0 = sched2 t0) →
highest_priority_job sched1 t = highest_priority_job sched2 t.
Corollary scheduler_uses_construction_function:
∀ t, scheduler t = highest_priority_job scheduler t.
End Implementation.
Section Proofs.
Context {Job: eqType}.
Variable job_arrival: Job → time.
Variable job_cost: Job → time.
Variable job_jitter: Job → time.
Variable arr_seq: arrival_sequence Job.
Hypothesis H_arrival_times_are_consistent: arrival_times_are_consistent job_arrival arr_seq.
Hypothesis H_no_duplicate_arrivals: arrival_sequence_is_a_set arr_seq.
Variable higher_eq_priority: JLDP_policy Job.
Hypothesis H_priority_is_reflexive: JLDP_is_reflexive higher_eq_priority.
Hypothesis H_priority_is_transitive: JLDP_is_transitive higher_eq_priority.
Hypothesis H_priority_is_total: JLDP_is_total arr_seq higher_eq_priority.
Let sched := scheduler job_arrival job_cost job_jitter arr_seq higher_eq_priority.
Lemma scheduler_jobs_come_from_arrival_sequence:
jobs_come_from_arrival_sequence sched arr_seq.
Theorem scheduler_jobs_execute_after_jitter:
jobs_execute_after_jitter job_arrival job_jitter sched.
Theorem scheduler_completed_jobs_dont_execute:
completed_jobs_dont_execute job_cost sched.
Theorem scheduler_work_conserving:
work_conserving job_arrival job_cost job_jitter arr_seq sched.
Theorem scheduler_respects_policy :
respects_JLDP_policy job_arrival job_cost job_jitter arr_seq sched higher_eq_priority.
End Proofs.
End ConcreteScheduler.
Require Import prosa.classic.model.arrival.basic.job prosa.classic.model.arrival.basic.arrival_sequence prosa.classic.model.priority.
Require Import prosa.classic.model.schedule.uni.jitter.schedule
prosa.classic.model.schedule.uni.jitter.platform.
Require Import prosa.classic.model.schedule.uni.transformation.construction.
From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype bigop seq path.
Module ConcreteScheduler.
Import Job ArrivalSequence UniprocessorScheduleWithJitter Platform Priority
ScheduleConstruction.
Section Implementation.
Context {Job: eqType}.
Variable job_arrival: Job → time.
Variable job_cost: Job → time.
Variable job_jitter: Job → time.
Variable arr_seq: arrival_sequence Job.
Variable higher_eq_priority: JLDP_policy Job.
Section ScheduleConstruction.
Variable sched_prefix: schedule Job.
Variable t: time.
Let is_pending := pending job_arrival job_cost job_jitter sched_prefix.
Definition pending_jobs :=
[seq j <- actual_arrivals_up_to job_arrival job_jitter arr_seq t | is_pending j t].
Definition highest_priority_job :=
seq_min (higher_eq_priority t) pending_jobs.
End ScheduleConstruction.
Let empty_schedule : schedule Job := fun t ⇒ None.
Definition scheduler :=
build_schedule_from_prefixes highest_priority_job empty_schedule.
Lemma scheduler_depends_only_on_prefix:
∀ sched1 sched2 t,
(∀ t0, t0 < t → sched1 t0 = sched2 t0) →
highest_priority_job sched1 t = highest_priority_job sched2 t.
Corollary scheduler_uses_construction_function:
∀ t, scheduler t = highest_priority_job scheduler t.
End Implementation.
Section Proofs.
Context {Job: eqType}.
Variable job_arrival: Job → time.
Variable job_cost: Job → time.
Variable job_jitter: Job → time.
Variable arr_seq: arrival_sequence Job.
Hypothesis H_arrival_times_are_consistent: arrival_times_are_consistent job_arrival arr_seq.
Hypothesis H_no_duplicate_arrivals: arrival_sequence_is_a_set arr_seq.
Variable higher_eq_priority: JLDP_policy Job.
Hypothesis H_priority_is_reflexive: JLDP_is_reflexive higher_eq_priority.
Hypothesis H_priority_is_transitive: JLDP_is_transitive higher_eq_priority.
Hypothesis H_priority_is_total: JLDP_is_total arr_seq higher_eq_priority.
Let sched := scheduler job_arrival job_cost job_jitter arr_seq higher_eq_priority.
Lemma scheduler_jobs_come_from_arrival_sequence:
jobs_come_from_arrival_sequence sched arr_seq.
Theorem scheduler_jobs_execute_after_jitter:
jobs_execute_after_jitter job_arrival job_jitter sched.
Theorem scheduler_completed_jobs_dont_execute:
completed_jobs_dont_execute job_cost sched.
Theorem scheduler_work_conserving:
work_conserving job_arrival job_cost job_jitter arr_seq sched.
Theorem scheduler_respects_policy :
respects_JLDP_policy job_arrival job_cost job_jitter arr_seq sched higher_eq_priority.
End Proofs.
End ConcreteScheduler.