Library prosa.classic.analysis.apa.bertogna_fp_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.response_time prosa.classic.model.schedule.global.schedulability
prosa.classic.model.schedule.global.workload.
Require Import prosa.classic.model.schedule.global.basic.schedule.
Require Import prosa.classic.model.schedule.apa.platform prosa.classic.model.schedule.apa.constrained_deadlines
prosa.classic.model.schedule.apa.interference prosa.classic.model.schedule.apa.affinity.
Require Import prosa.classic.analysis.apa.workload_bound
prosa.classic.analysis.apa.interference_bound_fp.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq fintype bigop div path.
Module ResponseTimeAnalysisFP.
Export Job SporadicTaskset ScheduleOfSporadicTask Workload Interference InterferenceBoundFP
Platform Schedulability ResponseTime Priority
TaskArrival WorkloadBound 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.
Variable higher_eq_priority: FP_policy sporadic_task.
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_respects_FP_policy:
respects_FP_policy_under_weak_APA job_arrival job_cost job_task arr_seq
sched alpha higher_eq_priority.
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 tsk: sporadic_task.
Hypothesis task_in_ts: tsk \in ts.
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 hp_task_in alpha' := higher_priority_task_in alpha higher_eq_priority tsk alpha'.
Let task_with_response_time := (sporadic_task × time)%type.
Variable hp_bounds: seq task_with_response_time.
Hypothesis H_response_time_of_interfering_tasks_is_known:
∀ hp_tsk R,
(hp_tsk, R) \in hp_bounds →
is_response_time_bound_of_task job_arrival job_cost job_task arr_seq sched hp_tsk R.
Hypothesis H_hp_bounds_has_interfering_tasks:
∀ hp_tsk,
hp_tsk \in ts →
hp_task_in (alpha tsk) hp_tsk →
∃ R,
(hp_tsk, R) \in hp_bounds.
Hypothesis H_response_time_bounds_ge_cost:
∀ hp_tsk R,
(hp_tsk, R) \in hp_bounds → R ≥ task_cost hp_tsk.
Hypothesis H_interfering_tasks_miss_no_deadlines:
∀ hp_tsk R,
(hp_tsk, R) \in hp_bounds → R ≤ task_deadline hp_tsk.
Variable R: time.
Hypothesis H_response_time_recurrence_holds :
R = task_cost tsk +
div_floor
(total_interference_bound_fp task_cost task_period alpha tsk
(alpha' tsk) hp_bounds R higher_eq_priority)
#|alpha' tsk|.
Hypothesis H_response_time_no_larger_than_deadline:
R ≤ task_deadline tsk.
Section Lemmas.
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_previous_jobs_of_tsk_completed :
∀ j0,
arrives_in arr_seq j0 →
job_task j0 = tsk →
job_arrival j0 < job_arrival j →
completed job_cost sched j0 (job_arrival j0 + R).
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 hp_tasks_in alpha' :=
[seq tsk_other <- ts | hp_task_in (alpha' tsk) tsk_other].
Section LemmasAboutHPTasks.
Variable tsk_other: sporadic_task.
Variable R_other: time.
Hypothesis H_tsk_other_already_processed: (tsk_other, R_other) \in hp_bounds.
Hypothesis H_tsk_other_has_higher_priority: hp_task_in (alpha tsk) tsk_other.
Lemma bertogna_fp_workload_bounds_interference :
x tsk_other ≤ workload_bound tsk_other R_other.
End LemmasAboutHPTasks.
Section DerivingContradiction.
Lemma bertogna_fp_too_much_interference : X ≥ R - task_cost tsk + 1.
Lemma bertogna_fp_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_fp_previous_interfering_jobs_complete_by_their_period:
∀ j0,
arrives_in arr_seq j0 →
hp_task_in (alpha tsk) (job_task j0) →
completed job_cost sched j0
(job_arrival j0 + task_period (job_task j0)).
Lemma bertogna_fp_all_cpus_in_affinity_busy :
\sum_(tsk_k <- hp_tasks_in alpha) x tsk_k = X × #|alpha tsk|.
Lemma bertogna_fp_all_cpus_in_subaffinity_busy :
\sum_(tsk_k <- hp_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_fp_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) (hp_tasks_in alpha') ≥ #|alpha' tsk|.
Let num_tasks_exceeding delta := count (fun i ⇒ x i ≥ delta) (hp_tasks_in alpha').
Lemma bertogna_fp_interference_in_non_full_processors :
∀ delta,
0 < num_tasks_exceeding delta < #|alpha' tsk| →
\sum_(i <- hp_tasks_in alpha' | x i < delta) x i ≥ delta × (#|alpha' tsk| - num_tasks_exceeding delta).
Lemma bertogna_fp_minimum_exceeds_interference :
∀ delta,
\sum_(tsk_k <- hp_tasks_in alpha') x tsk_k ≥ delta × #|alpha' tsk| →
\sum_(tsk_k <- hp_tasks_in alpha') minn (x tsk_k) delta ≥
delta × #|alpha' tsk|.
Lemma bertogna_fp_interference_on_subaffinity :
∀ delta,
\sum_(tsk_k <- hp_tasks_in alpha) x tsk_k ≥ delta × #|alpha tsk| →
\sum_(tsk_k <- hp_tasks_in alpha') x tsk_k ≥ delta × #|alpha' tsk|.
Lemma bertogna_fp_sum_exceeds_total_interference:
\sum_((tsk_other, R_other) <- hp_bounds | hp_task_in (alpha' tsk) tsk_other)
minn (x tsk_other) (R - task_cost tsk + 1) >
total_interference_bound_fp task_cost task_period alpha tsk
(alpha' tsk) hp_bounds R higher_eq_priority.
Lemma bertogna_fp_exists_task_that_exceeds_bound :
∃ tsk_k R_k,
(tsk_k, R_k) \in hp_bounds ∧
(minn (x tsk_k) (R - task_cost tsk + 1) >
minn (workload_bound tsk_k R_k) (R - task_cost tsk + 1)).
End DerivingContradiction.
End Lemmas.
Theorem bertogna_cirinei_response_time_bound_fp :
response_time_bounded_by tsk R.
End ResponseTimeBound.
End ResponseTimeAnalysisFP.
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.response_time prosa.classic.model.schedule.global.schedulability
prosa.classic.model.schedule.global.workload.
Require Import prosa.classic.model.schedule.global.basic.schedule.
Require Import prosa.classic.model.schedule.apa.platform prosa.classic.model.schedule.apa.constrained_deadlines
prosa.classic.model.schedule.apa.interference prosa.classic.model.schedule.apa.affinity.
Require Import prosa.classic.analysis.apa.workload_bound
prosa.classic.analysis.apa.interference_bound_fp.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq fintype bigop div path.
Module ResponseTimeAnalysisFP.
Export Job SporadicTaskset ScheduleOfSporadicTask Workload Interference InterferenceBoundFP
Platform Schedulability ResponseTime Priority
TaskArrival WorkloadBound 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.
Variable higher_eq_priority: FP_policy sporadic_task.
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_respects_FP_policy:
respects_FP_policy_under_weak_APA job_arrival job_cost job_task arr_seq
sched alpha higher_eq_priority.
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 tsk: sporadic_task.
Hypothesis task_in_ts: tsk \in ts.
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 hp_task_in alpha' := higher_priority_task_in alpha higher_eq_priority tsk alpha'.
Let task_with_response_time := (sporadic_task × time)%type.
Variable hp_bounds: seq task_with_response_time.
Hypothesis H_response_time_of_interfering_tasks_is_known:
∀ hp_tsk R,
(hp_tsk, R) \in hp_bounds →
is_response_time_bound_of_task job_arrival job_cost job_task arr_seq sched hp_tsk R.
Hypothesis H_hp_bounds_has_interfering_tasks:
∀ hp_tsk,
hp_tsk \in ts →
hp_task_in (alpha tsk) hp_tsk →
∃ R,
(hp_tsk, R) \in hp_bounds.
Hypothesis H_response_time_bounds_ge_cost:
∀ hp_tsk R,
(hp_tsk, R) \in hp_bounds → R ≥ task_cost hp_tsk.
Hypothesis H_interfering_tasks_miss_no_deadlines:
∀ hp_tsk R,
(hp_tsk, R) \in hp_bounds → R ≤ task_deadline hp_tsk.
Variable R: time.
Hypothesis H_response_time_recurrence_holds :
R = task_cost tsk +
div_floor
(total_interference_bound_fp task_cost task_period alpha tsk
(alpha' tsk) hp_bounds R higher_eq_priority)
#|alpha' tsk|.
Hypothesis H_response_time_no_larger_than_deadline:
R ≤ task_deadline tsk.
Section Lemmas.
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_previous_jobs_of_tsk_completed :
∀ j0,
arrives_in arr_seq j0 →
job_task j0 = tsk →
job_arrival j0 < job_arrival j →
completed job_cost sched j0 (job_arrival j0 + R).
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 hp_tasks_in alpha' :=
[seq tsk_other <- ts | hp_task_in (alpha' tsk) tsk_other].
Section LemmasAboutHPTasks.
Variable tsk_other: sporadic_task.
Variable R_other: time.
Hypothesis H_tsk_other_already_processed: (tsk_other, R_other) \in hp_bounds.
Hypothesis H_tsk_other_has_higher_priority: hp_task_in (alpha tsk) tsk_other.
Lemma bertogna_fp_workload_bounds_interference :
x tsk_other ≤ workload_bound tsk_other R_other.
End LemmasAboutHPTasks.
Section DerivingContradiction.
Lemma bertogna_fp_too_much_interference : X ≥ R - task_cost tsk + 1.
Lemma bertogna_fp_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_fp_previous_interfering_jobs_complete_by_their_period:
∀ j0,
arrives_in arr_seq j0 →
hp_task_in (alpha tsk) (job_task j0) →
completed job_cost sched j0
(job_arrival j0 + task_period (job_task j0)).
Lemma bertogna_fp_all_cpus_in_affinity_busy :
\sum_(tsk_k <- hp_tasks_in alpha) x tsk_k = X × #|alpha tsk|.
Lemma bertogna_fp_all_cpus_in_subaffinity_busy :
\sum_(tsk_k <- hp_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_fp_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) (hp_tasks_in alpha') ≥ #|alpha' tsk|.
Let num_tasks_exceeding delta := count (fun i ⇒ x i ≥ delta) (hp_tasks_in alpha').
Lemma bertogna_fp_interference_in_non_full_processors :
∀ delta,
0 < num_tasks_exceeding delta < #|alpha' tsk| →
\sum_(i <- hp_tasks_in alpha' | x i < delta) x i ≥ delta × (#|alpha' tsk| - num_tasks_exceeding delta).
Lemma bertogna_fp_minimum_exceeds_interference :
∀ delta,
\sum_(tsk_k <- hp_tasks_in alpha') x tsk_k ≥ delta × #|alpha' tsk| →
\sum_(tsk_k <- hp_tasks_in alpha') minn (x tsk_k) delta ≥
delta × #|alpha' tsk|.
Lemma bertogna_fp_interference_on_subaffinity :
∀ delta,
\sum_(tsk_k <- hp_tasks_in alpha) x tsk_k ≥ delta × #|alpha tsk| →
\sum_(tsk_k <- hp_tasks_in alpha') x tsk_k ≥ delta × #|alpha' tsk|.
Lemma bertogna_fp_sum_exceeds_total_interference:
\sum_((tsk_other, R_other) <- hp_bounds | hp_task_in (alpha' tsk) tsk_other)
minn (x tsk_other) (R - task_cost tsk + 1) >
total_interference_bound_fp task_cost task_period alpha tsk
(alpha' tsk) hp_bounds R higher_eq_priority.
Lemma bertogna_fp_exists_task_that_exceeds_bound :
∃ tsk_k R_k,
(tsk_k, R_k) \in hp_bounds ∧
(minn (x tsk_k) (R - task_cost tsk + 1) >
minn (workload_bound tsk_k R_k) (R - task_cost tsk + 1)).
End DerivingContradiction.
End Lemmas.
Theorem bertogna_cirinei_response_time_bound_fp :
response_time_bounded_by tsk R.
End ResponseTimeBound.
End ResponseTimeAnalysisFP.