Library rt.analysis.apa.bertogna_fp_theory
Require Import rt.util.all rt.util.divround.
Require Import rt.model.apa.task rt.model.apa.job rt.model.apa.task_arrival
rt.model.apa.schedule rt.model.apa.platform rt.model.apa.constrained_deadlines
rt.model.apa.workload rt.model.apa.schedulability rt.model.apa.priority
rt.model.apa.response_time rt.model.apa.interference
rt.model.apa.affinity rt.model.apa.constrained_deadlines.
Require Import rt.analysis.apa.workload_bound rt.analysis.apa.interference_bound_fp.
Module ResponseTimeAnalysisFP.
Export Job SporadicTaskset ScheduleOfSporadicTask Workload Interference InterferenceBoundFP
Platform Schedulability ResponseTime Priority
SporadicTaskArrival WorkloadBound Affinity ConstrainedDeadlines.
(* In this section, we prove that any fixed point in the APA-reduction of Bertogna
and Cirinei's RTA for FP scheduling with slack updates is a safe response-time
bound. This result corresponds to Lemma 9 in the revised version of the APA paper:
http://www.mpi-sws.org/~bbb/papers/pdf/ecrts13a-rev1.pdf *)
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.
(* 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 ∈ 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 ∈ ts.
(* Also assume that every task has a non-empty processor affinity alpha. *)
Context {num_cpus: nat}.
Variable alpha: task_affinity sporadic_task num_cpus.
(* Next, consider any schedule such that...*)
Variable sched: schedule num_cpus arr_seq.
(* ...jobs are sequential and do not execute before their
arrival times nor longer than their execution costs. *)
Hypothesis H_sequential_jobs: sequential_jobs sched.
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.
(* Consider a given FP policy, ... *)
Variable higher_eq_priority: FP_policy sporadic_task.
(* ... and assume that the schedule is an APA work-conserving
schedule that enforces this policy. *)
Hypothesis H_respects_affinity: respects_affinity job_task sched alpha.
Hypothesis H_work_conserving: apa_work_conserving job_cost job_task sched alpha.
Hypothesis H_enforces_FP_policy:
enforces_FP_policy_under_weak_APA job_cost job_task sched alpha higher_eq_priority.
(* 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_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.
(* Next, we consider the response-time recurrence.
Let tsk be a task in ts that is to be analyzed. *)
Variable tsk: sporadic_task.
Hypothesis task_in_ts: tsk ∈ ts.
(* When computing the response-time bounds, we assume that each task under
analysis has a non-empty subaffinity alpha'. *)
Variable alpha': task_affinity sporadic_task num_cpus.
Hypothesis H_affinity_subset: ∀ tsk, tsk ∈ ts → is_subaffinity (alpha' tsk) (alpha tsk).
Hypothesis H_at_least_one_cpu : ∀ tsk, tsk ∈ ts → #|alpha' tsk| > 0.
(* Let (hp_task_in alpha') denote whether a task is a higher-priority task
(with respect to tsk) that can execute on processors in (alpha' tsk). *)
Let hp_task_in alpha' := higher_priority_task_in alpha higher_eq_priority tsk alpha'.
(* Assume a response-time bound is known... *)
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) ∈ hp_bounds →
is_response_time_bound_of_task job_cost job_task hp_tsk sched R.
(* ... for every higher-priority task in (alpha tsk). *)
Hypothesis H_hp_bounds_has_interfering_tasks:
∀ hp_tsk,
hp_tsk ∈ ts →
hp_task_in (alpha tsk) hp_tsk →
∃ R,
(hp_tsk, R) ∈ hp_bounds.
(* Assume that the response-time bounds are larger than task costs. *)
Hypothesis H_response_time_bounds_ge_cost:
∀ hp_tsk R,
(hp_tsk, R) ∈ hp_bounds → R ≥ task_cost hp_tsk.
(* Assume that no deadline is missed by any higher-priority task. *)
Hypothesis H_interfering_tasks_miss_no_deadlines:
∀ hp_tsk R,
(hp_tsk, R) ∈ hp_bounds → R ≤ task_deadline hp_tsk.
(* Let R be the fixed point of the APA-reduction of Bertogna and
Cirinei's response-time recurrence (using subaffinity alpha'), ...*)
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|.
(* ... and assume that R is no larger than the deadline of tsk.*)
Hypothesis H_response_time_no_larger_than_deadline:
R ≤ task_deadline tsk.
(* In order to prove that R is a response-time bound, we first provide some lemmas. *)
Section Lemmas.
(* Consider any job j of tsk. *)
Variable j: JobIn arr_seq.
Hypothesis H_job_of_tsk: job_task j = tsk.
(* Assume that job j is the first job of tsk not to complete by the response time bound. *)
Hypothesis H_j_not_completed: ¬ completed job_cost sched j (job_arrival j + R).
Hypothesis H_previous_jobs_of_tsk_completed :
∀ (j0: JobIn arr_seq),
job_task j0 = tsk →
job_arrival j0 < job_arrival j →
completed job_cost sched j0 (job_arrival j0 + R).
(* 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 alpha 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.
(* Let (hp_tasks_in alpha') denote the set of higher-priority tasks
that can be scheduled on subaffinity alpha'. *)
Let hp_tasks_in alpha' :=
[seq tsk_other <- ts | hp_task_in (alpha' tsk) tsk_other].
(* Now we establish results the higher-priority tasks. *)
Section LemmasAboutHPTasks.
(* 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_tsk_other_already_processed: (tsk_other, R_other) ∈ hp_bounds.
Hypothesis H_tsk_other_has_higher_priority: hp_task_in (alpha tsk) 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_fp_workload_bounds_interference :
x tsk_other ≤ workload_bound tsk_other R_other.
End LemmasAboutHPTasks.
(* 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_fp_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_fp_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.
(* 2) In order to use the lemmas in constrained_deadlines.v, we show that
all jobs of higher-priority tasks released before the end of the
interval complete by their periods. This follows trivially from
the fact that all these tasks are known to be schedulable.
With this lemma, we can conclude that during job j's scheduling
window there cannot be multiple pending jobs of higher-priority tasks.*)
Lemma bertogna_fp_previous_interfering_jobs_complete_by_their_period:
∀ (j0: JobIn arr_seq),
hp_task_in (alpha tsk) (job_task j0) →
completed job_cost sched j0
(job_arrival j0 + task_period (job_task j0)).
(* 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 in tsk's *actual*
affinity. This holds because interference only occurs when all processors in
the affinity are busy.
With this lemma we can relate per-task interference with the total interference
incurred by j (backlogged time). *)
Lemma bertogna_fp_all_cpus_in_affinity_busy :
\sum_(tsk_k <- hp_tasks_in alpha) x tsk_k = X × #|alpha tsk|.
(* 4) Recall that the reduction-based analysis considers only the interfering tasks
within subaffinity (alpha' tsk), as opposed to (alpha tsk). Therefore, we must
reason about the task interference wihin (alpha' tsk).
We begin by showing that the cumulative interference within (alpha' tsk) is at
least the total interference multiplied by the number of processors in (alpha' tsk). *)
Lemma bertogna_fp_all_cpus_in_subaffinity_busy :
\sum_(tsk_k <- hp_tasks_in alpha') x tsk_k ≥ X × #|alpha' tsk|.
(* Let's define a predicate for whether a task is scheduled on (alpha tsk). *)
Let scheduled_on_alpha_tsk := fun t tsk_k ⇒
task_scheduled_on_affinity job_task sched (alpha tsk) tsk_k t.
(* 5) Now we prove that, at all times that j is backlogged, the number
of tasks whose affinity intersects (alpha' tsk) that are in fact
scheduled on (alpha' tsk) is at least the size of (alpha' tsk).
This is required to prove lemma (6). *)
Lemma bertogna_fp_alpha'_is_full:
∀ t,
job_arrival j ≤ t < job_arrival j + R →
backlogged job_cost sched j t →
count (scheduled_on_alpha_tsk t) (hp_tasks_in alpha') ≥ #|alpha' tsk|.
(* Before stating the next lemma, let (num_tasks_exceeding delta) be the
number of interfering tasks that can execute on (alpha' tsk) whose
interference x is larger than delta. *)
Let num_tasks_exceeding delta := count (fun i ⇒ x i ≥ delta) (hp_tasks_in alpha').
(* 6) 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 (|alpha' tsk| - num_tasks_exceeding delta) processors. *)
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).
(* 7) Based on lemma (6), we prove that, for any interval delta, if the sum of per-task
interference exceeds (delta * |alpha' tsk|), the same applies for the
sum of the minimum of the interference and 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|.
(* 8) Note that lemma (7) only refers to interference within subaffinity (alpha' tsk).
To reason about the interference incurred by job j in its complete affinity,
we prove that an exceeding interference on affinity (alpha tsk) also
implies an exceeding interference on the subaffinity (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|.
(* 9) Next, using lemmas (0), (3), (7) and (8), 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_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.
(* 10) 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, R - e_i + 1).
This implies that x_k > W_k, which is of course a contradiction,
since W_k is a valid task interference bound. *)
Lemma bertogna_fp_exists_task_that_exceeds_bound :
∃ tsk_k R_k,
(tsk_k, R_k) ∈ 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.
(* Using the lemmas above, we prove that R bounds the response time of tsk. *)
Theorem bertogna_cirinei_response_time_bound_fp :
response_time_bounded_by tsk R.
End ResponseTimeBound.
End ResponseTimeAnalysisFP.
Require Import rt.model.apa.task rt.model.apa.job rt.model.apa.task_arrival
rt.model.apa.schedule rt.model.apa.platform rt.model.apa.constrained_deadlines
rt.model.apa.workload rt.model.apa.schedulability rt.model.apa.priority
rt.model.apa.response_time rt.model.apa.interference
rt.model.apa.affinity rt.model.apa.constrained_deadlines.
Require Import rt.analysis.apa.workload_bound rt.analysis.apa.interference_bound_fp.
Module ResponseTimeAnalysisFP.
Export Job SporadicTaskset ScheduleOfSporadicTask Workload Interference InterferenceBoundFP
Platform Schedulability ResponseTime Priority
SporadicTaskArrival WorkloadBound Affinity ConstrainedDeadlines.
(* In this section, we prove that any fixed point in the APA-reduction of Bertogna
and Cirinei's RTA for FP scheduling with slack updates is a safe response-time
bound. This result corresponds to Lemma 9 in the revised version of the APA paper:
http://www.mpi-sws.org/~bbb/papers/pdf/ecrts13a-rev1.pdf *)
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.
(* 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 ∈ 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 ∈ ts.
(* Also assume that every task has a non-empty processor affinity alpha. *)
Context {num_cpus: nat}.
Variable alpha: task_affinity sporadic_task num_cpus.
(* Next, consider any schedule such that...*)
Variable sched: schedule num_cpus arr_seq.
(* ...jobs are sequential and do not execute before their
arrival times nor longer than their execution costs. *)
Hypothesis H_sequential_jobs: sequential_jobs sched.
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.
(* Consider a given FP policy, ... *)
Variable higher_eq_priority: FP_policy sporadic_task.
(* ... and assume that the schedule is an APA work-conserving
schedule that enforces this policy. *)
Hypothesis H_respects_affinity: respects_affinity job_task sched alpha.
Hypothesis H_work_conserving: apa_work_conserving job_cost job_task sched alpha.
Hypothesis H_enforces_FP_policy:
enforces_FP_policy_under_weak_APA job_cost job_task sched alpha higher_eq_priority.
(* 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_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.
(* Next, we consider the response-time recurrence.
Let tsk be a task in ts that is to be analyzed. *)
Variable tsk: sporadic_task.
Hypothesis task_in_ts: tsk ∈ ts.
(* When computing the response-time bounds, we assume that each task under
analysis has a non-empty subaffinity alpha'. *)
Variable alpha': task_affinity sporadic_task num_cpus.
Hypothesis H_affinity_subset: ∀ tsk, tsk ∈ ts → is_subaffinity (alpha' tsk) (alpha tsk).
Hypothesis H_at_least_one_cpu : ∀ tsk, tsk ∈ ts → #|alpha' tsk| > 0.
(* Let (hp_task_in alpha') denote whether a task is a higher-priority task
(with respect to tsk) that can execute on processors in (alpha' tsk). *)
Let hp_task_in alpha' := higher_priority_task_in alpha higher_eq_priority tsk alpha'.
(* Assume a response-time bound is known... *)
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) ∈ hp_bounds →
is_response_time_bound_of_task job_cost job_task hp_tsk sched R.
(* ... for every higher-priority task in (alpha tsk). *)
Hypothesis H_hp_bounds_has_interfering_tasks:
∀ hp_tsk,
hp_tsk ∈ ts →
hp_task_in (alpha tsk) hp_tsk →
∃ R,
(hp_tsk, R) ∈ hp_bounds.
(* Assume that the response-time bounds are larger than task costs. *)
Hypothesis H_response_time_bounds_ge_cost:
∀ hp_tsk R,
(hp_tsk, R) ∈ hp_bounds → R ≥ task_cost hp_tsk.
(* Assume that no deadline is missed by any higher-priority task. *)
Hypothesis H_interfering_tasks_miss_no_deadlines:
∀ hp_tsk R,
(hp_tsk, R) ∈ hp_bounds → R ≤ task_deadline hp_tsk.
(* Let R be the fixed point of the APA-reduction of Bertogna and
Cirinei's response-time recurrence (using subaffinity alpha'), ...*)
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|.
(* ... and assume that R is no larger than the deadline of tsk.*)
Hypothesis H_response_time_no_larger_than_deadline:
R ≤ task_deadline tsk.
(* In order to prove that R is a response-time bound, we first provide some lemmas. *)
Section Lemmas.
(* Consider any job j of tsk. *)
Variable j: JobIn arr_seq.
Hypothesis H_job_of_tsk: job_task j = tsk.
(* Assume that job j is the first job of tsk not to complete by the response time bound. *)
Hypothesis H_j_not_completed: ¬ completed job_cost sched j (job_arrival j + R).
Hypothesis H_previous_jobs_of_tsk_completed :
∀ (j0: JobIn arr_seq),
job_task j0 = tsk →
job_arrival j0 < job_arrival j →
completed job_cost sched j0 (job_arrival j0 + R).
(* 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 alpha 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.
(* Let (hp_tasks_in alpha') denote the set of higher-priority tasks
that can be scheduled on subaffinity alpha'. *)
Let hp_tasks_in alpha' :=
[seq tsk_other <- ts | hp_task_in (alpha' tsk) tsk_other].
(* Now we establish results the higher-priority tasks. *)
Section LemmasAboutHPTasks.
(* 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_tsk_other_already_processed: (tsk_other, R_other) ∈ hp_bounds.
Hypothesis H_tsk_other_has_higher_priority: hp_task_in (alpha tsk) 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_fp_workload_bounds_interference :
x tsk_other ≤ workload_bound tsk_other R_other.
End LemmasAboutHPTasks.
(* 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_fp_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_fp_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.
(* 2) In order to use the lemmas in constrained_deadlines.v, we show that
all jobs of higher-priority tasks released before the end of the
interval complete by their periods. This follows trivially from
the fact that all these tasks are known to be schedulable.
With this lemma, we can conclude that during job j's scheduling
window there cannot be multiple pending jobs of higher-priority tasks.*)
Lemma bertogna_fp_previous_interfering_jobs_complete_by_their_period:
∀ (j0: JobIn arr_seq),
hp_task_in (alpha tsk) (job_task j0) →
completed job_cost sched j0
(job_arrival j0 + task_period (job_task j0)).
(* 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 in tsk's *actual*
affinity. This holds because interference only occurs when all processors in
the affinity are busy.
With this lemma we can relate per-task interference with the total interference
incurred by j (backlogged time). *)
Lemma bertogna_fp_all_cpus_in_affinity_busy :
\sum_(tsk_k <- hp_tasks_in alpha) x tsk_k = X × #|alpha tsk|.
(* 4) Recall that the reduction-based analysis considers only the interfering tasks
within subaffinity (alpha' tsk), as opposed to (alpha tsk). Therefore, we must
reason about the task interference wihin (alpha' tsk).
We begin by showing that the cumulative interference within (alpha' tsk) is at
least the total interference multiplied by the number of processors in (alpha' tsk). *)
Lemma bertogna_fp_all_cpus_in_subaffinity_busy :
\sum_(tsk_k <- hp_tasks_in alpha') x tsk_k ≥ X × #|alpha' tsk|.
(* Let's define a predicate for whether a task is scheduled on (alpha tsk). *)
Let scheduled_on_alpha_tsk := fun t tsk_k ⇒
task_scheduled_on_affinity job_task sched (alpha tsk) tsk_k t.
(* 5) Now we prove that, at all times that j is backlogged, the number
of tasks whose affinity intersects (alpha' tsk) that are in fact
scheduled on (alpha' tsk) is at least the size of (alpha' tsk).
This is required to prove lemma (6). *)
Lemma bertogna_fp_alpha'_is_full:
∀ t,
job_arrival j ≤ t < job_arrival j + R →
backlogged job_cost sched j t →
count (scheduled_on_alpha_tsk t) (hp_tasks_in alpha') ≥ #|alpha' tsk|.
(* Before stating the next lemma, let (num_tasks_exceeding delta) be the
number of interfering tasks that can execute on (alpha' tsk) whose
interference x is larger than delta. *)
Let num_tasks_exceeding delta := count (fun i ⇒ x i ≥ delta) (hp_tasks_in alpha').
(* 6) 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 (|alpha' tsk| - num_tasks_exceeding delta) processors. *)
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).
(* 7) Based on lemma (6), we prove that, for any interval delta, if the sum of per-task
interference exceeds (delta * |alpha' tsk|), the same applies for the
sum of the minimum of the interference and 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|.
(* 8) Note that lemma (7) only refers to interference within subaffinity (alpha' tsk).
To reason about the interference incurred by job j in its complete affinity,
we prove that an exceeding interference on affinity (alpha tsk) also
implies an exceeding interference on the subaffinity (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|.
(* 9) Next, using lemmas (0), (3), (7) and (8), 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_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.
(* 10) 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, R - e_i + 1).
This implies that x_k > W_k, which is of course a contradiction,
since W_k is a valid task interference bound. *)
Lemma bertogna_fp_exists_task_that_exceeds_bound :
∃ tsk_k R_k,
(tsk_k, R_k) ∈ 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.
(* Using the lemmas above, we prove that R bounds the response time of tsk. *)
Theorem bertogna_cirinei_response_time_bound_fp :
response_time_bounded_by tsk R.
End ResponseTimeBound.
End ResponseTimeAnalysisFP.