Library prosa.classic.analysis.global.jitter.bertogna_fp_theory
Require Import prosa.classic.util.all.
Require Import prosa.classic.model.arrival.basic.task 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.jitter.job prosa.classic.model.schedule.global.jitter.interference
prosa.classic.model.schedule.global.jitter.schedule prosa.classic.model.schedule.global.jitter.platform
prosa.classic.model.schedule.global.jitter.constrained_deadlines.
Require Import prosa.classic.analysis.global.jitter.workload_bound
prosa.classic.analysis.global.jitter.interference_bound_fp.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq fintype bigop div path.
Module ResponseTimeAnalysisFP.
Export JobWithJitter SporadicTaskset ScheduleOfSporadicTaskWithJitter
Workload Interference Platform ConstrainedDeadlines Schedulability
ResponseTime Priority TaskArrival WorkloadBoundJitter
Interference InterferenceBoundFP.
Section ResponseTimeBound.
Context {sporadic_task: eqType}.
Variable task_cost: sporadic_task → time.
Variable task_period: sporadic_task → time.
Variable task_deadline: sporadic_task → time.
Variable task_jitter: 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 job_jitter: Job → time.
Context {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_with_jitter task_cost task_deadline task_jitter job_cost
job_deadline job_task job_jitter 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_sequential_jobs: sequential_jobs sched.
Hypothesis H_jobs_execute_after_jitter:
jobs_execute_after_jitter job_arrival job_jitter sched.
Hypothesis H_completed_jobs_dont_execute:
completed_jobs_dont_execute job_cost sched.
Hypothesis H_at_least_one_cpu: num_cpus > 0.
Variable higher_eq_priority: FP_policy sporadic_task.
Hypothesis H_work_conserving: work_conserving job_arrival job_cost job_jitter arr_seq sched.
Hypothesis H_respects_priority:
respects_FP_policy job_arrival job_cost job_task job_jitter arr_seq sched 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.
Let is_hp_task := higher_priority_task higher_eq_priority tsk.
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 →
response_time_bounded_by hp_tsk (task_jitter hp_tsk + R).
Hypothesis H_hp_bounds_has_interfering_tasks:
∀ hp_tsk,
hp_tsk \in ts →
is_hp_task 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 →
task_jitter hp_tsk + 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 task_jitter
tsk hp_bounds R)
num_cpus.
Hypothesis H_response_time_no_larger_than_deadline:
task_jitter tsk + R ≤ task_deadline tsk.
Section Lemmas.
Variable j: Job.
Hypothesis H_job_arrives: arrives_in arr_seq j.
Hypothesis H_job_of_tsk: job_task j = tsk.
Let t1 := job_arrival j + job_jitter j.
Hypothesis H_j_not_completed: ~~ completed job_cost sched j (t1 + 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 + task_jitter tsk + R).
Let x (tsk_other: sporadic_task) :=
task_interference job_arrival job_cost job_task job_jitter sched j tsk_other t1 (t1 + R).
Let X := total_interference job_arrival job_cost job_jitter sched j t1 (t1 + R).
Let workload_bound (tsk_other: sporadic_task) (R_other: time) :=
W_jitter task_cost task_period task_jitter tsk_other R_other R.
Let hp_tasks := [seq tsk_other <- ts | is_hp_task tsk_other].
Section LemmasAboutHPTasks.
Variable tsk_other: sporadic_task.
Variable R_other: time.
Hypothesis H_response_time_of_tsk_other: (tsk_other, R_other) \in hp_bounds.
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,
t1 ≤ t < t1 + R →
arrives_in arr_seq j_other →
backlogged job_arrival job_cost job_jitter sched j t →
scheduled sched j_other t →
job_task j_other != tsk.
Let other_scheduled_task (t: time) (tsk_other: sporadic_task) :=
task_is_scheduled job_task sched tsk_other t &&
is_hp_task tsk_other.
Lemma bertogna_fp_all_cpus_are_busy:
∀ t,
t1 ≤ t < t1 + R →
backlogged job_arrival job_cost job_jitter sched j t →
count (other_scheduled_task t) ts = num_cpus.
Lemma bertogna_fp_interference_on_all_cpus:
\sum_(tsk_k <- hp_tasks) x tsk_k = X × num_cpus.
Let num_tasks_exceeding delta := count (fun i ⇒ x i ≥ delta) (hp_tasks).
Lemma bertogna_fp_interference_in_non_full_processors :
∀ delta,
0 < num_tasks_exceeding delta < num_cpus →
\sum_(i <- hp_tasks | x i < delta) x i ≥ delta × (num_cpus - num_tasks_exceeding delta).
Lemma bertogna_fp_minimum_exceeds_interference :
∀ delta,
\sum_(tsk_k <- hp_tasks) x tsk_k ≥ delta × num_cpus →
\sum_(tsk_k <- hp_tasks) minn (x tsk_k) delta ≥
delta × num_cpus.
Lemma bertogna_fp_sum_exceeds_total_interference:
\sum_((tsk_k, R_k) <- hp_bounds)
minn (x tsk_k) (R - task_cost tsk + 1) >
total_interference_bound_fp task_cost task_period task_jitter tsk hp_bounds R.
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 (task_jitter tsk + R).
End ResponseTimeBound.
End ResponseTimeAnalysisFP.
Require Import prosa.classic.model.arrival.basic.task 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.jitter.job prosa.classic.model.schedule.global.jitter.interference
prosa.classic.model.schedule.global.jitter.schedule prosa.classic.model.schedule.global.jitter.platform
prosa.classic.model.schedule.global.jitter.constrained_deadlines.
Require Import prosa.classic.analysis.global.jitter.workload_bound
prosa.classic.analysis.global.jitter.interference_bound_fp.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq fintype bigop div path.
Module ResponseTimeAnalysisFP.
Export JobWithJitter SporadicTaskset ScheduleOfSporadicTaskWithJitter
Workload Interference Platform ConstrainedDeadlines Schedulability
ResponseTime Priority TaskArrival WorkloadBoundJitter
Interference InterferenceBoundFP.
Section ResponseTimeBound.
Context {sporadic_task: eqType}.
Variable task_cost: sporadic_task → time.
Variable task_period: sporadic_task → time.
Variable task_deadline: sporadic_task → time.
Variable task_jitter: 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 job_jitter: Job → time.
Context {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_with_jitter task_cost task_deadline task_jitter job_cost
job_deadline job_task job_jitter 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_sequential_jobs: sequential_jobs sched.
Hypothesis H_jobs_execute_after_jitter:
jobs_execute_after_jitter job_arrival job_jitter sched.
Hypothesis H_completed_jobs_dont_execute:
completed_jobs_dont_execute job_cost sched.
Hypothesis H_at_least_one_cpu: num_cpus > 0.
Variable higher_eq_priority: FP_policy sporadic_task.
Hypothesis H_work_conserving: work_conserving job_arrival job_cost job_jitter arr_seq sched.
Hypothesis H_respects_priority:
respects_FP_policy job_arrival job_cost job_task job_jitter arr_seq sched 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.
Let is_hp_task := higher_priority_task higher_eq_priority tsk.
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 →
response_time_bounded_by hp_tsk (task_jitter hp_tsk + R).
Hypothesis H_hp_bounds_has_interfering_tasks:
∀ hp_tsk,
hp_tsk \in ts →
is_hp_task 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 →
task_jitter hp_tsk + 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 task_jitter
tsk hp_bounds R)
num_cpus.
Hypothesis H_response_time_no_larger_than_deadline:
task_jitter tsk + R ≤ task_deadline tsk.
Section Lemmas.
Variable j: Job.
Hypothesis H_job_arrives: arrives_in arr_seq j.
Hypothesis H_job_of_tsk: job_task j = tsk.
Let t1 := job_arrival j + job_jitter j.
Hypothesis H_j_not_completed: ~~ completed job_cost sched j (t1 + 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 + task_jitter tsk + R).
Let x (tsk_other: sporadic_task) :=
task_interference job_arrival job_cost job_task job_jitter sched j tsk_other t1 (t1 + R).
Let X := total_interference job_arrival job_cost job_jitter sched j t1 (t1 + R).
Let workload_bound (tsk_other: sporadic_task) (R_other: time) :=
W_jitter task_cost task_period task_jitter tsk_other R_other R.
Let hp_tasks := [seq tsk_other <- ts | is_hp_task tsk_other].
Section LemmasAboutHPTasks.
Variable tsk_other: sporadic_task.
Variable R_other: time.
Hypothesis H_response_time_of_tsk_other: (tsk_other, R_other) \in hp_bounds.
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,
t1 ≤ t < t1 + R →
arrives_in arr_seq j_other →
backlogged job_arrival job_cost job_jitter sched j t →
scheduled sched j_other t →
job_task j_other != tsk.
Let other_scheduled_task (t: time) (tsk_other: sporadic_task) :=
task_is_scheduled job_task sched tsk_other t &&
is_hp_task tsk_other.
Lemma bertogna_fp_all_cpus_are_busy:
∀ t,
t1 ≤ t < t1 + R →
backlogged job_arrival job_cost job_jitter sched j t →
count (other_scheduled_task t) ts = num_cpus.
Lemma bertogna_fp_interference_on_all_cpus:
\sum_(tsk_k <- hp_tasks) x tsk_k = X × num_cpus.
Let num_tasks_exceeding delta := count (fun i ⇒ x i ≥ delta) (hp_tasks).
Lemma bertogna_fp_interference_in_non_full_processors :
∀ delta,
0 < num_tasks_exceeding delta < num_cpus →
\sum_(i <- hp_tasks | x i < delta) x i ≥ delta × (num_cpus - num_tasks_exceeding delta).
Lemma bertogna_fp_minimum_exceeds_interference :
∀ delta,
\sum_(tsk_k <- hp_tasks) x tsk_k ≥ delta × num_cpus →
\sum_(tsk_k <- hp_tasks) minn (x tsk_k) delta ≥
delta × num_cpus.
Lemma bertogna_fp_sum_exceeds_total_interference:
\sum_((tsk_k, R_k) <- hp_bounds)
minn (x tsk_k) (R - task_cost tsk + 1) >
total_interference_bound_fp task_cost task_period task_jitter tsk hp_bounds R.
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 (task_jitter tsk + R).
End ResponseTimeBound.
End ResponseTimeAnalysisFP.