Library rt.restructuring.analysis.fixed_priority.rta.nonpr_reg.response_time_bound
Require Export rt.restructuring.analysis.schedulability.
Require Export rt.restructuring.analysis.arrival.workload_bound.
Require Export rt.restructuring.analysis.fixed_priority.rta.response_time_bound.
Require Export rt.restructuring.analysis.facts.priority_inversion_is_bounded.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq path fintype bigop.
Require Export rt.restructuring.analysis.arrival.workload_bound.
Require Export rt.restructuring.analysis.fixed_priority.rta.response_time_bound.
Require Export rt.restructuring.analysis.facts.priority_inversion_is_bounded.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq path fintype bigop.
Throughout this file, we assume ideal uniprocessor schedules.
Throughout this file, we assume the basic (i.e., Liu & Layland) readiness model.
RTA for FP-schedulers with Bounded Non-Preemprive 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 `{JobArrival Job}.
Context `{JobCost Job}.
Context `{JobTask Job Task}.
Context `{JobArrival Job}.
Context `{JobCost Job}.
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 uniprocessor 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 maping jobs
to theirs preemption points ...
... and assume that it defines a valid preemption
model with bounded nonpreemptive 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.
Consider an FP policy that indicates a higher-or-equal priority
relation, and assume that the relation is reflexive and
transitive.
Variable higher_eq_priority : 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.
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 thejob_preemptable
function (i.e., jobs have bounded nonpreemptive 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.
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 | ~~ higher_eq_priority tsk_other tsk)
(task_max_nonpreemptive_segment tsk_other - ε).
\max_(tsk_other <- ts | ~~ higher_eq_priority 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 nonpreemtive 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 nonpreemptive 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 →
max_length_of_priority_inversion j t ≤ blocking_bound.
Proof.
intros j t ARR TSK.
rewrite /max_length_of_priority_inversion /blocking_bound /FP_to_JLFP
/priority_inversion_is_bounded.max_length_of_priority_inversion.
apply leq_trans with
(\max_(j_lp <- arrivals_between arr_seq 0 t
| ~~ higher_eq_priority (job_task j_lp) tsk)
(task_max_nonpreemptive_segment (job_task j_lp) - ε)).
{ rewrite TSK.
apply leq_big_max.
intros 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.
intros j' JINB NOTHEP.
apply leq_bigmax_cond_seq with
(i0 := (job_task j')) (F := fun tsk ⇒ task_max_nonpreemptive_segment tsk - 1); last by done.
apply H_all_jobs_from_taskset.
apply mem_bigcat_nat_exists in JINB.
by inversion JINB as [ta' [JIN' _]]; ∃ ta'.
}
Qed.
∀ j t,
arrives_in arr_seq j →
job_task j = tsk →
max_length_of_priority_inversion j t ≤ blocking_bound.
Proof.
intros j t ARR TSK.
rewrite /max_length_of_priority_inversion /blocking_bound /FP_to_JLFP
/priority_inversion_is_bounded.max_length_of_priority_inversion.
apply leq_trans with
(\max_(j_lp <- arrivals_between arr_seq 0 t
| ~~ higher_eq_priority (job_task j_lp) tsk)
(task_max_nonpreemptive_segment (job_task j_lp) - ε)).
{ rewrite TSK.
apply leq_big_max.
intros 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.
intros j' JINB NOTHEP.
apply leq_bigmax_cond_seq with
(i0 := (job_task j')) (F := fun tsk ⇒ task_max_nonpreemptive_segment tsk - 1); last by done.
apply H_all_jobs_from_taskset.
apply mem_bigcat_nat_exists in JINB.
by inversion JINB as [ta' [JIN' _]]; ∃ ta'.
}
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
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 /is_priority_inversion.
rewrite -[X in _ ≤ X]addn0 -[t2 - t1]mul1n -iter_addn -big_const_nat leq_sum //.
intros t _; case: (sched t); last by done.
by intros s; case: (FP_to_JLFP Job Task s j).
}
move: NEQ ⇒ /negP /negP; rewrite -ltnNge; move ⇒ BOUND.
edestruct (@preemption_time_exists) as [ppt [PPT NEQ]]; eauto 2; move: NEQ ⇒ /andP [GE LE].
apply leq_trans with (cumulative_priority_inversion 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 /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; case: (FP_to_JLFP Job Task s j).
- rewrite /cumulative_priority_inversion /is_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.
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) (t1 := t1) (t2 := t2) (t := t) as [j_hp [ARRB [HP SCHEDHP]]]; eauto 2.
{ by ∃ ppt; split; [done | rewrite subnKC //; apply/andP]. }
{ by rewrite subnKC //; apply/andP; split. }
apply/eqP; rewrite eqb0 Bool.negb_involutive.
enough (EQef : s = j_hp); first by subst;auto.
eapply ideal_proc_model_is_a_uniprocessor_model; eauto 2.
by rewrite scheduled_at_def SCHED.
Qed.
End PriorityInversionIsBounded.
priority_inversion_is_bounded_by
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 /is_priority_inversion.
rewrite -[X in _ ≤ X]addn0 -[t2 - t1]mul1n -iter_addn -big_const_nat leq_sum //.
intros t _; case: (sched t); last by done.
by intros s; case: (FP_to_JLFP Job Task s j).
}
move: NEQ ⇒ /negP /negP; rewrite -ltnNge; move ⇒ BOUND.
edestruct (@preemption_time_exists) as [ppt [PPT NEQ]]; eauto 2; move: NEQ ⇒ /andP [GE LE].
apply leq_trans with (cumulative_priority_inversion 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 /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; case: (FP_to_JLFP Job Task s j).
- rewrite /cumulative_priority_inversion /is_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.
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) (t1 := t1) (t2 := t2) (t := t) as [j_hp [ARRB [HP SCHEDHP]]]; eauto 2.
{ by ∃ ppt; split; [done | rewrite subnKC //; apply/andP]. }
{ by rewrite subnKC //; apply/andP; split. }
apply/eqP; rewrite eqb0 Bool.negb_involutive.
enough (EQef : s = j_hp); first by subst;auto.
eapply ideal_proc_model_is_a_uniprocessor_model; eauto 2.
by rewrite scheduled_at_def SCHED.
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_run_to_completion_threshold tsk))
+ total_ohep_rbf (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))
+ total_ohep_rbf (A + F) ∧
F + (task_cost tsk - task_run_to_completion_threshold 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.
Qed.
End ResponseTimeBound.
End RTAforFPwithBoundedNonpreemptiveSegmentsWithArrivalCurves.
response_time_bounded_by tsk R.
Proof.
eapply uniprocessor_response_time_bound_fp;
eauto using priority_inversion_is_bounded.
Qed.
End ResponseTimeBound.
End RTAforFPwithBoundedNonpreemptiveSegmentsWithArrivalCurves.