Library prosa.classic.analysis.global.parallel.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.arrival.basic.task_arrival prosa.classic.model.priority.
Require Import prosa.classic.model.schedule.global.workload prosa.classic.model.schedule.global.schedulability
prosa.classic.model.schedule.global.response_time.
Require Import prosa.classic.model.schedule.global.basic.schedule prosa.classic.model.schedule.global.basic.interference
prosa.classic.model.schedule.global.basic.platform.
Require Import prosa.classic.analysis.global.parallel.workload_bound
prosa.classic.analysis.global.parallel.interference_bound_edf.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq fintype bigop div path.
Module ResponseTimeAnalysisEDF.
Export Job SporadicTaskset Schedule ScheduleOfSporadicTask Workload Schedulability ResponseTime
Priority TaskArrival WorkloadBound InterferenceBoundEDF
Interference Platform.
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.
Variable num_cpus: nat.
Variable sched: schedule Job num_cpus.
Hypothesis H_jobs_come_from_arrival_sequence:
jobs_come_from_arrival_sequence sched arr_seq.
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_at_least_one_cpu: num_cpus > 0.
Hypothesis H_work_conserving: work_conserving job_arrival job_cost arr_seq sched.
Hypothesis H_edf_policy: respects_JLFP_policy job_arrival job_cost arr_seq sched
(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.
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 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) num_cpus.
Hypothesis H_tasks_miss_no_deadlines:
∀ tsk_other R,
(tsk_other, R) \in rt_bounds → R ≤ task_deadline tsk_other.
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 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 := different_task tsk.
Let other_tasks :=
[seq tsk_other <- ts | other_task 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_on_all_cpus :
\sum_(tsk_k <- other_tasks) x tsk_k = X × num_cpus.
Lemma bertogna_edf_sum_exceeds_total_interference:
\sum_((tsk_other, R_other) <- rt_bounds | other_task tsk_other)
x tsk_other > I tsk R.
Lemma bertogna_edf_exists_task_that_exceeds_bound :
∃ tsk_other R_other,
(tsk_other, R_other) \in rt_bounds ∧
x tsk_other > 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.arrival.basic.task_arrival prosa.classic.model.priority.
Require Import prosa.classic.model.schedule.global.workload prosa.classic.model.schedule.global.schedulability
prosa.classic.model.schedule.global.response_time.
Require Import prosa.classic.model.schedule.global.basic.schedule prosa.classic.model.schedule.global.basic.interference
prosa.classic.model.schedule.global.basic.platform.
Require Import prosa.classic.analysis.global.parallel.workload_bound
prosa.classic.analysis.global.parallel.interference_bound_edf.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq fintype bigop div path.
Module ResponseTimeAnalysisEDF.
Export Job SporadicTaskset Schedule ScheduleOfSporadicTask Workload Schedulability ResponseTime
Priority TaskArrival WorkloadBound InterferenceBoundEDF
Interference Platform.
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.
Variable num_cpus: nat.
Variable sched: schedule Job num_cpus.
Hypothesis H_jobs_come_from_arrival_sequence:
jobs_come_from_arrival_sequence sched arr_seq.
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_at_least_one_cpu: num_cpus > 0.
Hypothesis H_work_conserving: work_conserving job_arrival job_cost arr_seq sched.
Hypothesis H_edf_policy: respects_JLFP_policy job_arrival job_cost arr_seq sched
(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.
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 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) num_cpus.
Hypothesis H_tasks_miss_no_deadlines:
∀ tsk_other R,
(tsk_other, R) \in rt_bounds → R ≤ task_deadline tsk_other.
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 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 := different_task tsk.
Let other_tasks :=
[seq tsk_other <- ts | other_task 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_on_all_cpus :
\sum_(tsk_k <- other_tasks) x tsk_k = X × num_cpus.
Lemma bertogna_edf_sum_exceeds_total_interference:
\sum_((tsk_other, R_other) <- rt_bounds | other_task tsk_other)
x tsk_other > I tsk R.
Lemma bertogna_edf_exists_task_that_exceeds_bound :
∃ tsk_other R_other,
(tsk_other, R_other) \in rt_bounds ∧
x tsk_other > 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.