Library rt.analysis.basic.bertogna_edf_theory
Require Import
rt.util.all.
Require Import rt.model.basic.task rt.model.basic.job rt.model.basic.task_arrival
rt.model.basic.schedule rt.model.basic.platform rt.model.basic.interference
rt.model.basic.workload rt.model.basic.schedulability rt.model.basic.priority
rt.model.basic.platform rt.model.basic.response_time.
Require Import rt.analysis.basic.workload_bound rt.analysis.basic.interference_bound_edf.
Module ResponseTimeAnalysisEDF.
Export Job SporadicTaskset Schedule ScheduleOfSporadicTask Workload Schedulability ResponseTime
Priority SporadicTaskArrival WorkloadBound InterferenceBoundEDF
Interference Platform.
(* In this section, we prove that Bertogna and Cirinei's RTA yields
safe response-time bounds. *)
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_cost: Job → time.
Variable job_deadline: Job → time.
Variable job_task: Job → sporadic_task.
(* Assume any job arrival sequence... *)
Context {arr_seq: arrival_sequence Job}.
(* ... in which jobs arrive sporadically and have valid parameters. *)
Hypothesis H_sporadic_tasks:
sporadic_task_model task_period arr_seq job_task.
Hypothesis H_valid_job_parameters:
∀ (j: JobIn arr_seq),
valid_sporadic_job task_cost task_deadline job_cost job_deadline job_task j.
(* Consider any schedule such that...*)
Variable num_cpus: nat.
Variable sched: schedule num_cpus arr_seq.
(* ...jobs do not execute before their arrival times nor longer
than their execution costs. *)
Hypothesis H_jobs_must_arrive_to_execute:
jobs_must_arrive_to_execute sched.
Hypothesis H_completed_jobs_dont_execute:
completed_jobs_dont_execute job_cost sched.
(* Also assume that jobs are sequential and that
there exists at least one processor. *)
Hypothesis H_sequential_jobs: sequential_jobs sched.
Hypothesis H_at_least_one_cpu : num_cpus > 0.
(* Assume that we have a task set where all tasks have valid
parameters and constrained deadlines, ... *)
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.
(* ... and that all jobs in the arrival sequence come from the task set. *)
Hypothesis H_all_jobs_from_taskset:
∀ (j: JobIn arr_seq), job_task j \in ts.
Let no_deadline_is_missed_by_tsk (tsk: sporadic_task) :=
task_misses_no_deadline job_cost job_deadline job_task sched tsk.
Let response_time_bounded_by (tsk: sporadic_task) :=
is_response_time_bound_of_task job_cost job_task tsk sched.
(* Assume that a response-time bound R is known... *)
Let task_with_response_time := (sporadic_task × time)%type.
Variable rt_bounds: seq task_with_response_time.
(* ...for any task in the task set, ... *)
Hypothesis H_rt_bounds_contains_all_tasks: unzip1 rt_bounds = ts.
(* ... such that R is a fixed-point of the response-time recurrence, ... *)
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.
(* ..., and R is no larger than the deadline. *)
Hypothesis H_tasks_miss_no_deadlines:
∀ tsk_other R,
(tsk_other, R) \in rt_bounds → R ≤ task_deadline tsk_other.
(* Assume that we have a work-conserving EDF scheduler. *)
Hypothesis H_work_conserving: work_conserving job_cost sched.
Hypothesis H_edf_policy: enforces_JLDP_policy job_cost sched (EDF job_deadline).
(* Assume that the task set has no duplicates. This is required to
avoid problems when counting tasks (for example, when stating
that the number of interfering tasks is at most num_cpus). *)
Hypothesis H_ts_is_a_set : uniq ts.
(* In order to prove that R is a response-time bound, we first present some lemmas. *)
Section Lemmas.
(* Let (tsk, R) be any task to be analyzed, with its response-time bound R. *)
Variable tsk: sporadic_task.
Variable R: time.
Hypothesis H_tsk_R_in_rt_bounds: (tsk, R) \in rt_bounds.
(* Consider any job j of tsk ... *)
Variable j: JobIn arr_seq.
Hypothesis H_job_of_tsk: job_task j = tsk.
(* ... that did not complete on time, ... *)
Hypothesis H_j_not_completed: ~~ completed job_cost sched j (job_arrival j + R).
(* ... and that is the first job not to satisfy its response-time bound. *)
Hypothesis H_all_previous_jobs_completed_on_time :
∀ (j_other: JobIn arr_seq) tsk_other R_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's call x the interference incurred by job j due to tsk_other, ...*)
Let x (tsk_other: sporadic_task) :=
task_interference job_cost job_task sched j tsk_other
(job_arrival j) (job_arrival j + R).
(* and X the total interference incurred by job j due to any task. *)
Let X := total_interference job_cost sched j (job_arrival j) (job_arrival j + R).
(* Recall Bertogna and Cirinei's workload bound ... *)
Let workload_bound (tsk_other: sporadic_task) (R_other: time) :=
W task_cost task_period tsk_other R_other R.
(*... and the EDF-specific bound, ... *)
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.
(* ... which combined form the interference bound. *)
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).
(* Also, let ts_interf be the subset of tasks that interfere with tsk. *)
Let ts_interf := [seq tsk_other <- ts | jldp_can_interfere_with tsk tsk_other].
Section LemmasAboutInterferingTasks.
(* Let (tsk_other, R_other) be any pair of higher-priority task and
response-time bound computed in previous iterations. *)
Variable tsk_other: sporadic_task.
Variable R_other: time.
Hypothesis H_response_time_of_tsk_other: (tsk_other, R_other) \in rt_bounds.
(* Note that tsk_other is in the task set, ...*)
Lemma bertogna_edf_tsk_other_in_ts: tsk_other \in ts.
(* ... and R_other is larger than the cost of tsk_other. *)
Lemma bertogna_edf_R_other_ge_cost :
R_other ≥ task_cost tsk_other.
(* Since tsk_other cannot interfere more than it executes, we show that
the interference caused by tsk_other is no larger than workload bound W. *)
Lemma bertogna_edf_workload_bounds_interference :
x tsk_other ≤ workload_bound tsk_other R_other.
(* Recall that the edf-specific interference bound also holds for tsk_other. *)
Lemma bertogna_edf_specific_bound_holds :
x tsk_other ≤ edf_specific_bound tsk_other R_other.
End LemmasAboutInterferingTasks.
(* Next we prove some lemmas that help to derive a contradiction.*)
Section DerivingContradiction.
(* 0) Since job j did not complete by its response time bound, it follows that
the total interference X >= R - e_k + 1. *)
Lemma bertogna_edf_too_much_interference : X ≥ R - task_cost tsk + 1.
Let scheduled_task_other_than_tsk (t: time) (tsk_other: sporadic_task) :=
task_is_scheduled job_task sched tsk_other t &&
jldp_can_interfere_with tsk tsk_other.
(* 1) At all times that j is backlogged, all processors are busy with other tasks. *)
Lemma bertogna_edf_scheduling_invariant:
∀ t,
job_arrival j ≤ t < job_arrival j + R →
backlogged job_cost sched j t →
count (scheduled_task_other_than_tsk t) ts = num_cpus.
(* 2) Prove that during the scheduling window of j, any job that is scheduled while j is
backlogged comes from a different task. *)
Lemma bertogna_edf_interference_by_different_tasks :
∀ t j_other,
job_arrival j ≤ t < job_arrival j + R →
backlogged job_cost sched j t →
scheduled sched j_other t →
job_task j_other != tsk.
(* 3) Next, we prove that the sum of the interference of each task is equal
to the total interference multiplied by the number of processors. This
holds because interference only occurs when all processors are busy. *)
Lemma bertogna_edf_all_cpus_busy :
\sum_(tsk_k <- ts_interf) x tsk_k = X × num_cpus.
(* 4) Show that by the induction hypothesis, all jobs released
before the end of the interval complete by their periods. *)
Lemma bertogna_edf_all_previous_jobs_complete_by_their_period:
∀ t (j0: JobIn arr_seq),
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)).
(* Let (cardGE delta) be the number of interfering tasks whose interference
is larger than delta. *)
Let cardGE (delta: time) := count (fun i ⇒ x i ≥ delta) ts_interf.
(* 5) If there is at least one of such tasks (e.g., cardGE > 0), then the
cumulative interference caused by the complementary set of interfering
tasks fills at least (num_cpus - cardGE) processors. *)
Lemma bertogna_edf_helper_lemma :
∀ delta,
0 < cardGE delta < num_cpus →
\sum_(i <- ts_interf | x i < delta) x i ≥ delta × (num_cpus - cardGE delta).
(* 6) Next, we prove that, for any interval delta, if the sum of per-task
interference exceeds delta * num_cpus, the same applies for the
sum of the minimum between the interference and delta. *)
Lemma bertogna_edf_minimum_exceeds_interference :
∀ delta,
\sum_(tsk_k <- ts_interf) x tsk_k ≥ delta × num_cpus →
\sum_(tsk_k <- ts_interf) minn (x tsk_k) delta ≥
delta × num_cpus.
(* 7) Now, we prove that the Bertogna's interference bound
is not enough to cover the sum of the "minimum" term over
all tasks (artifact of the proof by contradiction). *)
Lemma bertogna_edf_sum_exceeds_total_interference:
\sum_((tsk_other, R_other) <- rt_bounds | jldp_can_interfere_with tsk tsk_other)
minn (x tsk_other) (R - task_cost tsk + 1) > I tsk R.
(* 8) After concluding that the sum of the minimum exceeds (R - e_i + 1),
we prove that there exists a tuple (tsk_k, R_k) such that
min (x_k, R - e_i + 1) > min (W_k, edf_bound, R - e_i + 1). *)
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.
(* Let (tsk, R) be any task to be analyzed, with its response-time bound R. *)
Variable tsk: sporadic_task.
Variable R: time.
Hypothesis H_tsk_R_in_rt_bounds: (tsk, R) \in rt_bounds.
(* Using the lemmas above, we prove that R bounds the response time of task tsk. *)
Theorem bertogna_cirinei_response_time_bound_edf :
response_time_bounded_by tsk R.
End MainProof.
End ResponseTimeBound.
End ResponseTimeAnalysisEDF.
Require Import rt.model.basic.task rt.model.basic.job rt.model.basic.task_arrival
rt.model.basic.schedule rt.model.basic.platform rt.model.basic.interference
rt.model.basic.workload rt.model.basic.schedulability rt.model.basic.priority
rt.model.basic.platform rt.model.basic.response_time.
Require Import rt.analysis.basic.workload_bound rt.analysis.basic.interference_bound_edf.
Module ResponseTimeAnalysisEDF.
Export Job SporadicTaskset Schedule ScheduleOfSporadicTask Workload Schedulability ResponseTime
Priority SporadicTaskArrival WorkloadBound InterferenceBoundEDF
Interference Platform.
(* In this section, we prove that Bertogna and Cirinei's RTA yields
safe response-time bounds. *)
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_cost: Job → time.
Variable job_deadline: Job → time.
Variable job_task: Job → sporadic_task.
(* Assume any job arrival sequence... *)
Context {arr_seq: arrival_sequence Job}.
(* ... in which jobs arrive sporadically and have valid parameters. *)
Hypothesis H_sporadic_tasks:
sporadic_task_model task_period arr_seq job_task.
Hypothesis H_valid_job_parameters:
∀ (j: JobIn arr_seq),
valid_sporadic_job task_cost task_deadline job_cost job_deadline job_task j.
(* Consider any schedule such that...*)
Variable num_cpus: nat.
Variable sched: schedule num_cpus arr_seq.
(* ...jobs do not execute before their arrival times nor longer
than their execution costs. *)
Hypothesis H_jobs_must_arrive_to_execute:
jobs_must_arrive_to_execute sched.
Hypothesis H_completed_jobs_dont_execute:
completed_jobs_dont_execute job_cost sched.
(* Also assume that jobs are sequential and that
there exists at least one processor. *)
Hypothesis H_sequential_jobs: sequential_jobs sched.
Hypothesis H_at_least_one_cpu : num_cpus > 0.
(* Assume that we have a task set where all tasks have valid
parameters and constrained deadlines, ... *)
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.
(* ... and that all jobs in the arrival sequence come from the task set. *)
Hypothesis H_all_jobs_from_taskset:
∀ (j: JobIn arr_seq), job_task j \in ts.
Let no_deadline_is_missed_by_tsk (tsk: sporadic_task) :=
task_misses_no_deadline job_cost job_deadline job_task sched tsk.
Let response_time_bounded_by (tsk: sporadic_task) :=
is_response_time_bound_of_task job_cost job_task tsk sched.
(* Assume that a response-time bound R is known... *)
Let task_with_response_time := (sporadic_task × time)%type.
Variable rt_bounds: seq task_with_response_time.
(* ...for any task in the task set, ... *)
Hypothesis H_rt_bounds_contains_all_tasks: unzip1 rt_bounds = ts.
(* ... such that R is a fixed-point of the response-time recurrence, ... *)
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.
(* ..., and R is no larger than the deadline. *)
Hypothesis H_tasks_miss_no_deadlines:
∀ tsk_other R,
(tsk_other, R) \in rt_bounds → R ≤ task_deadline tsk_other.
(* Assume that we have a work-conserving EDF scheduler. *)
Hypothesis H_work_conserving: work_conserving job_cost sched.
Hypothesis H_edf_policy: enforces_JLDP_policy job_cost sched (EDF job_deadline).
(* Assume that the task set has no duplicates. This is required to
avoid problems when counting tasks (for example, when stating
that the number of interfering tasks is at most num_cpus). *)
Hypothesis H_ts_is_a_set : uniq ts.
(* In order to prove that R is a response-time bound, we first present some lemmas. *)
Section Lemmas.
(* Let (tsk, R) be any task to be analyzed, with its response-time bound R. *)
Variable tsk: sporadic_task.
Variable R: time.
Hypothesis H_tsk_R_in_rt_bounds: (tsk, R) \in rt_bounds.
(* Consider any job j of tsk ... *)
Variable j: JobIn arr_seq.
Hypothesis H_job_of_tsk: job_task j = tsk.
(* ... that did not complete on time, ... *)
Hypothesis H_j_not_completed: ~~ completed job_cost sched j (job_arrival j + R).
(* ... and that is the first job not to satisfy its response-time bound. *)
Hypothesis H_all_previous_jobs_completed_on_time :
∀ (j_other: JobIn arr_seq) tsk_other R_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's call x the interference incurred by job j due to tsk_other, ...*)
Let x (tsk_other: sporadic_task) :=
task_interference job_cost job_task sched j tsk_other
(job_arrival j) (job_arrival j + R).
(* and X the total interference incurred by job j due to any task. *)
Let X := total_interference job_cost sched j (job_arrival j) (job_arrival j + R).
(* Recall Bertogna and Cirinei's workload bound ... *)
Let workload_bound (tsk_other: sporadic_task) (R_other: time) :=
W task_cost task_period tsk_other R_other R.
(*... and the EDF-specific bound, ... *)
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.
(* ... which combined form the interference bound. *)
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).
(* Also, let ts_interf be the subset of tasks that interfere with tsk. *)
Let ts_interf := [seq tsk_other <- ts | jldp_can_interfere_with tsk tsk_other].
Section LemmasAboutInterferingTasks.
(* Let (tsk_other, R_other) be any pair of higher-priority task and
response-time bound computed in previous iterations. *)
Variable tsk_other: sporadic_task.
Variable R_other: time.
Hypothesis H_response_time_of_tsk_other: (tsk_other, R_other) \in rt_bounds.
(* Note that tsk_other is in the task set, ...*)
Lemma bertogna_edf_tsk_other_in_ts: tsk_other \in ts.
(* ... and R_other is larger than the cost of tsk_other. *)
Lemma bertogna_edf_R_other_ge_cost :
R_other ≥ task_cost tsk_other.
(* Since tsk_other cannot interfere more than it executes, we show that
the interference caused by tsk_other is no larger than workload bound W. *)
Lemma bertogna_edf_workload_bounds_interference :
x tsk_other ≤ workload_bound tsk_other R_other.
(* Recall that the edf-specific interference bound also holds for tsk_other. *)
Lemma bertogna_edf_specific_bound_holds :
x tsk_other ≤ edf_specific_bound tsk_other R_other.
End LemmasAboutInterferingTasks.
(* Next we prove some lemmas that help to derive a contradiction.*)
Section DerivingContradiction.
(* 0) Since job j did not complete by its response time bound, it follows that
the total interference X >= R - e_k + 1. *)
Lemma bertogna_edf_too_much_interference : X ≥ R - task_cost tsk + 1.
Let scheduled_task_other_than_tsk (t: time) (tsk_other: sporadic_task) :=
task_is_scheduled job_task sched tsk_other t &&
jldp_can_interfere_with tsk tsk_other.
(* 1) At all times that j is backlogged, all processors are busy with other tasks. *)
Lemma bertogna_edf_scheduling_invariant:
∀ t,
job_arrival j ≤ t < job_arrival j + R →
backlogged job_cost sched j t →
count (scheduled_task_other_than_tsk t) ts = num_cpus.
(* 2) Prove that during the scheduling window of j, any job that is scheduled while j is
backlogged comes from a different task. *)
Lemma bertogna_edf_interference_by_different_tasks :
∀ t j_other,
job_arrival j ≤ t < job_arrival j + R →
backlogged job_cost sched j t →
scheduled sched j_other t →
job_task j_other != tsk.
(* 3) Next, we prove that the sum of the interference of each task is equal
to the total interference multiplied by the number of processors. This
holds because interference only occurs when all processors are busy. *)
Lemma bertogna_edf_all_cpus_busy :
\sum_(tsk_k <- ts_interf) x tsk_k = X × num_cpus.
(* 4) Show that by the induction hypothesis, all jobs released
before the end of the interval complete by their periods. *)
Lemma bertogna_edf_all_previous_jobs_complete_by_their_period:
∀ t (j0: JobIn arr_seq),
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)).
(* Let (cardGE delta) be the number of interfering tasks whose interference
is larger than delta. *)
Let cardGE (delta: time) := count (fun i ⇒ x i ≥ delta) ts_interf.
(* 5) If there is at least one of such tasks (e.g., cardGE > 0), then the
cumulative interference caused by the complementary set of interfering
tasks fills at least (num_cpus - cardGE) processors. *)
Lemma bertogna_edf_helper_lemma :
∀ delta,
0 < cardGE delta < num_cpus →
\sum_(i <- ts_interf | x i < delta) x i ≥ delta × (num_cpus - cardGE delta).
(* 6) Next, we prove that, for any interval delta, if the sum of per-task
interference exceeds delta * num_cpus, the same applies for the
sum of the minimum between the interference and delta. *)
Lemma bertogna_edf_minimum_exceeds_interference :
∀ delta,
\sum_(tsk_k <- ts_interf) x tsk_k ≥ delta × num_cpus →
\sum_(tsk_k <- ts_interf) minn (x tsk_k) delta ≥
delta × num_cpus.
(* 7) Now, we prove that the Bertogna's interference bound
is not enough to cover the sum of the "minimum" term over
all tasks (artifact of the proof by contradiction). *)
Lemma bertogna_edf_sum_exceeds_total_interference:
\sum_((tsk_other, R_other) <- rt_bounds | jldp_can_interfere_with tsk tsk_other)
minn (x tsk_other) (R - task_cost tsk + 1) > I tsk R.
(* 8) After concluding that the sum of the minimum exceeds (R - e_i + 1),
we prove that there exists a tuple (tsk_k, R_k) such that
min (x_k, R - e_i + 1) > min (W_k, edf_bound, R - e_i + 1). *)
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.
(* Let (tsk, R) be any task to be analyzed, with its response-time bound R. *)
Variable tsk: sporadic_task.
Variable R: time.
Hypothesis H_tsk_R_in_rt_bounds: (tsk, R) \in rt_bounds.
(* Using the lemmas above, we prove that R bounds the response time of task tsk. *)
Theorem bertogna_cirinei_response_time_bound_edf :
response_time_bounded_by tsk R.
End MainProof.
End ResponseTimeBound.
End ResponseTimeAnalysisEDF.