Library prosa.classic.analysis.apa.bertogna_edf_theory
Require Import prosa.classic.util.all.
Require Import prosa.classic.model.arrival.basic.task prosa.classic.model.arrival.basic.job prosa.classic.model.priority prosa.classic.model.arrival.basic.task_arrival.
Require Import prosa.classic.model.schedule.global.workload prosa.classic.model.schedule.global.response_time
prosa.classic.model.schedule.global.schedulability.
Require Import prosa.classic.model.schedule.global.basic.schedule.
Require Import prosa.classic.model.schedule.apa.platform prosa.classic.model.schedule.apa.interference
prosa.classic.model.schedule.apa.affinity prosa.classic.model.schedule.apa.constrained_deadlines.
Require Import prosa.classic.analysis.apa.workload_bound prosa.classic.analysis.apa.interference_bound_edf.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq fintype bigop div path.
Module ResponseTimeAnalysisEDF.
Export Job SporadicTaskset ScheduleOfSporadicTask Workload Schedulability ResponseTime
Priority TaskArrival WorkloadBound InterferenceBoundEDF
Interference Platform Affinity ConstrainedDeadlines.
Section ResponseTimeBound.
Context {sporadic_task: eqType}.
Variable task_cost: sporadic_task → time.
Variable task_period: sporadic_task → time.
Variable task_deadline: sporadic_task → time.
Context {Job: eqType}.
Variable job_arrival: Job → time.
Variable job_cost: Job → time.
Variable job_deadline: Job → time.
Variable job_task: Job → sporadic_task.
Variable arr_seq: arrival_sequence Job.
Hypothesis H_sporadic_tasks:
sporadic_task_model task_period job_arrival job_task arr_seq.
Hypothesis H_valid_job_parameters:
∀ j,
arrives_in arr_seq j →
valid_sporadic_job task_cost task_deadline job_cost job_deadline job_task j.
Variable ts: taskset_of sporadic_task.
Hypothesis H_valid_task_parameters:
valid_sporadic_taskset task_cost task_period task_deadline ts.
Hypothesis H_constrained_deadlines:
∀ tsk, tsk \in ts → task_deadline tsk ≤ task_period tsk.
Hypothesis H_all_jobs_from_taskset:
∀ j, arrives_in arr_seq j → job_task j \in ts.
Context {num_cpus: nat}.
Variable alpha: task_affinity sporadic_task num_cpus.
Variable sched: schedule Job num_cpus.
Hypothesis H_jobs_come_from_arrival_sequence:
jobs_come_from_arrival_sequence sched arr_seq.
Hypothesis H_sequential_jobs: sequential_jobs sched.
Hypothesis H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute job_arrival sched.
Hypothesis H_completed_jobs_dont_execute: completed_jobs_dont_execute job_cost sched.
Hypothesis H_respects_affinity: respects_affinity job_task sched alpha.
Hypothesis H_work_conserving: apa_work_conserving job_arrival job_cost job_task arr_seq
sched alpha.
Hypothesis H_edf_policy:
respects_JLFP_policy_under_weak_APA job_arrival job_cost job_task arr_seq
sched alpha (EDF job_arrival job_deadline).
Let no_deadline_is_missed_by_tsk (tsk: sporadic_task) :=
task_misses_no_deadline job_arrival job_cost job_deadline job_task arr_seq sched tsk.
Let response_time_bounded_by (tsk: sporadic_task) :=
is_response_time_bound_of_task job_arrival job_cost job_task arr_seq sched tsk.
Variable alpha': task_affinity sporadic_task num_cpus.
Hypothesis H_affinity_subset: ∀ tsk, tsk \in ts → is_subaffinity (alpha' tsk) (alpha tsk).
Hypothesis H_at_least_one_cpu : ∀ tsk, tsk \in ts → #|alpha' tsk| > 0.
Let task_with_response_time := (sporadic_task × time)%type.
Variable rt_bounds: seq task_with_response_time.
Hypothesis H_rt_bounds_contains_all_tasks: unzip1 rt_bounds = ts.
Let I (tsk: sporadic_task) (delta: time) :=
total_interference_bound_edf task_cost task_period task_deadline alpha
tsk (alpha' tsk) rt_bounds delta.
Hypothesis H_response_time_is_fixed_point :
∀ tsk R,
(tsk, R) \in rt_bounds →
R = task_cost tsk + div_floor (I tsk R) #|alpha' tsk|.
Hypothesis H_tasks_miss_no_deadlines:
∀ tsk R,
(tsk, R) \in rt_bounds → R ≤ task_deadline tsk.
Section Lemmas.
Variable tsk: sporadic_task.
Variable R: time.
Hypothesis H_tsk_R_in_rt_bounds: (tsk, R) \in rt_bounds.
Variable j: Job.
Hypothesis H_j_arrives: arrives_in arr_seq j.
Hypothesis H_job_of_tsk: job_task j = tsk.
Hypothesis H_j_not_completed: ~~ completed job_cost sched j (job_arrival j + R).
Hypothesis H_all_previous_jobs_completed_on_time :
∀ j_other tsk_other R_other,
arrives_in arr_seq j_other →
job_task j_other = tsk_other →
(tsk_other, R_other) \in rt_bounds →
job_arrival j_other + R_other < job_arrival j + R →
completed job_cost sched j_other (job_arrival j_other + R_other).
Let x (tsk_other: sporadic_task) :=
task_interference job_arrival job_cost job_task sched alpha j
tsk_other (job_arrival j) (job_arrival j + R).
Let X := total_interference job_arrival job_cost sched j (job_arrival j) (job_arrival j + R).
Let workload_bound (tsk_other: sporadic_task) (R_other: time) :=
W task_cost task_period tsk_other R_other R.
Let edf_specific_bound (tsk_other: sporadic_task) (R_other: time) :=
edf_specific_interference_bound task_cost task_period task_deadline tsk tsk_other R_other.
Let interference_bound (tsk_other: sporadic_task) (R_other: time) :=
interference_bound_edf task_cost task_period task_deadline tsk R (tsk_other, R_other).
Let other_task_in alpha' := different_task_in alpha tsk alpha'.
Let other_tasks_in alpha' :=
[seq tsk_other <- ts | other_task_in (alpha' tsk) tsk_other].
Section LemmasAboutInterferingTasks.
Variable tsk_other: sporadic_task.
Variable R_other: time.
Hypothesis H_response_time_of_tsk_other: (tsk_other, R_other) \in rt_bounds.
Lemma bertogna_edf_tsk_other_in_ts: tsk_other \in ts.
Lemma bertogna_edf_R_other_ge_cost :
R_other ≥ task_cost tsk_other.
Lemma bertogna_edf_workload_bounds_interference :
x tsk_other ≤ workload_bound tsk_other R_other.
Lemma bertogna_edf_specific_bound_holds :
x tsk_other ≤ edf_specific_bound tsk_other R_other.
End LemmasAboutInterferingTasks.
Section DerivingContradiction.
Lemma bertogna_edf_too_much_interference : X ≥ R - task_cost tsk + 1.
Lemma bertogna_edf_interference_by_different_tasks :
∀ t j_other,
job_arrival j ≤ t < job_arrival j + R →
arrives_in arr_seq j_other →
backlogged job_arrival job_cost sched j t →
scheduled sched j_other t →
job_task j_other != tsk.
Lemma bertogna_edf_all_previous_jobs_complete_by_their_period:
∀ t j0,
arrives_in arr_seq j0 →
t < job_arrival j + R →
job_arrival j0 + task_period (job_task j0) ≤ t →
completed job_cost sched j0
(job_arrival j0 + task_period (job_task j0)).
Lemma bertogna_edf_all_cpus_in_affinity_busy :
\sum_(tsk_k <- other_tasks_in alpha) x tsk_k = X × #|alpha tsk|.
Lemma bertogna_edf_all_cpus_in_subaffinity_busy :
\sum_(tsk_k <- other_tasks_in alpha') x tsk_k ≥ X × #|alpha' tsk|.
Let scheduled_on_alpha_tsk := fun t tsk_k ⇒
task_scheduled_on_affinity job_task sched (alpha tsk) tsk_k t.
Lemma bertogna_edf_alpha'_is_full:
∀ t,
job_arrival j ≤ t < job_arrival j + R →
backlogged job_arrival job_cost sched j t →
count (scheduled_on_alpha_tsk t) (other_tasks_in alpha') ≥ #|alpha' tsk|.
Let num_tasks_exceeding delta := count (fun i ⇒ x i ≥ delta) (other_tasks_in alpha').
Lemma bertogna_edf_interference_in_non_full_processors :
∀ delta,
0 < num_tasks_exceeding delta < #|alpha' tsk| →
\sum_(i <- other_tasks_in alpha' | x i < delta) x i ≥ delta × (#|alpha' tsk| - num_tasks_exceeding delta).
Lemma bertogna_edf_minimum_exceeds_interference :
∀ delta,
\sum_(tsk_k <- other_tasks_in alpha') x tsk_k ≥ delta × #|alpha' tsk| →
\sum_(tsk_k <- other_tasks_in alpha') minn (x tsk_k) delta ≥
delta × #|alpha' tsk|.
Lemma bertogna_edf_interference_on_subaffinity :
∀ delta,
\sum_(tsk_k <- other_tasks_in alpha) x tsk_k ≥ delta × #|alpha tsk| →
\sum_(tsk_k <- other_tasks_in alpha') x tsk_k ≥ delta × #|alpha' tsk|.
Lemma bertogna_edf_sum_exceeds_total_interference:
\sum_((tsk_other, R_other) <- rt_bounds | other_task_in (alpha' tsk) tsk_other)
minn (x tsk_other) (R - task_cost tsk + 1) > I tsk R.
Lemma bertogna_edf_exists_task_that_exceeds_bound :
∃ tsk_other R_other,
(tsk_other, R_other) \in rt_bounds ∧
(minn (x tsk_other) (R - task_cost tsk + 1) > interference_bound tsk_other R_other).
End DerivingContradiction.
End Lemmas.
Section MainProof.
Variable tsk: sporadic_task.
Variable R: time.
Hypothesis H_tsk_R_in_rt_bounds: (tsk, R) \in rt_bounds.
Theorem bertogna_cirinei_response_time_bound_edf :
response_time_bounded_by tsk R.
End MainProof.
End ResponseTimeBound.
End ResponseTimeAnalysisEDF.
Require Import prosa.classic.model.arrival.basic.task prosa.classic.model.arrival.basic.job prosa.classic.model.priority prosa.classic.model.arrival.basic.task_arrival.
Require Import prosa.classic.model.schedule.global.workload prosa.classic.model.schedule.global.response_time
prosa.classic.model.schedule.global.schedulability.
Require Import prosa.classic.model.schedule.global.basic.schedule.
Require Import prosa.classic.model.schedule.apa.platform prosa.classic.model.schedule.apa.interference
prosa.classic.model.schedule.apa.affinity prosa.classic.model.schedule.apa.constrained_deadlines.
Require Import prosa.classic.analysis.apa.workload_bound prosa.classic.analysis.apa.interference_bound_edf.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq fintype bigop div path.
Module ResponseTimeAnalysisEDF.
Export Job SporadicTaskset ScheduleOfSporadicTask Workload Schedulability ResponseTime
Priority TaskArrival WorkloadBound InterferenceBoundEDF
Interference Platform Affinity ConstrainedDeadlines.
Section ResponseTimeBound.
Context {sporadic_task: eqType}.
Variable task_cost: sporadic_task → time.
Variable task_period: sporadic_task → time.
Variable task_deadline: sporadic_task → time.
Context {Job: eqType}.
Variable job_arrival: Job → time.
Variable job_cost: Job → time.
Variable job_deadline: Job → time.
Variable job_task: Job → sporadic_task.
Variable arr_seq: arrival_sequence Job.
Hypothesis H_sporadic_tasks:
sporadic_task_model task_period job_arrival job_task arr_seq.
Hypothesis H_valid_job_parameters:
∀ j,
arrives_in arr_seq j →
valid_sporadic_job task_cost task_deadline job_cost job_deadline job_task j.
Variable ts: taskset_of sporadic_task.
Hypothesis H_valid_task_parameters:
valid_sporadic_taskset task_cost task_period task_deadline ts.
Hypothesis H_constrained_deadlines:
∀ tsk, tsk \in ts → task_deadline tsk ≤ task_period tsk.
Hypothesis H_all_jobs_from_taskset:
∀ j, arrives_in arr_seq j → job_task j \in ts.
Context {num_cpus: nat}.
Variable alpha: task_affinity sporadic_task num_cpus.
Variable sched: schedule Job num_cpus.
Hypothesis H_jobs_come_from_arrival_sequence:
jobs_come_from_arrival_sequence sched arr_seq.
Hypothesis H_sequential_jobs: sequential_jobs sched.
Hypothesis H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute job_arrival sched.
Hypothesis H_completed_jobs_dont_execute: completed_jobs_dont_execute job_cost sched.
Hypothesis H_respects_affinity: respects_affinity job_task sched alpha.
Hypothesis H_work_conserving: apa_work_conserving job_arrival job_cost job_task arr_seq
sched alpha.
Hypothesis H_edf_policy:
respects_JLFP_policy_under_weak_APA job_arrival job_cost job_task arr_seq
sched alpha (EDF job_arrival job_deadline).
Let no_deadline_is_missed_by_tsk (tsk: sporadic_task) :=
task_misses_no_deadline job_arrival job_cost job_deadline job_task arr_seq sched tsk.
Let response_time_bounded_by (tsk: sporadic_task) :=
is_response_time_bound_of_task job_arrival job_cost job_task arr_seq sched tsk.
Variable alpha': task_affinity sporadic_task num_cpus.
Hypothesis H_affinity_subset: ∀ tsk, tsk \in ts → is_subaffinity (alpha' tsk) (alpha tsk).
Hypothesis H_at_least_one_cpu : ∀ tsk, tsk \in ts → #|alpha' tsk| > 0.
Let task_with_response_time := (sporadic_task × time)%type.
Variable rt_bounds: seq task_with_response_time.
Hypothesis H_rt_bounds_contains_all_tasks: unzip1 rt_bounds = ts.
Let I (tsk: sporadic_task) (delta: time) :=
total_interference_bound_edf task_cost task_period task_deadline alpha
tsk (alpha' tsk) rt_bounds delta.
Hypothesis H_response_time_is_fixed_point :
∀ tsk R,
(tsk, R) \in rt_bounds →
R = task_cost tsk + div_floor (I tsk R) #|alpha' tsk|.
Hypothesis H_tasks_miss_no_deadlines:
∀ tsk R,
(tsk, R) \in rt_bounds → R ≤ task_deadline tsk.
Section Lemmas.
Variable tsk: sporadic_task.
Variable R: time.
Hypothesis H_tsk_R_in_rt_bounds: (tsk, R) \in rt_bounds.
Variable j: Job.
Hypothesis H_j_arrives: arrives_in arr_seq j.
Hypothesis H_job_of_tsk: job_task j = tsk.
Hypothesis H_j_not_completed: ~~ completed job_cost sched j (job_arrival j + R).
Hypothesis H_all_previous_jobs_completed_on_time :
∀ j_other tsk_other R_other,
arrives_in arr_seq j_other →
job_task j_other = tsk_other →
(tsk_other, R_other) \in rt_bounds →
job_arrival j_other + R_other < job_arrival j + R →
completed job_cost sched j_other (job_arrival j_other + R_other).
Let x (tsk_other: sporadic_task) :=
task_interference job_arrival job_cost job_task sched alpha j
tsk_other (job_arrival j) (job_arrival j + R).
Let X := total_interference job_arrival job_cost sched j (job_arrival j) (job_arrival j + R).
Let workload_bound (tsk_other: sporadic_task) (R_other: time) :=
W task_cost task_period tsk_other R_other R.
Let edf_specific_bound (tsk_other: sporadic_task) (R_other: time) :=
edf_specific_interference_bound task_cost task_period task_deadline tsk tsk_other R_other.
Let interference_bound (tsk_other: sporadic_task) (R_other: time) :=
interference_bound_edf task_cost task_period task_deadline tsk R (tsk_other, R_other).
Let other_task_in alpha' := different_task_in alpha tsk alpha'.
Let other_tasks_in alpha' :=
[seq tsk_other <- ts | other_task_in (alpha' tsk) tsk_other].
Section LemmasAboutInterferingTasks.
Variable tsk_other: sporadic_task.
Variable R_other: time.
Hypothesis H_response_time_of_tsk_other: (tsk_other, R_other) \in rt_bounds.
Lemma bertogna_edf_tsk_other_in_ts: tsk_other \in ts.
Lemma bertogna_edf_R_other_ge_cost :
R_other ≥ task_cost tsk_other.
Lemma bertogna_edf_workload_bounds_interference :
x tsk_other ≤ workload_bound tsk_other R_other.
Lemma bertogna_edf_specific_bound_holds :
x tsk_other ≤ edf_specific_bound tsk_other R_other.
End LemmasAboutInterferingTasks.
Section DerivingContradiction.
Lemma bertogna_edf_too_much_interference : X ≥ R - task_cost tsk + 1.
Lemma bertogna_edf_interference_by_different_tasks :
∀ t j_other,
job_arrival j ≤ t < job_arrival j + R →
arrives_in arr_seq j_other →
backlogged job_arrival job_cost sched j t →
scheduled sched j_other t →
job_task j_other != tsk.
Lemma bertogna_edf_all_previous_jobs_complete_by_their_period:
∀ t j0,
arrives_in arr_seq j0 →
t < job_arrival j + R →
job_arrival j0 + task_period (job_task j0) ≤ t →
completed job_cost sched j0
(job_arrival j0 + task_period (job_task j0)).
Lemma bertogna_edf_all_cpus_in_affinity_busy :
\sum_(tsk_k <- other_tasks_in alpha) x tsk_k = X × #|alpha tsk|.
Lemma bertogna_edf_all_cpus_in_subaffinity_busy :
\sum_(tsk_k <- other_tasks_in alpha') x tsk_k ≥ X × #|alpha' tsk|.
Let scheduled_on_alpha_tsk := fun t tsk_k ⇒
task_scheduled_on_affinity job_task sched (alpha tsk) tsk_k t.
Lemma bertogna_edf_alpha'_is_full:
∀ t,
job_arrival j ≤ t < job_arrival j + R →
backlogged job_arrival job_cost sched j t →
count (scheduled_on_alpha_tsk t) (other_tasks_in alpha') ≥ #|alpha' tsk|.
Let num_tasks_exceeding delta := count (fun i ⇒ x i ≥ delta) (other_tasks_in alpha').
Lemma bertogna_edf_interference_in_non_full_processors :
∀ delta,
0 < num_tasks_exceeding delta < #|alpha' tsk| →
\sum_(i <- other_tasks_in alpha' | x i < delta) x i ≥ delta × (#|alpha' tsk| - num_tasks_exceeding delta).
Lemma bertogna_edf_minimum_exceeds_interference :
∀ delta,
\sum_(tsk_k <- other_tasks_in alpha') x tsk_k ≥ delta × #|alpha' tsk| →
\sum_(tsk_k <- other_tasks_in alpha') minn (x tsk_k) delta ≥
delta × #|alpha' tsk|.
Lemma bertogna_edf_interference_on_subaffinity :
∀ delta,
\sum_(tsk_k <- other_tasks_in alpha) x tsk_k ≥ delta × #|alpha tsk| →
\sum_(tsk_k <- other_tasks_in alpha') x tsk_k ≥ delta × #|alpha' tsk|.
Lemma bertogna_edf_sum_exceeds_total_interference:
\sum_((tsk_other, R_other) <- rt_bounds | other_task_in (alpha' tsk) tsk_other)
minn (x tsk_other) (R - task_cost tsk + 1) > I tsk R.
Lemma bertogna_edf_exists_task_that_exceeds_bound :
∃ tsk_other R_other,
(tsk_other, R_other) \in rt_bounds ∧
(minn (x tsk_other) (R - task_cost tsk + 1) > interference_bound tsk_other R_other).
End DerivingContradiction.
End Lemmas.
Section MainProof.
Variable tsk: sporadic_task.
Variable R: time.
Hypothesis H_tsk_R_in_rt_bounds: (tsk, R) \in rt_bounds.
Theorem bertogna_cirinei_response_time_bound_edf :
response_time_bounded_by tsk R.
End MainProof.
End ResponseTimeBound.
End ResponseTimeAnalysisEDF.