Library prosa.classic.analysis.global.jitter.bertogna_edf_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.schedulability
prosa.classic.model.schedule.global.response_time.
Require Import prosa.classic.model.schedule.global.jitter.job prosa.classic.model.schedule.global.jitter.schedule
prosa.classic.model.schedule.global.jitter.platform prosa.classic.model.schedule.global.jitter.interference
prosa.classic.model.schedule.global.jitter.constrained_deadlines.
Require Import prosa.classic.analysis.global.jitter.workload_bound
prosa.classic.analysis.global.jitter.interference_bound_edf.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq fintype bigop div path.
Module ResponseTimeAnalysisEDFJitter.
Export JobWithJitter SporadicTaskset ScheduleOfSporadicTaskWithJitter Workload
Schedulability ResponseTime Priority TaskArrival WorkloadBoundJitter
InterferenceBoundEDFJitter Platform Interference ConstrainedDeadlines.
(* In this section, we prove that any fixed point in Bertogna and
Cirinei's RTA for EDF scheduling modified to account for jitter
yields a safe response-time bound. This is an extension of the
analysis found in Chapter 17.1.2 of Baruah et al.'s book
Multiprocessor Scheduling for Real-time Systems. *)
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.
(* Assume any job arrival sequence... *)
Context {arr_seq: arrival_sequence Job}.
(* ... in which jobs arrive sporadically and have valid parameters.
Note: the jitter of a valid job is bounded by the jitter of its task. *)
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.
(* 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,
arrives_in arr_seq j →
job_task j \in ts.
(* Next, consider any schedule such that...*)
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.
(* ...jobs are sequential, do not execute before the
jitter has passed and nor longer than their execution costs. *)
Hypothesis H_sequential_jobs: sequential_jobs sched.
Hypothesis H_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.
(* Assume that there exists at least one processor. *)
Hypothesis H_at_least_one_cpu: num_cpus > 0.
(* Assume that the schedule is a work-conserving EDF schedule. *)
Hypothesis H_work_conserving: work_conserving job_arrival job_cost job_jitter arr_seq sched.
Hypothesis H_edf_policy: respects_JLFP_policy job_arrival job_cost job_jitter arr_seq sched
(EDF job_arrival job_deadline).
(* Let's define some local names to avoid passing many parameters. *)
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.
(* Next we consider the response-time recurrence.
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.
(* Also, assume 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 task_jitter 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 R,
(tsk, R) \in rt_bounds →
task_jitter tsk + R ≤ task_deadline tsk.
(* In order to prove that R is a response-time bound, we first provide 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: Job.
Hypothesis H_j_arrives: arrives_in arr_seq j.
Hypothesis H_job_of_tsk: job_task j = tsk.
(* Let t1 be the first point in time where j can actually be scheduled. *)
Let t1 := job_arrival j + job_jitter j.
(* Assume that job j did not complete on time, ... *)
Hypothesis H_j_not_completed: ~~ completed job_cost sched j (t1 + R).
(* ...and that it is the first job not to satisfy its response-time bound. *)
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 + task_jitter tsk_other + R_other < job_arrival j + task_jitter tsk + R →
completed job_cost sched j_other (job_arrival j_other + task_jitter tsk_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_arrival job_cost job_task job_jitter sched j tsk_other t1 (t1 + R).
(* and X the total interference incurred by job j due to any task. *)
Let X := total_interference job_arrival job_cost job_jitter sched j t1 (t1 + R).
(* Recall Bertogna and Cirinei's workload bound ... *)
Let workload_bound (tsk_other: sporadic_task) (R_other: time) :=
W_jitter task_cost task_period task_jitter 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 task_jitter 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 task_jitter tsk R (tsk_other, R_other).
(* Based on the definition of a different task, ... *)
Let other_task := different_task tsk.
(* ...let other_tasks denote the set of tasks that are different from tsk. *)
Let other_tasks :=
[seq tsk_other <- ts | other_task tsk_other].
(* Now we establish results the interfering tasks. *)
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 task set ts ...*)
Lemma bertogna_edf_tsk_other_in_ts: tsk_other \in ts.
(* Also, 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. *)
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.
(* 1) Next, we prove that during the scheduling window of j, any job that is
scheduled while j is backlogged comes from a different task.
This follows from the fact that j is the first job not to complete
by its response-time bound, so previous jobs of j's task must have
completed by their periods and cannot be pending. *)
Lemma bertogna_edf_interference_by_different_tasks :
∀ t j_other,
t1 ≤ t < t1 + R →
backlogged job_arrival job_cost job_jitter sched j t →
scheduled sched j_other t →
job_task j_other != tsk.
(* 2) In order to use the lemmas in constrained_deadlines.v, we show that
all jobs released before the end of the interval complete by their
periods. This follows trivially from the hypothesis that all jobs
before (t1 + R) complete by their response-time bounds.
With this lemma, we can conclude that during job j's scheduling
window there cannot be multiple pending jobs of each task.*)
Lemma bertogna_edf_all_previous_jobs_complete_by_their_period:
∀ t j0,
t < t1 + R →
arrives_in arr_seq j0 →
job_arrival j0 + task_period (job_task j0) ≤ t →
completed job_cost sched j0
(job_arrival j0 + task_period (job_task j0)).
(* Let's define a predicate to identify the other tasks that are scheduled. *)
Let other_scheduled_task (t: time) (tsk_other: sporadic_task) :=
task_is_scheduled job_task sched tsk_other t &&
other_task tsk_other.
(* 3) Now we prove that, at all times that j is backlogged, the number
of tasks other than tsk that are scheduled is exactly the number
of processors in the system. This is required to prove lemma (4). *)
Lemma bertogna_edf_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.
(* 4) 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.
With this lemma we can relate per-task interference with the total
interference incurred by j (backlogged time). *)
Lemma bertogna_edf_interference_on_all_cpus :
\sum_(tsk_k <- other_tasks) x tsk_k = X × num_cpus.
(* Before stating the next lemma, let (num_tasks_exceeding delta) be the
number of interfering tasks whose interference x is larger than delta. *)
Let num_tasks_exceeding delta := count (fun i ⇒ x i ≥ delta) (other_tasks).
(* 5) Now we prove that, for any delta, if (num_task_exceeding delta > 0), then the
cumulative interference caused by the complementary set of interfering tasks fills
the remaining, not-completely-full (num_cpus - num_tasks_exceeding delta)
processors. *)
Lemma bertogna_edf_interference_in_non_full_processors :
∀ delta,
0 < num_tasks_exceeding delta < num_cpus →
\sum_(i <- other_tasks | x i < delta) x i ≥ delta × (num_cpus - num_tasks_exceeding delta).
(* 6) Based on lemma (5), 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 of the interference and delta. *)
Lemma bertogna_edf_minimum_exceeds_interference :
∀ delta,
\sum_(tsk_k <- other_tasks) x tsk_k ≥ delta × num_cpus →
\sum_(tsk_k <- other_tasks) minn (x tsk_k) delta ≥
delta × num_cpus.
(* 7) Next, using lemmas (0), (4) and (6) we prove that the reduction-based
interference bound is not enough to cover the sum of the minima over all tasks
(artifact of the proof by contradiction). *)
Lemma bertogna_edf_sum_exceeds_total_interference:
\sum_((tsk_other, R_other) <- rt_bounds | other_task tsk_other)
minn (x tsk_other) (R - task_cost tsk + 1) > I tsk R.
(* 8) After concluding that the sum of the minima exceeds (R - e_i + 1),
we prove that there exists a tuple (tsk_k, R_k) that satisfies
min (x_k, R - e_i + 1) > min (W_k, I_edf, R - e_i + 1).
This implies that either x_k > W_k or x_k > I_edf, which is a contradiction,
since both W_k and I_edf are valid task interference bounds. *)
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 (task_jitter tsk + R).
End MainProof.
End ResponseTimeBound.
End ResponseTimeAnalysisEDFJitter.
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.schedulability
prosa.classic.model.schedule.global.response_time.
Require Import prosa.classic.model.schedule.global.jitter.job prosa.classic.model.schedule.global.jitter.schedule
prosa.classic.model.schedule.global.jitter.platform prosa.classic.model.schedule.global.jitter.interference
prosa.classic.model.schedule.global.jitter.constrained_deadlines.
Require Import prosa.classic.analysis.global.jitter.workload_bound
prosa.classic.analysis.global.jitter.interference_bound_edf.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq fintype bigop div path.
Module ResponseTimeAnalysisEDFJitter.
Export JobWithJitter SporadicTaskset ScheduleOfSporadicTaskWithJitter Workload
Schedulability ResponseTime Priority TaskArrival WorkloadBoundJitter
InterferenceBoundEDFJitter Platform Interference ConstrainedDeadlines.
(* In this section, we prove that any fixed point in Bertogna and
Cirinei's RTA for EDF scheduling modified to account for jitter
yields a safe response-time bound. This is an extension of the
analysis found in Chapter 17.1.2 of Baruah et al.'s book
Multiprocessor Scheduling for Real-time Systems. *)
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.
(* Assume any job arrival sequence... *)
Context {arr_seq: arrival_sequence Job}.
(* ... in which jobs arrive sporadically and have valid parameters.
Note: the jitter of a valid job is bounded by the jitter of its task. *)
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.
(* 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,
arrives_in arr_seq j →
job_task j \in ts.
(* Next, consider any schedule such that...*)
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.
(* ...jobs are sequential, do not execute before the
jitter has passed and nor longer than their execution costs. *)
Hypothesis H_sequential_jobs: sequential_jobs sched.
Hypothesis H_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.
(* Assume that there exists at least one processor. *)
Hypothesis H_at_least_one_cpu: num_cpus > 0.
(* Assume that the schedule is a work-conserving EDF schedule. *)
Hypothesis H_work_conserving: work_conserving job_arrival job_cost job_jitter arr_seq sched.
Hypothesis H_edf_policy: respects_JLFP_policy job_arrival job_cost job_jitter arr_seq sched
(EDF job_arrival job_deadline).
(* Let's define some local names to avoid passing many parameters. *)
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.
(* Next we consider the response-time recurrence.
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.
(* Also, assume 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 task_jitter 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 R,
(tsk, R) \in rt_bounds →
task_jitter tsk + R ≤ task_deadline tsk.
(* In order to prove that R is a response-time bound, we first provide 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: Job.
Hypothesis H_j_arrives: arrives_in arr_seq j.
Hypothesis H_job_of_tsk: job_task j = tsk.
(* Let t1 be the first point in time where j can actually be scheduled. *)
Let t1 := job_arrival j + job_jitter j.
(* Assume that job j did not complete on time, ... *)
Hypothesis H_j_not_completed: ~~ completed job_cost sched j (t1 + R).
(* ...and that it is the first job not to satisfy its response-time bound. *)
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 + task_jitter tsk_other + R_other < job_arrival j + task_jitter tsk + R →
completed job_cost sched j_other (job_arrival j_other + task_jitter tsk_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_arrival job_cost job_task job_jitter sched j tsk_other t1 (t1 + R).
(* and X the total interference incurred by job j due to any task. *)
Let X := total_interference job_arrival job_cost job_jitter sched j t1 (t1 + R).
(* Recall Bertogna and Cirinei's workload bound ... *)
Let workload_bound (tsk_other: sporadic_task) (R_other: time) :=
W_jitter task_cost task_period task_jitter 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 task_jitter 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 task_jitter tsk R (tsk_other, R_other).
(* Based on the definition of a different task, ... *)
Let other_task := different_task tsk.
(* ...let other_tasks denote the set of tasks that are different from tsk. *)
Let other_tasks :=
[seq tsk_other <- ts | other_task tsk_other].
(* Now we establish results the interfering tasks. *)
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 task set ts ...*)
Lemma bertogna_edf_tsk_other_in_ts: tsk_other \in ts.
(* Also, 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. *)
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.
(* 1) Next, we prove that during the scheduling window of j, any job that is
scheduled while j is backlogged comes from a different task.
This follows from the fact that j is the first job not to complete
by its response-time bound, so previous jobs of j's task must have
completed by their periods and cannot be pending. *)
Lemma bertogna_edf_interference_by_different_tasks :
∀ t j_other,
t1 ≤ t < t1 + R →
backlogged job_arrival job_cost job_jitter sched j t →
scheduled sched j_other t →
job_task j_other != tsk.
(* 2) In order to use the lemmas in constrained_deadlines.v, we show that
all jobs released before the end of the interval complete by their
periods. This follows trivially from the hypothesis that all jobs
before (t1 + R) complete by their response-time bounds.
With this lemma, we can conclude that during job j's scheduling
window there cannot be multiple pending jobs of each task.*)
Lemma bertogna_edf_all_previous_jobs_complete_by_their_period:
∀ t j0,
t < t1 + R →
arrives_in arr_seq j0 →
job_arrival j0 + task_period (job_task j0) ≤ t →
completed job_cost sched j0
(job_arrival j0 + task_period (job_task j0)).
(* Let's define a predicate to identify the other tasks that are scheduled. *)
Let other_scheduled_task (t: time) (tsk_other: sporadic_task) :=
task_is_scheduled job_task sched tsk_other t &&
other_task tsk_other.
(* 3) Now we prove that, at all times that j is backlogged, the number
of tasks other than tsk that are scheduled is exactly the number
of processors in the system. This is required to prove lemma (4). *)
Lemma bertogna_edf_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.
(* 4) 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.
With this lemma we can relate per-task interference with the total
interference incurred by j (backlogged time). *)
Lemma bertogna_edf_interference_on_all_cpus :
\sum_(tsk_k <- other_tasks) x tsk_k = X × num_cpus.
(* Before stating the next lemma, let (num_tasks_exceeding delta) be the
number of interfering tasks whose interference x is larger than delta. *)
Let num_tasks_exceeding delta := count (fun i ⇒ x i ≥ delta) (other_tasks).
(* 5) Now we prove that, for any delta, if (num_task_exceeding delta > 0), then the
cumulative interference caused by the complementary set of interfering tasks fills
the remaining, not-completely-full (num_cpus - num_tasks_exceeding delta)
processors. *)
Lemma bertogna_edf_interference_in_non_full_processors :
∀ delta,
0 < num_tasks_exceeding delta < num_cpus →
\sum_(i <- other_tasks | x i < delta) x i ≥ delta × (num_cpus - num_tasks_exceeding delta).
(* 6) Based on lemma (5), 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 of the interference and delta. *)
Lemma bertogna_edf_minimum_exceeds_interference :
∀ delta,
\sum_(tsk_k <- other_tasks) x tsk_k ≥ delta × num_cpus →
\sum_(tsk_k <- other_tasks) minn (x tsk_k) delta ≥
delta × num_cpus.
(* 7) Next, using lemmas (0), (4) and (6) we prove that the reduction-based
interference bound is not enough to cover the sum of the minima over all tasks
(artifact of the proof by contradiction). *)
Lemma bertogna_edf_sum_exceeds_total_interference:
\sum_((tsk_other, R_other) <- rt_bounds | other_task tsk_other)
minn (x tsk_other) (R - task_cost tsk + 1) > I tsk R.
(* 8) After concluding that the sum of the minima exceeds (R - e_i + 1),
we prove that there exists a tuple (tsk_k, R_k) that satisfies
min (x_k, R - e_i + 1) > min (W_k, I_edf, R - e_i + 1).
This implies that either x_k > W_k or x_k > I_edf, which is a contradiction,
since both W_k and I_edf are valid task interference bounds. *)
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 (task_jitter tsk + R).
End MainProof.
End ResponseTimeBound.
End ResponseTimeAnalysisEDFJitter.