# Library prosa.results.edf.rta.bounded_nps

Require Import prosa.model.priority.edf.

Require Export prosa.analysis.facts.model.rbf.

Require Export prosa.analysis.facts.model.sequential.

Require Export prosa.results.edf.rta.bounded_pi.

Require Export prosa.analysis.facts.busy_interval.priority_inversion.

Require Export prosa.analysis.facts.model.rbf.

Require Export prosa.analysis.facts.model.sequential.

Require Export prosa.results.edf.rta.bounded_pi.

Require Export prosa.analysis.facts.busy_interval.priority_inversion.

Throughout this file, we assume ideal uni-processor schedules.

Throughout this file, we assume the basic (i.e., Liu & Layland) readiness model.

# RTA for EDF with Bounded Non-Preemptive Segments

*and*bounded non-preemptive segments.

Consider any type of tasks ...

Context {Task : TaskType}.

Context `{TaskCost Task}.

Context `{TaskDeadline Task}.

Context `{TaskRunToCompletionThreshold Task}.

Context `{TaskMaxNonpreemptiveSegment Task}.

Context `{TaskCost Task}.

Context `{TaskDeadline Task}.

Context `{TaskRunToCompletionThreshold Task}.

Context `{TaskMaxNonpreemptiveSegment Task}.

... and any type of jobs associated with these tasks.

Context {Job : JobType}.

Context `{JobTask Job Task}.

Context `{JobArrival Job}.

Context `{JobCost Job}.

Context `{JobTask Job Task}.

Context `{JobArrival Job}.

Context `{JobCost Job}.

For clarity, let's denote the relative deadline of a task as D.

Consider the EDF policy that indicates a higher-or-equal priority relation.
Note that we do not relate the EDF policy with the scheduler. However, we
define functions for Interference and Interfering Workload that actively use
the concept of priorities.

Consider any arrival sequence with consistent, non-duplicate arrivals.

Variable arr_seq : arrival_sequence Job.

Hypothesis H_arrival_times_are_consistent : consistent_arrival_times arr_seq.

Hypothesis H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq.

Hypothesis H_arrival_times_are_consistent : consistent_arrival_times arr_seq.

Hypothesis H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq.

Next, consider any ideal uni-processor schedule of this arrival sequence ...

Variable sched : schedule (ideal.processor_state Job).

Hypothesis H_jobs_come_from_arrival_sequence:

jobs_come_from_arrival_sequence sched arr_seq.

Hypothesis H_jobs_come_from_arrival_sequence:

jobs_come_from_arrival_sequence sched arr_seq.

... where jobs do not execute before their arrival or after completion.

Hypothesis H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched.

Hypothesis H_completed_jobs_dont_execute : completed_jobs_dont_execute sched.

Hypothesis H_completed_jobs_dont_execute : completed_jobs_dont_execute sched.

In addition, we assume the existence of a function mapping jobs
to theirs preemption points ...

... and assume that it defines a valid preemption
model with bounded non-preemptive segments.

Hypothesis H_valid_model_with_bounded_nonpreemptive_segments:

valid_model_with_bounded_nonpreemptive_segments

arr_seq sched.

valid_model_with_bounded_nonpreemptive_segments

arr_seq sched.

Assume we have sequential tasks, i.e, jobs from the
same task execute in the order of their arrival.

Next, we assume that the schedule is a work-conserving schedule...

... and the schedule respects the policy defined by the job_preemptable
function (i.e., jobs have bounded non-preemptive segments).

Consider an arbitrary task set ts, ...

... assume that all jobs come from the task set, ...

... and the cost of a job cannot be larger than the task cost.

Let max_arrivals be a family of valid arrival curves, i.e., for
any task tsk in ts max_arrival tsk is (1) an arrival bound of
tsk, and (2) it is a monotonic function that equals 0 for the
empty interval delta = 0.

Context `{MaxArrivals Task}.

Hypothesis H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals.

Hypothesis H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts.

Hypothesis H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals.

Hypothesis H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts.

Let tsk be any task in ts that is to be analyzed.

Consider a valid preemption model...

...and a valid task run-to-completion threshold function. That is,
task_run_to_completion_threshold tsk is (1) no bigger than tsk's
cost, (2) for any job of task tsk job_run_to_completion_threshold
is bounded by task_run_to_completion_threshold.

We introduce as an abbreviation rbf for the task request bound function,
which is defined as task_cost(T) × max_arrivals(T,Δ) for a task T.

Using the sum of individual request bound functions, we define the request bound
function of all tasks (total request bound function).

Next, we define an upper bound on interfering workload received from jobs
of other tasks with higher-than-or-equal priority.

Let bound_on_total_hep_workload A Δ :=

\sum_(tsk_o <- ts | tsk_o != tsk)

rbf tsk_o (minn ((A + ε) + D tsk - D tsk_o) Δ).

\sum_(tsk_o <- ts | tsk_o != tsk)

rbf tsk_o (minn ((A + ε) + D tsk - D tsk_o) Δ).

Let's define some local names for clarity.

Let max_length_of_priority_inversion :=

max_length_of_priority_inversion arr_seq.

Let task_rbf_changes_at A := task_rbf_changes_at tsk A.

Let bound_on_total_hep_workload_changes_at :=

bound_on_total_hep_workload_changes_at ts tsk.

Let response_time_bounded_by := task_response_time_bound arr_seq sched.

max_length_of_priority_inversion arr_seq.

Let task_rbf_changes_at A := task_rbf_changes_at tsk A.

Let bound_on_total_hep_workload_changes_at :=

bound_on_total_hep_workload_changes_at ts tsk.

Let response_time_bounded_by := task_response_time_bound arr_seq sched.

We also define a bound for the priority inversion caused by jobs with lower priority.

Definition blocking_bound :=

\max_(tsk_o <- ts | (tsk_o != tsk) && (D tsk < D tsk_o))

(task_max_nonpreemptive_segment tsk_o - ε).

\max_(tsk_o <- ts | (tsk_o != tsk) && (D tsk < D tsk_o))

(task_max_nonpreemptive_segment tsk_o - ε).

## Priority inversion is bounded

In this section, we prove that a priority inversion for task tsk is bounded by the maximum length of non-preemptive segments among the tasks with lower priority.
First, we prove that the maximum length of a priority
inversion of job j is bounded by the maximum length of a
non-preemptive section of a task with lower-priority task
(i.e., the blocking term).

Lemma priority_inversion_is_bounded_by_blocking:

∀ j t,

arrives_in arr_seq j →

job_task j = tsk →

t ≤ job_arrival j →

max_length_of_priority_inversion j t ≤ blocking_bound.

Proof.

intros j t ARR TSK LE; unfold max_length_of_priority_inversion, blocking_bound.

apply leq_trans with

(\max_(j_lp <- arrivals_between arr_seq 0 t | ~~ EDF j_lp j)

(task_max_nonpreemptive_segment (job_task j_lp) - ε)).

- apply leq_big_max.

intros j' JINB NOTHEP.

rewrite leq_sub2r //.

apply in_arrivals_implies_arrived in JINB.

by apply H_valid_model_with_bounded_nonpreemptive_segments.

- apply /bigmax_leq_seqP.

intros j' JINB NOTHEP.

apply leq_bigmax_cond_seq with (i0 := (job_task j')) (F := fun tsk ⇒ task_max_nonpreemptive_segment tsk - 1).

{ apply H_all_jobs_from_taskset.

apply mem_bigcat_nat_exists in JINB.

by inversion JINB as [ta' [JIN' _]]; ∃ ta'. }

{ have NINTSK: job_task j' != tsk.

{ apply/eqP; intros TSKj'.

rewrite /EDF -ltnNge in NOTHEP.

rewrite /job_deadline /absolute_deadline.job_deadline_from_task_deadline in NOTHEP.

rewrite TSKj' TSK ltn_add2r in NOTHEP.

move: NOTHEP; rewrite ltnNge; move ⇒ /negP T; apply: T.

apply leq_trans with t; last by done.

eapply in_arrivals_implies_arrived_between in JINB; last by eauto 2.

move: JINB; move ⇒ /andP [_ T].

by apply ltnW.

}

apply/andP; split; first by done.

rewrite /EDF -ltnNge in NOTHEP.

rewrite -TSK.

have ARRLE: job_arrival j' < job_arrival j.

{ apply leq_trans with t; last by done.

eapply in_arrivals_implies_arrived_between in JINB; last by eauto 2.

by move: JINB; move ⇒ /andP [_ T].

}

rewrite /job_deadline /absolute_deadline.job_deadline_from_task_deadline in NOTHEP.

rewrite /D; ssromega.

}

Qed.

∀ j t,

arrives_in arr_seq j →

job_task j = tsk →

t ≤ job_arrival j →

max_length_of_priority_inversion j t ≤ blocking_bound.

Proof.

intros j t ARR TSK LE; unfold max_length_of_priority_inversion, blocking_bound.

apply leq_trans with

(\max_(j_lp <- arrivals_between arr_seq 0 t | ~~ EDF j_lp j)

(task_max_nonpreemptive_segment (job_task j_lp) - ε)).

- apply leq_big_max.

intros j' JINB NOTHEP.

rewrite leq_sub2r //.

apply in_arrivals_implies_arrived in JINB.

by apply H_valid_model_with_bounded_nonpreemptive_segments.

- apply /bigmax_leq_seqP.

intros j' JINB NOTHEP.

apply leq_bigmax_cond_seq with (i0 := (job_task j')) (F := fun tsk ⇒ task_max_nonpreemptive_segment tsk - 1).

{ apply H_all_jobs_from_taskset.

apply mem_bigcat_nat_exists in JINB.

by inversion JINB as [ta' [JIN' _]]; ∃ ta'. }

{ have NINTSK: job_task j' != tsk.

{ apply/eqP; intros TSKj'.

rewrite /EDF -ltnNge in NOTHEP.

rewrite /job_deadline /absolute_deadline.job_deadline_from_task_deadline in NOTHEP.

rewrite TSKj' TSK ltn_add2r in NOTHEP.

move: NOTHEP; rewrite ltnNge; move ⇒ /negP T; apply: T.

apply leq_trans with t; last by done.

eapply in_arrivals_implies_arrived_between in JINB; last by eauto 2.

move: JINB; move ⇒ /andP [_ T].

by apply ltnW.

}

apply/andP; split; first by done.

rewrite /EDF -ltnNge in NOTHEP.

rewrite -TSK.

have ARRLE: job_arrival j' < job_arrival j.

{ apply leq_trans with t; last by done.

eapply in_arrivals_implies_arrived_between in JINB; last by eauto 2.

by move: JINB; move ⇒ /andP [_ T].

}

rewrite /job_deadline /absolute_deadline.job_deadline_from_task_deadline in NOTHEP.

rewrite /D; ssromega.

}

Qed.

Using the lemma above, we prove that the priority inversion of the task is bounded by
the maximum length of a nonpreemptive section of lower-priority tasks.

Lemma priority_inversion_is_bounded:

priority_inversion_is_bounded_by arr_seq sched tsk blocking_bound.

Proof.

move ⇒ j ARR TSK POS t1 t2 PREF; move: (PREF) ⇒ [_ [_ [_ /andP [T _]]]].

destruct (leqP (t2 - t1) blocking_bound) as [NEQ|NEQ].

{ apply leq_trans with (t2 - t1); last by done.

rewrite /cumulative_priority_inversion /is_priority_inversion.

rewrite -[X in _ ≤ X]addn0 -[t2 - t1]mul1n -iter_addn -big_const_nat.

rewrite leq_sum //.

intros t _; case: (sched t); last by done.

by intros s; destruct (hep_job s j).

}

edestruct @preemption_time_exists as [ppt [PPT NEQ2]]; eauto 2 with basic_facts.

move: NEQ2 ⇒ /andP [GE LE].

apply leq_trans with (cumulative_priority_inversion sched j t1 ppt);

last apply leq_trans with (ppt - t1).

- rewrite /cumulative_priority_inversion /is_priority_inversion.

rewrite (@big_cat_nat _ _ _ ppt) //=; last first.

{ rewrite ltn_subRL in NEQ.

apply leq_trans with (t1 + blocking_bound); last by apply ltnW.

apply leq_trans with (t1 + max_length_of_priority_inversion j t1); first by done.

by rewrite leq_add2l; eapply priority_inversion_is_bounded_by_blocking; eauto 2. }

rewrite -[X in _ ≤ X]addn0 leq_add2l leqn0.

rewrite big_nat_cond big1 //; move ⇒ t /andP [/andP [GEt LTt] _ ].

case SCHED: (sched t) ⇒ [s | ]; last by done.

edestruct @not_quiet_implies_exists_scheduled_hp_job

with (K := ppt - t1) (t := t) as [j_hp [ARRB [HP SCHEDHP]]]; eauto 2 with basic_facts.

{ ∃ ppt; split. by done. by rewrite subnKC //; apply/andP; split. }

{ by rewrite subnKC //; apply/andP; split. }

apply/eqP; rewrite eqb0 Bool.negb_involutive.

enough (EQ : s = j_hp); first by subst.

move: SCHED ⇒ /eqP SCHED; rewrite -scheduled_at_def in SCHED.

by eapply ideal_proc_model_is_a_uniprocessor_model; [exact SCHED | exact SCHEDHP].

- rewrite /cumulative_priority_inversion /is_priority_inversion.

rewrite -[X in _ ≤ X]addn0 -[ppt - t1]mul1n -iter_addn -big_const_nat.

rewrite leq_sum //.

intros t _; case: (sched t); last by done.

by intros s; destruct (hep_job s j).

- rewrite leq_subLR.

apply leq_trans with (t1 + max_length_of_priority_inversion j t1); first by done.

by rewrite leq_add2l; eapply priority_inversion_is_bounded_by_blocking; eauto 2.

Qed.

End PriorityInversionIsBounded.

priority_inversion_is_bounded_by arr_seq sched tsk blocking_bound.

Proof.

move ⇒ j ARR TSK POS t1 t2 PREF; move: (PREF) ⇒ [_ [_ [_ /andP [T _]]]].

destruct (leqP (t2 - t1) blocking_bound) as [NEQ|NEQ].

{ apply leq_trans with (t2 - t1); last by done.

rewrite /cumulative_priority_inversion /is_priority_inversion.

rewrite -[X in _ ≤ X]addn0 -[t2 - t1]mul1n -iter_addn -big_const_nat.

rewrite leq_sum //.

intros t _; case: (sched t); last by done.

by intros s; destruct (hep_job s j).

}

edestruct @preemption_time_exists as [ppt [PPT NEQ2]]; eauto 2 with basic_facts.

move: NEQ2 ⇒ /andP [GE LE].

apply leq_trans with (cumulative_priority_inversion sched j t1 ppt);

last apply leq_trans with (ppt - t1).

- rewrite /cumulative_priority_inversion /is_priority_inversion.

rewrite (@big_cat_nat _ _ _ ppt) //=; last first.

{ rewrite ltn_subRL in NEQ.

apply leq_trans with (t1 + blocking_bound); last by apply ltnW.

apply leq_trans with (t1 + max_length_of_priority_inversion j t1); first by done.

by rewrite leq_add2l; eapply priority_inversion_is_bounded_by_blocking; eauto 2. }

rewrite -[X in _ ≤ X]addn0 leq_add2l leqn0.

rewrite big_nat_cond big1 //; move ⇒ t /andP [/andP [GEt LTt] _ ].

case SCHED: (sched t) ⇒ [s | ]; last by done.

edestruct @not_quiet_implies_exists_scheduled_hp_job

with (K := ppt - t1) (t := t) as [j_hp [ARRB [HP SCHEDHP]]]; eauto 2 with basic_facts.

{ ∃ ppt; split. by done. by rewrite subnKC //; apply/andP; split. }

{ by rewrite subnKC //; apply/andP; split. }

apply/eqP; rewrite eqb0 Bool.negb_involutive.

enough (EQ : s = j_hp); first by subst.

move: SCHED ⇒ /eqP SCHED; rewrite -scheduled_at_def in SCHED.

by eapply ideal_proc_model_is_a_uniprocessor_model; [exact SCHED | exact SCHEDHP].

- rewrite /cumulative_priority_inversion /is_priority_inversion.

rewrite -[X in _ ≤ X]addn0 -[ppt - t1]mul1n -iter_addn -big_const_nat.

rewrite leq_sum //.

intros t _; case: (sched t); last by done.

by intros s; destruct (hep_job s j).

- rewrite leq_subLR.

apply leq_trans with (t1 + max_length_of_priority_inversion j t1); first by done.

by rewrite leq_add2l; eapply priority_inversion_is_bounded_by_blocking; eauto 2.

Qed.

End PriorityInversionIsBounded.

## Response-Time Bound

In this section, we prove that the maximum among the solutions of the response-time bound recurrence is a response-time bound for tsk.
Let L be any positive fixed point of the busy interval recurrence.

To reduce the time complexity of the analysis, recall the notion of search space.

Let is_in_search_space A :=

(A < L) && (task_rbf_changes_at A || bound_on_total_hep_workload_changes_at A).

(A < L) && (task_rbf_changes_at A || bound_on_total_hep_workload_changes_at A).

Consider any value R, and assume that for any given arrival offset A in the search space,
there is a solution of the response-time bound recurrence which is bounded by R.

Variable R : duration.

Hypothesis H_R_is_maximum:

∀ (A : duration),

is_in_search_space A →

∃ (F : duration),

A + F = blocking_bound

+ (task_rbf (A + ε) - (task_cost tsk - task_run_to_completion_threshold tsk))

+ bound_on_total_hep_workload A (A + F) ∧

F + (task_cost tsk - task_run_to_completion_threshold tsk) ≤ R.

Hypothesis H_R_is_maximum:

∀ (A : duration),

is_in_search_space A →

∃ (F : duration),

A + F = blocking_bound

+ (task_rbf (A + ε) - (task_cost tsk - task_run_to_completion_threshold tsk))

+ bound_on_total_hep_workload A (A + F) ∧

F + (task_cost tsk - task_run_to_completion_threshold tsk) ≤ R.

Then, using the results for the general RTA for EDF-schedulers, we establish a
response-time bound for the more concrete model of bounded nonpreemptive segments.
Note that in case of the general RTA for EDF-schedulers, we just

*assume*that the priority inversion is bounded. In this module we provide the preemption model with bounded nonpreemptive segments and*prove*that the priority inversion is bounded.
Theorem uniprocessor_response_time_bound_edf_with_bounded_nonpreemptive_segments:

response_time_bounded_by tsk R.

Proof.

eapply uniprocessor_response_time_bound_edf; eauto 2.

by apply priority_inversion_is_bounded.

Qed.

End ResponseTimeBound.

End RTAforEDFwithBoundedNonpreemptiveSegmentsWithArrivalCurves.

response_time_bounded_by tsk R.

Proof.

eapply uniprocessor_response_time_bound_edf; eauto 2.

by apply priority_inversion_is_bounded.

Qed.

End ResponseTimeBound.

End RTAforEDFwithBoundedNonpreemptiveSegmentsWithArrivalCurves.