# Library prosa.results.fixed_priority.rta.bounded_nps

Require Export prosa.analysis.definitions.schedulability.

Require Export prosa.analysis.definitions.request_bound_function.

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

Require Export prosa.analysis.facts.busy_interval.ideal.priority_inversion_bounded.

Require Export prosa.results.fixed_priority.rta.bounded_pi.

Require Export prosa.analysis.definitions.request_bound_function.

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

Require Export prosa.analysis.facts.busy_interval.ideal.priority_inversion_bounded.

Require Export prosa.results.fixed_priority.rta.bounded_pi.

# RTA for FP-schedulers with Bounded Non-Preemptive Segments

*and*bounded non-preemptive segments.

Consider any type of tasks ...

Context {Task : TaskType}.

Context `{TaskCost Task}.

Context `{TaskRunToCompletionThreshold Task}.

Context `{TaskMaxNonpreemptiveSegment Task}.

Context `{TaskCost Task}.

Context `{TaskRunToCompletionThreshold Task}.

Context `{TaskMaxNonpreemptiveSegment Task}.

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

Context {Job : JobType}.

Context `{JobTask Job Task}.

Context `{Arrival : JobArrival Job}.

Context `{Cost : JobCost Job}.

Context `{JobTask Job Task}.

Context `{Arrival : JobArrival Job}.

Context `{Cost : JobCost Job}.

Consider an FP policy that indicates a higher-or-equal priority
relation, and assume that the relation is reflexive and
transitive.

Context {FP : FP_policy Task}.

Hypothesis H_priority_is_reflexive : reflexive_priorities.

Hypothesis H_priority_is_transitive : transitive_priorities.

Hypothesis H_priority_is_reflexive : reflexive_priorities.

Hypothesis H_priority_is_transitive : transitive_priorities.

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

Variable arr_seq : arrival_sequence Job.

Hypothesis H_valid_arrival_sequence : valid_arrival_sequence arr_seq.

Hypothesis H_valid_arrival_sequence : valid_arrival_sequence arr_seq.

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

... allow for any work-bearing notion of job readiness, ...

Context `{@JobReady Job (ideal.processor_state Job) Cost Arrival}.

Hypothesis H_job_ready : work_bearing_readiness arr_seq sched.

Hypothesis H_job_ready : work_bearing_readiness arr_seq sched.

... and assume that the schedule is valid.

In addition, we assume the existence of a function mapping jobs
to their 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.

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

... and the schedule respects the scheduling policy.

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

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_rtct tsk is (1) no bigger than tsk's cost, (2) for
any job of task tsk job_rtct is bounded by task_rtct.

Let's define some local names for clarity.

Let max_length_of_priority_inversion :=

max_length_of_priority_inversion arr_seq.

Let task_rbf := task_request_bound_function tsk.

Let total_hep_rbf := total_hep_request_bound_function_FP ts tsk.

Let total_ohep_rbf := total_ohep_request_bound_function_FP ts tsk.

Let response_time_bounded_by := task_response_time_bound arr_seq sched.

max_length_of_priority_inversion arr_seq.

Let task_rbf := task_request_bound_function tsk.

Let total_hep_rbf := total_hep_request_bound_function_FP ts tsk.

Let total_ohep_rbf := total_ohep_request_bound_function_FP 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_other <- ts | ~~ hep_task tsk_other tsk)

(task_max_nonpreemptive_segment tsk_other - ε).

\max_(tsk_other <- ts | ~~ hep_task tsk_other tsk)

(task_max_nonpreemptive_segment tsk_other - 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 a 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_of_task tsk j →

max_length_of_priority_inversion j t ≤ blocking_bound.

Proof.

intros j t ARR TSK; move: TSK ⇒ /eqP TSK.

rewrite /max_length_of_priority_inversion /blocking_bound /max_length_of_priority_inversion.

apply: (@leq_trans (\max_(j_lp <- arrivals_before arr_seq t | ~~ hep_job j_lp j)

(job_max_nonpreemptive_segment j_lp - ε)));

first by apply: bigmax_subset ⇒ j' IN /andP [not_hep _].

apply: (@leq_trans (\max_(j_lp <- arrivals_between arr_seq 0 t

| ~~ hep_task (job_task j_lp) tsk)

(task_max_nonpreemptive_segment (job_task j_lp) - ε))).

{ rewrite /hep_job /FP_to_JLFP TSK.

apply leq_big_max ⇒ j' JINB NOTHEP.

rewrite leq_sub2r //.

apply H_valid_model_with_bounded_nonpreemptive_segments.

by eapply in_arrivals_implies_arrived; eauto 2. }

{ apply /bigmax_leq_seqP ⇒ j' JINB NOTHEP.

apply leq_bigmax_cond_seq with

(x := (job_task j')) (F := fun tsk ⇒ task_max_nonpreemptive_segment tsk - 1);

last by done.

apply H_all_jobs_from_taskset.

by apply: in_arrivals_implies_arrived (JINB). }

Qed.

∀ j t,

arrives_in arr_seq j →

job_of_task tsk j →

max_length_of_priority_inversion j t ≤ blocking_bound.

Proof.

intros j t ARR TSK; move: TSK ⇒ /eqP TSK.

rewrite /max_length_of_priority_inversion /blocking_bound /max_length_of_priority_inversion.

apply: (@leq_trans (\max_(j_lp <- arrivals_before arr_seq t | ~~ hep_job j_lp j)

(job_max_nonpreemptive_segment j_lp - ε)));

first by apply: bigmax_subset ⇒ j' IN /andP [not_hep _].

apply: (@leq_trans (\max_(j_lp <- arrivals_between arr_seq 0 t

| ~~ hep_task (job_task j_lp) tsk)

(task_max_nonpreemptive_segment (job_task j_lp) - ε))).

{ rewrite /hep_job /FP_to_JLFP TSK.

apply leq_big_max ⇒ j' JINB NOTHEP.

rewrite leq_sub2r //.

apply H_valid_model_with_bounded_nonpreemptive_segments.

by eapply in_arrivals_implies_arrived; eauto 2. }

{ apply /bigmax_leq_seqP ⇒ j' JINB NOTHEP.

apply leq_bigmax_cond_seq with

(x := (job_task j')) (F := fun tsk ⇒ task_max_nonpreemptive_segment tsk - 1);

last by done.

apply H_all_jobs_from_taskset.

by apply: in_arrivals_implies_arrived (JINB). }

Qed.

Using the above lemma, we prove that the priority inversion of the task is bounded by blocking_bound.

Lemma priority_inversion_is_bounded:

priority_inversion_is_bounded_by_constant

arr_seq sched tsk blocking_bound.

Proof.

intros j ARR TSK POS t1 t2 PREF.

case NEQ: (t2 - t1 ≤ blocking_bound).

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

rewrite /cumulative_priority_inversion -[X in _ ≤ X]addn0

-[t2 - t1]mul1n -iter_addn -big_const_nat leq_sum //.

by intros t _; case: (priority_inversion_dec _ _ _).

}

move: NEQ ⇒ /negP /negP; rewrite -ltnNge; move ⇒ BOUND.

edestruct (@preemption_time_exists) as [ppt [PPT NEQ]]; rt_eauto.

move: NEQ ⇒ /andP [GE LE].

apply leq_trans with (cumulative_priority_inversion arr_seq sched j t1 ppt);

last apply leq_trans with (ppt - t1); first last.

- 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.

rewrite /cumulative_priority_inversion -[X in _ ≤ X]addn0

-[ppt - t1]mul1n -iter_addn -big_const_nat leq_sum //.

by intros t _; case: (priority_inversion_dec _ _ _).

- rewrite /cumulative_priority_inversion.

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

{ rewrite ltn_subRL in BOUND.

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 // ⇒ t /andP [/andP [GEt LTt] _ ].

apply/eqP; rewrite eqb0; apply/negP ⇒ /priority_inversion_P PI; feed_n 3 PI; rt_eauto.

move: PI ⇒ [NSCHED [j__lp /andP [SCHED HEP]]].

edestruct (@not_quiet_implies_exists_scheduled_hp_job)

with (K := ppt - t1) (t1 := t1) (t2 := t2) (t := t)

as [j_hp [ARRB [HP SCHEDHP]]]; rt_eauto.

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

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

enough (EQef : j__lp = j_hp); first by subst; rewrite HP in HEP.

by eapply ideal_proc_model_is_a_uniprocessor_model; rt_eauto.

Qed.

End PriorityInversionIsBounded.

priority_inversion_is_bounded_by_constant

arr_seq sched tsk blocking_bound.

Proof.

intros j ARR TSK POS t1 t2 PREF.

case NEQ: (t2 - t1 ≤ blocking_bound).

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

rewrite /cumulative_priority_inversion -[X in _ ≤ X]addn0

-[t2 - t1]mul1n -iter_addn -big_const_nat leq_sum //.

by intros t _; case: (priority_inversion_dec _ _ _).

}

move: NEQ ⇒ /negP /negP; rewrite -ltnNge; move ⇒ BOUND.

edestruct (@preemption_time_exists) as [ppt [PPT NEQ]]; rt_eauto.

move: NEQ ⇒ /andP [GE LE].

apply leq_trans with (cumulative_priority_inversion arr_seq sched j t1 ppt);

last apply leq_trans with (ppt - t1); first last.

- 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.

rewrite /cumulative_priority_inversion -[X in _ ≤ X]addn0

-[ppt - t1]mul1n -iter_addn -big_const_nat leq_sum //.

by intros t _; case: (priority_inversion_dec _ _ _).

- rewrite /cumulative_priority_inversion.

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

{ rewrite ltn_subRL in BOUND.

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 // ⇒ t /andP [/andP [GEt LTt] _ ].

apply/eqP; rewrite eqb0; apply/negP ⇒ /priority_inversion_P PI; feed_n 3 PI; rt_eauto.

move: PI ⇒ [NSCHED [j__lp /andP [SCHED HEP]]].

edestruct (@not_quiet_implies_exists_scheduled_hp_job)

with (K := ppt - t1) (t1 := t1) (t2 := t2) (t := t)

as [j_hp [ARRB [HP SCHEDHP]]]; rt_eauto.

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

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

enough (EQef : j__lp = j_hp); first by subst; rewrite HP in HEP.

by eapply ideal_proc_model_is_a_uniprocessor_model; rt_eauto.

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.

Variable L : duration.

Hypothesis H_L_positive : L > 0.

Hypothesis H_fixed_point : L = blocking_bound + total_hep_rbf L.

Hypothesis H_L_positive : L > 0.

Hypothesis H_fixed_point : L = blocking_bound + total_hep_rbf L.

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

Next, consider any value R, and assume that for any given arrival offset A from the search
space there is a solution of the response-time bound recurrence that 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_rtct tsk))

+ total_ohep_rbf (A + F) ∧

F + (task_cost tsk - task_rtct 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_rtct tsk))

+ total_ohep_rbf (A + F) ∧

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

Then, using the results for the general RTA for FP-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 FP-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_fp_with_bounded_nonpreemptive_segments:

response_time_bounded_by tsk R.

Proof.

eapply uniprocessor_response_time_bound_fp;

eauto using priority_inversion_is_bounded with basic_rt_facts.

Qed.

End ResponseTimeBound.

End RTAforFPwithBoundedNonpreemptiveSegmentsWithArrivalCurves.

response_time_bounded_by tsk R.

Proof.

eapply uniprocessor_response_time_bound_fp;

eauto using priority_inversion_is_bounded with basic_rt_facts.

Qed.

End ResponseTimeBound.

End RTAforFPwithBoundedNonpreemptiveSegmentsWithArrivalCurves.