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.
(* In this section, we implement a jitter-aware priority-based uniprocessor scheduler *)
Section Implementation.
Context {Job: eqType}.
Variable job_arrival: Job → time.
Variable job_cost: Job → time.
Variable job_jitter: Job → time.
(* Let arr_seq be any job arrival sequence. *)
Variable arr_seq: arrival_sequence Job.
(* Also, assume that a JLDP policy is given. *)
Variable higher_eq_priority: JLDP_policy Job.
(* Next, we show how to recursively construct the schedule. *)
Section ScheduleConstruction.
(* For any time t, suppose that we have generated the schedule prefix in the
interval 0, t). Then, we must define what should be scheduled at time t. *)
Variable sched_prefix: schedule Job.
Variable t: time.
(* For simplicity, let's use some local names. *)
Let is_pending := pending job_arrival job_cost job_jitter sched_prefix.
(* Consider the list of pending jobs with actual arrivals up to t. *)
Definition pending_jobs :=
[seq j <- actual_arrivals_up_to job_arrival job_jitter arr_seq t | is_pending j t].
(* To make the scheduling decision, we just pick one of the highest-priority jobs
that are pending (if it exists). *)
Definition highest_priority_job :=
seq_min (higher_eq_priority t) pending_jobs.
End ScheduleConstruction.
(* Starting from the empty schedule, the final schedule is obtained by iteratively
picking the highest-priority job. *)
Let empty_schedule : schedule Job := fun t ⇒ None.
Definition scheduler :=
build_schedule_from_prefixes highest_priority_job empty_schedule.
(* Then, by showing that the construction function depends only on the prefix, ... *)
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.
Proof.
intros sched1 sched2 t ALL.
rewrite /highest_priority_job.
suff SAME: pending_jobs sched1 t = pending_jobs sched2 t by rewrite SAME.
apply eq_in_filter.
intros j IN; rewrite mem_filter in IN.
move: IN ⇒ /andP [ARR IN].
rewrite ltnS /= in ARR.
rewrite /pending /jitter_has_passed ARR 2!andTb; f_equal.
rewrite /completed_by; f_equal.
apply eq_big_nat; move ⇒ i /= LTi.
by rewrite /service_at /scheduled_at ALL.
Qed.
(* ...we infer that the generated schedule is indeed based on the construction function. *)
Corollary scheduler_uses_construction_function:
∀ t, scheduler t = highest_priority_job scheduler t.
Proof.
by ins; apply prefix_dependent_schedule_construction,
scheduler_depends_only_on_prefix.
Qed.
End Implementation.
(* In this section, we prove the properties of the scheduler that are used
as hypotheses in the analyses. *)
Section Proofs.
Context {Job: eqType}.
Variable job_arrival: Job → time.
Variable job_cost: Job → time.
Variable job_jitter: Job → time.
(* Assume any job arrival sequence with consistent, duplicate-free arrivals. *)
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.
(* Consider any JLDP policy that is reflexive, transitive and total. *)
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 denote our concrete scheduler implementation. *)
Let sched := scheduler job_arrival job_cost job_jitter arr_seq higher_eq_priority.
(* To conclude, we prove the important properties of the scheduler implementation. *)
(* First, we show that scheduled jobs come from the arrival sequence. *)
Lemma scheduler_jobs_come_from_arrival_sequence:
jobs_come_from_arrival_sequence sched arr_seq.
Proof.
move ⇒ j t /eqP SCHED.
rewrite /sched scheduler_uses_construction_function // /highest_priority_job in SCHED.
apply seq_min_in_seq in SCHED.
rewrite mem_filter in SCHED.
move: SCHED ⇒ /andP [_ ARR].
rewrite /actual_arrivals_up_to in ARR.
by apply in_actual_arrivals_between_implies_arrived in ARR.
Qed.
(* Next, we show that jobs do not execute before they arrive...*)
Theorem scheduler_jobs_execute_after_jitter:
jobs_execute_after_jitter job_arrival job_jitter sched.
Proof.
move ⇒ j t /eqP SCHED.
rewrite /sched scheduler_uses_construction_function // /highest_priority_job in SCHED.
apply seq_min_in_seq in SCHED.
rewrite mem_filter in SCHED.
by move: SCHED ⇒ /andP [/andP [ARR _] _].
Qed.
(* ... nor after completion. *)
Theorem scheduler_completed_jobs_dont_execute:
completed_jobs_dont_execute job_cost sched.
Proof.
intros j t.
induction t;
first by rewrite /service /service_during big_geq //.
rewrite /service /service_during big_nat_recr //=.
rewrite leq_eqVlt in IHt; move: IHt ⇒ /orP [/eqP EQ | LT]; last first.
{
apply: leq_trans LT; rewrite -addn1.
by apply leq_add; last by apply leq_b1.
}
rewrite -[job_cost _]addn0; apply leq_add; first by rewrite -EQ.
rewrite leqn0 eqb0 /scheduled_at.
rewrite /sched scheduler_uses_construction_function /highest_priority_job.
apply/eqP; intro HP.
apply seq_min_in_seq in HP.
by rewrite mem_filter /pending /completed_by -EQ leqnn andbF /= in HP.
Qed.
(* In addition, the scheduler is work conserving... *)
Theorem scheduler_work_conserving:
work_conserving job_arrival job_cost job_jitter arr_seq sched.
Proof.
intros j t IN BACK.
move: BACK ⇒ /andP [/andP [ARR NOTCOMP] NOTSCHED].
rewrite /scheduled_at /sched scheduler_uses_construction_function /= in NOTSCHED.
rewrite /scheduled_at /sched scheduler_uses_construction_function /=.
case HP: (highest_priority_job _ _ _ _ _ ) ⇒ [j_hp|]; first by ∃ j_hp.
set hp := highest_priority_job _ _ _ _ _ _ _ in NOTSCHED HP.
suff BUG: hp != None by rewrite HP in BUG.
apply seq_min_exists with (x := j).
rewrite mem_filter /pending ARR NOTCOMP /=.
by eapply arrived_between_implies_in_actual_arrivals, ARR.
Qed.
(* ... and respects the JLDP policy. *)
Theorem scheduler_respects_policy :
respects_JLDP_policy job_arrival job_cost job_jitter arr_seq sched higher_eq_priority.
Proof.
rename H_priority_is_transitive into TRANS, H_priority_is_total into TOTAL.
move ⇒ j1 j2 t ARR1 BACK /eqP SCHED.
move: BACK ⇒ /andP [/andP [ARR NOTCOMP] NOTSCHED].
rewrite /scheduled_at /sched scheduler_uses_construction_function // in NOTSCHED.
rewrite /scheduled_at /sched scheduler_uses_construction_function // in SCHED.
rewrite /highest_priority_job in SCHED NOTSCHED.
set jobs := pending_jobs _ _ _ _ _ _ in SCHED NOTSCHED.
have IN: j1 \in jobs.
{
rewrite mem_filter /pending ARR NOTCOMP /=.
by eapply arrived_between_implies_in_actual_arrivals, ARR.
}
apply seq_min_computes_min with (y := j1) in SCHED; try (by done).
intros x y; rewrite /jobs mem_filter [y \in _]mem_filter.
move ⇒ /andP [_ INx] /andP [_ INy].
rewrite /actual_arrivals_up_to in INx INy.
by apply TOTAL; eapply in_actual_arrivals_between_implies_arrived; eauto 2.
Qed.
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.
(* In this section, we implement a jitter-aware priority-based uniprocessor scheduler *)
Section Implementation.
Context {Job: eqType}.
Variable job_arrival: Job → time.
Variable job_cost: Job → time.
Variable job_jitter: Job → time.
(* Let arr_seq be any job arrival sequence. *)
Variable arr_seq: arrival_sequence Job.
(* Also, assume that a JLDP policy is given. *)
Variable higher_eq_priority: JLDP_policy Job.
(* Next, we show how to recursively construct the schedule. *)
Section ScheduleConstruction.
(* For any time t, suppose that we have generated the schedule prefix in the
interval 0, t). Then, we must define what should be scheduled at time t. *)
Variable sched_prefix: schedule Job.
Variable t: time.
(* For simplicity, let's use some local names. *)
Let is_pending := pending job_arrival job_cost job_jitter sched_prefix.
(* Consider the list of pending jobs with actual arrivals up to t. *)
Definition pending_jobs :=
[seq j <- actual_arrivals_up_to job_arrival job_jitter arr_seq t | is_pending j t].
(* To make the scheduling decision, we just pick one of the highest-priority jobs
that are pending (if it exists). *)
Definition highest_priority_job :=
seq_min (higher_eq_priority t) pending_jobs.
End ScheduleConstruction.
(* Starting from the empty schedule, the final schedule is obtained by iteratively
picking the highest-priority job. *)
Let empty_schedule : schedule Job := fun t ⇒ None.
Definition scheduler :=
build_schedule_from_prefixes highest_priority_job empty_schedule.
(* Then, by showing that the construction function depends only on the prefix, ... *)
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.
Proof.
intros sched1 sched2 t ALL.
rewrite /highest_priority_job.
suff SAME: pending_jobs sched1 t = pending_jobs sched2 t by rewrite SAME.
apply eq_in_filter.
intros j IN; rewrite mem_filter in IN.
move: IN ⇒ /andP [ARR IN].
rewrite ltnS /= in ARR.
rewrite /pending /jitter_has_passed ARR 2!andTb; f_equal.
rewrite /completed_by; f_equal.
apply eq_big_nat; move ⇒ i /= LTi.
by rewrite /service_at /scheduled_at ALL.
Qed.
(* ...we infer that the generated schedule is indeed based on the construction function. *)
Corollary scheduler_uses_construction_function:
∀ t, scheduler t = highest_priority_job scheduler t.
Proof.
by ins; apply prefix_dependent_schedule_construction,
scheduler_depends_only_on_prefix.
Qed.
End Implementation.
(* In this section, we prove the properties of the scheduler that are used
as hypotheses in the analyses. *)
Section Proofs.
Context {Job: eqType}.
Variable job_arrival: Job → time.
Variable job_cost: Job → time.
Variable job_jitter: Job → time.
(* Assume any job arrival sequence with consistent, duplicate-free arrivals. *)
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.
(* Consider any JLDP policy that is reflexive, transitive and total. *)
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 denote our concrete scheduler implementation. *)
Let sched := scheduler job_arrival job_cost job_jitter arr_seq higher_eq_priority.
(* To conclude, we prove the important properties of the scheduler implementation. *)
(* First, we show that scheduled jobs come from the arrival sequence. *)
Lemma scheduler_jobs_come_from_arrival_sequence:
jobs_come_from_arrival_sequence sched arr_seq.
Proof.
move ⇒ j t /eqP SCHED.
rewrite /sched scheduler_uses_construction_function // /highest_priority_job in SCHED.
apply seq_min_in_seq in SCHED.
rewrite mem_filter in SCHED.
move: SCHED ⇒ /andP [_ ARR].
rewrite /actual_arrivals_up_to in ARR.
by apply in_actual_arrivals_between_implies_arrived in ARR.
Qed.
(* Next, we show that jobs do not execute before they arrive...*)
Theorem scheduler_jobs_execute_after_jitter:
jobs_execute_after_jitter job_arrival job_jitter sched.
Proof.
move ⇒ j t /eqP SCHED.
rewrite /sched scheduler_uses_construction_function // /highest_priority_job in SCHED.
apply seq_min_in_seq in SCHED.
rewrite mem_filter in SCHED.
by move: SCHED ⇒ /andP [/andP [ARR _] _].
Qed.
(* ... nor after completion. *)
Theorem scheduler_completed_jobs_dont_execute:
completed_jobs_dont_execute job_cost sched.
Proof.
intros j t.
induction t;
first by rewrite /service /service_during big_geq //.
rewrite /service /service_during big_nat_recr //=.
rewrite leq_eqVlt in IHt; move: IHt ⇒ /orP [/eqP EQ | LT]; last first.
{
apply: leq_trans LT; rewrite -addn1.
by apply leq_add; last by apply leq_b1.
}
rewrite -[job_cost _]addn0; apply leq_add; first by rewrite -EQ.
rewrite leqn0 eqb0 /scheduled_at.
rewrite /sched scheduler_uses_construction_function /highest_priority_job.
apply/eqP; intro HP.
apply seq_min_in_seq in HP.
by rewrite mem_filter /pending /completed_by -EQ leqnn andbF /= in HP.
Qed.
(* In addition, the scheduler is work conserving... *)
Theorem scheduler_work_conserving:
work_conserving job_arrival job_cost job_jitter arr_seq sched.
Proof.
intros j t IN BACK.
move: BACK ⇒ /andP [/andP [ARR NOTCOMP] NOTSCHED].
rewrite /scheduled_at /sched scheduler_uses_construction_function /= in NOTSCHED.
rewrite /scheduled_at /sched scheduler_uses_construction_function /=.
case HP: (highest_priority_job _ _ _ _ _ ) ⇒ [j_hp|]; first by ∃ j_hp.
set hp := highest_priority_job _ _ _ _ _ _ _ in NOTSCHED HP.
suff BUG: hp != None by rewrite HP in BUG.
apply seq_min_exists with (x := j).
rewrite mem_filter /pending ARR NOTCOMP /=.
by eapply arrived_between_implies_in_actual_arrivals, ARR.
Qed.
(* ... and respects the JLDP policy. *)
Theorem scheduler_respects_policy :
respects_JLDP_policy job_arrival job_cost job_jitter arr_seq sched higher_eq_priority.
Proof.
rename H_priority_is_transitive into TRANS, H_priority_is_total into TOTAL.
move ⇒ j1 j2 t ARR1 BACK /eqP SCHED.
move: BACK ⇒ /andP [/andP [ARR NOTCOMP] NOTSCHED].
rewrite /scheduled_at /sched scheduler_uses_construction_function // in NOTSCHED.
rewrite /scheduled_at /sched scheduler_uses_construction_function // in SCHED.
rewrite /highest_priority_job in SCHED NOTSCHED.
set jobs := pending_jobs _ _ _ _ _ _ in SCHED NOTSCHED.
have IN: j1 \in jobs.
{
rewrite mem_filter /pending ARR NOTCOMP /=.
by eapply arrived_between_implies_in_actual_arrivals, ARR.
}
apply seq_min_computes_min with (y := j1) in SCHED; try (by done).
intros x y; rewrite /jobs mem_filter [y \in _]mem_filter.
move ⇒ /andP [_ INx] /andP [_ INy].
rewrite /actual_arrivals_up_to in INx INy.
by apply TOTAL; eapply in_actual_arrivals_between_implies_arrived; eauto 2.
Qed.
End Proofs.
End ConcreteScheduler.