Library prosa.analysis.abstract.abstract_seq_rta
Require Export prosa.analysis.definitions.task_schedule.
Require Export prosa.analysis.facts.model.rbf.
Require Export prosa.analysis.facts.model.task_arrivals.
Require Export prosa.analysis.facts.model.sequential.
Require Export prosa.analysis.abstract.abstract_rta.
Require Export prosa.analysis.facts.model.rbf.
Require Export prosa.analysis.facts.model.task_arrivals.
Require Export prosa.analysis.facts.model.sequential.
Require Export prosa.analysis.abstract.abstract_rta.
Abstract Response-Time Analysis with sequential tasks
In this section we propose the general framework for response-time analysis (RTA) of uni-processor scheduling of real-time tasks with arbitrary arrival models and sequential tasks.
Consider any type of tasks ...
... and any type of jobs associated with these tasks.
Context {Job : JobType}.
Context `{JobTask Job Task}.
Context `{JobArrival Job}.
Context `{JobCost Job}.
Context `{JobPreemptable Job}.
Context `{JobTask Job Task}.
Context `{JobArrival Job}.
Context `{JobCost Job}.
Context `{JobPreemptable 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 nor 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.
Assume that the job costs are no larger than the task costs.
Consider an arbitrary task set.
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 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.
Assume we are provided with abstract functions for interference and interfering workload.
Variable interference : Job → instant → bool.
Variable interfering_workload : Job → instant → duration.
Variable interfering_workload : Job → instant → duration.
Let's define some local names for clarity.
Let task_rbf := task_request_bound_function tsk.
Let busy_interval := busy_interval sched interference interfering_workload.
Let arrivals_between := arrivals_between arr_seq.
Let service_of_jobs_at := service_of_jobs_at sched.
Let task_workload_between := task_workload_between arr_seq tsk.
Let task_service_of_jobs_in := task_service_of_jobs_in sched tsk.
Let response_time_bounded_by := task_response_time_bound arr_seq sched.
Let busy_interval := busy_interval sched interference interfering_workload.
Let arrivals_between := arrivals_between arr_seq.
Let service_of_jobs_at := service_of_jobs_at sched.
Let task_workload_between := task_workload_between arr_seq tsk.
Let task_service_of_jobs_in := task_service_of_jobs_in sched tsk.
Let response_time_bounded_by := task_response_time_bound arr_seq sched.
In this section, we introduce a few new definitions to make it easier
to express the new bound of the worst-case execution time.
When assuming sequential tasks, we can introduce an additional hypothesis that
ensures that the values of interference and workload remain consistent. It states
that any of tsk's job, that arrived before the busy interval, should be
completed by the beginning of the busy interval.
Definition interference_and_workload_consistent_with_sequential_tasks :=
∀ (j : Job) (t1 t2 : instant),
arrives_in arr_seq j →
job_task j = tsk →
job_cost j > 0 →
busy_interval j t1 t2 →
task_workload_between 0 t1 = task_service_of_jobs_in (arrivals_between 0 t1) 0 t1.
∀ (j : Job) (t1 t2 : instant),
arrives_in arr_seq j →
job_task j = tsk →
job_cost j > 0 →
busy_interval j t1 t2 →
task_workload_between 0 t1 = task_service_of_jobs_in (arrivals_between 0 t1) 0 t1.
Next we introduce the notion of task interference. Intuitively, task tsk incurs
interference when some of the jobs of task tsk incur interference. As a result,
tsk cannot make any progress. More formally, task tsk experiences interference at
a time instant time t, if at time t task tsk is not scheduled and there exists
a job of tsk that (1) experiences interference and (2) has arrived before some
time instant upper_bound.
It is important to note two subtle points: according to our semantics of the
interference function, jobs from the same task can cause interference to each other.
In the definition of interference of a task we want to avoid such situations. That
is why we use the term ~~ task_scheduled_at tsk t.
Moreover, in order to make the definition constructive, we introduce an upper bound
on the arrival time of jobs from task tsk. As a result, we need to consider only a
finite number of jobs. For the function to produce the correct values it is enough
to specify a sufficiently large upper_bound. Usually as upper_bound one can use the
end of the corresponding busy interval.
Definition task_interference_received_before (tsk : Task) (upper_bound : instant) (t : instant) :=
(~~ task_scheduled_at sched tsk t)
&& has (fun j ⇒ interference j t) (task_arrivals_before arr_seq tsk upper_bound).
(~~ task_scheduled_at sched tsk t)
&& has (fun j ⇒ interference j t) (task_arrivals_before arr_seq tsk upper_bound).
Next we define the cumulative task interference.
Definition cumul_task_interference tsk upper_bound t1 t2 :=
\sum_(t1 ≤ t < t2) task_interference_received_before tsk upper_bound t.
\sum_(t1 ≤ t < t2) task_interference_received_before tsk upper_bound t.
We say that task interference is bounded by task_interference_bound_function (tIBF)
iff for any job j of task tsk cumulative task interference within the interval
t1, t1 + R) is bounded by function [tIBF(tsk, A, R)].
Note that this definition is almost the same as the definition of job_interference_is_bounded_by
from the non-necessary-sequential case. However, in this case we ignore the
interference that comes from jobs from the same task.
Definition task_interference_is_bounded_by
(task_interference_bound_function : Task → duration → duration → duration) :=
∀ j R t1 t2,
arrives_in arr_seq j →
job_task j = tsk →
t1 + R < t2 →
~~ completed_by sched j (t1 + R) →
busy_interval j t1 t2 →
let offset := job_arrival j - t1 in
cumul_task_interference tsk t2 t1 (t1 + R) ≤ task_interference_bound_function tsk offset R.
End Definitions.
(task_interference_bound_function : Task → duration → duration → duration) :=
∀ j R t1 t2,
arrives_in arr_seq j →
job_task j = tsk →
t1 + R < t2 →
~~ completed_by sched j (t1 + R) →
busy_interval j t1 t2 →
let offset := job_arrival j - t1 in
cumul_task_interference tsk t2 t1 (t1 + R) ≤ task_interference_bound_function tsk offset R.
End Definitions.
In this section, we prove that the maximum among the solutions of the
response-time bound recurrence is a response-time bound for tsk.
For simplicity, let's define some local names.
Let cumul_interference := cumul_interference interference.
Let cumul_workload := cumul_interfering_workload interfering_workload.
Let cumul_task_interference := cumul_task_interference tsk.
Let cumul_workload := cumul_interfering_workload interfering_workload.
Let cumul_task_interference := cumul_task_interference tsk.
We assume that the schedule is work-conserving.
Unlike the previous theorem uniprocessor_response_time_bound, we assume
that (1) tasks are sequential, moreover (2) functions interference and
interfering_workload are consistent with the hypothesis of sequential tasks.
Hypothesis H_sequential_tasks : sequential_tasks sched.
Hypothesis H_interference_and_workload_consistent_with_sequential_tasks:
interference_and_workload_consistent_with_sequential_tasks.
Hypothesis H_interference_and_workload_consistent_with_sequential_tasks:
interference_and_workload_consistent_with_sequential_tasks.
Assume we have a constant L which bounds the busy interval of any of tsk's jobs.
Variable L : duration.
Hypothesis H_busy_interval_exists:
busy_intervals_are_bounded_by arr_seq sched tsk interference interfering_workload L.
Hypothesis H_busy_interval_exists:
busy_intervals_are_bounded_by arr_seq sched tsk interference interfering_workload L.
Next, we assume that task_interference_bound_function is a bound on interference incurred by the task.
Variable task_interference_bound_function : Task → duration → duration → duration.
Hypothesis H_task_interference_is_bounded:
task_interference_is_bounded_by task_interference_bound_function.
Hypothesis H_task_interference_is_bounded:
task_interference_is_bounded_by task_interference_bound_function.
Given any job j of task tsk that arrives exactly A units after the beginning of the busy
interval, the bound on the total interference incurred by j within an interval of length Δ
is no greater than task_rbf (A + ε) - task_cost tsk + task's IBF Δ. Note that in case of
sequential tasks the bound consists of two parts: (1) the part that bounds the interference
received from other jobs of task tsk -- task_rbf (A + ε) - task_cost tsk and (2) any other
interference that is bounded by task_IBF(tsk, A, Δ).
Let total_interference_bound (tsk : Task) (A Δ : duration) :=
task_rbf (A + ε) - task_cost tsk + task_interference_bound_function tsk A Δ.
task_rbf (A + ε) - task_cost tsk + task_interference_bound_function tsk A Δ.
Note that since we consider the modified interference bound function, the search space has
also changed. One can see that the new search space is guaranteed to include any A for which
task_rbf (A) ≠ task_rbf (A + ε), since this implies the fact that
total_interference_bound (tsk, A, Δ) ≠ total_interference_bound (tsk, A + ε, Δ).
Consider any value R, and assume that for any relative arrival time A from the search
space there is a solution F of the response-time recurrence that is bounded by R. In
contrast to the formula in "non-sequential" Abstract RTA, assuming that tasks are
sequential leads to a more precise response-time bound. Now we can explicitly express
the interference caused by other jobs of the task under consideration.
To understand the right part of the fix-point in the equation it is helpful to note
that the bound on the total interference (bound_of_total_interference) is equal to
task_rbf (A + ε) - task_cost tsk + tIBF tsk A Δ. Besides, a job must receive
enough service to become non-preemptive task_lock_in_service tsk. The sum of
these two quantities is exactly the right-hand side of the equation.
Variable R : nat.
Hypothesis H_R_is_maximum_seq:
∀ (A : duration),
is_in_search_space_seq A →
∃ (F : duration),
A + F = (task_rbf (A + ε) - (task_cost tsk - task_run_to_completion_threshold tsk))
+ task_interference_bound_function tsk A (A + F) ∧
F + (task_cost tsk - task_run_to_completion_threshold tsk) ≤ R.
Hypothesis H_R_is_maximum_seq:
∀ (A : duration),
is_in_search_space_seq A →
∃ (F : duration),
A + F = (task_rbf (A + ε) - (task_cost tsk - task_run_to_completion_threshold tsk))
+ task_interference_bound_function tsk A (A + F) ∧
F + (task_cost tsk - task_run_to_completion_threshold tsk) ≤ R.
In this section we prove a few simple lemmas about the completion of jobs from the task
considering the busy interval of the job under consideration.
Variable j1 j2 : Job.
Hypothesis H_j1_arrives: arrives_in arr_seq j1.
Hypothesis H_j2_arrives: arrives_in arr_seq j2.
Hypothesis H_j1_from_tsk: job_task j1 = tsk.
Hypothesis H_j2_from_tsk: job_task j2 = tsk.
Hypothesis H_j1_cost_positive: job_cost_positive j1.
Hypothesis H_j1_arrives: arrives_in arr_seq j1.
Hypothesis H_j2_arrives: arrives_in arr_seq j2.
Hypothesis H_j1_from_tsk: job_task j1 = tsk.
Hypothesis H_j2_from_tsk: job_task j2 = tsk.
Hypothesis H_j1_cost_positive: job_cost_positive j1.
We prove that if a job from task tsk arrived before the beginning of the busy
interval, then it must be completed before the beginning of the busy interval
Lemma completed_before_beginning_of_busy_interval:
job_arrival j2 < t1 →
completed_by sched j2 t1.
Proof.
move ⇒ JA; move: (H_j2_from_tsk) ⇒ /eqP TSK2eq.
move: (posnP (@job_cost _ H3 j2)) ⇒ [ZERO|POS].
{ by rewrite /completed_by /service.completed_by ZERO. }
move: (H_interference_and_workload_consistent_with_sequential_tasks
j1 t1 t2 H_j1_arrives H_j1_from_tsk H_j1_cost_positive H_busy_interval) ⇒ SWEQ.
eapply all_jobs_have_completed_equiv_workload_eq_service
with (j := j2) in SWEQ; eauto 2; try done.
by apply arrived_between_implies_in_arrivals.
Qed.
job_arrival j2 < t1 →
completed_by sched j2 t1.
Proof.
move ⇒ JA; move: (H_j2_from_tsk) ⇒ /eqP TSK2eq.
move: (posnP (@job_cost _ H3 j2)) ⇒ [ZERO|POS].
{ by rewrite /completed_by /service.completed_by ZERO. }
move: (H_interference_and_workload_consistent_with_sequential_tasks
j1 t1 t2 H_j1_arrives H_j1_from_tsk H_j1_cost_positive H_busy_interval) ⇒ SWEQ.
eapply all_jobs_have_completed_equiv_workload_eq_service
with (j := j2) in SWEQ; eauto 2; try done.
by apply arrived_between_implies_in_arrivals.
Qed.
Next we prove that if a job is pending after the beginning
of the busy interval t1, t2) then it arrives after t1.
Lemma arrives_after_beginning_of_busy_interval:
∀ t,
t1 ≤ t →
pending sched j2 t →
arrived_between j2 t1 t.+1.
Proof.
intros t GE PEND.
rewrite /arrived_between; apply/andP; split; last first.
{ by move: PEND ⇒ /andP [ARR _]; rewrite ltnS. }
rewrite leqNgt; apply/negP; intros LT.
move: (H_busy_interval) ⇒ [[/andP [AFR1 AFR2] [QT _]] _].
have L12 := completed_before_beginning_of_busy_interval LT.
apply completion_monotonic with (t' := t) in L12; try done.
by move: PEND ⇒ /andP [_ /negP T2].
Qed.
End CompletionOfJobsFromSameTask.
∀ t,
t1 ≤ t →
pending sched j2 t →
arrived_between j2 t1 t.+1.
Proof.
intros t GE PEND.
rewrite /arrived_between; apply/andP; split; last first.
{ by move: PEND ⇒ /andP [ARR _]; rewrite ltnS. }
rewrite leqNgt; apply/negP; intros LT.
move: (H_busy_interval) ⇒ [[/andP [AFR1 AFR2] [QT _]] _].
have L12 := completed_before_beginning_of_busy_interval LT.
apply completion_monotonic with (t' := t) in L12; try done.
by move: PEND ⇒ /andP [_ /negP T2].
Qed.
End CompletionOfJobsFromSameTask.
Since we are going to use the uniprocessor_response_time_bound theorem to prove
the theorem of this section, we have to show that all the hypotheses are satisfied.
Namely, we need to show that hypotheses H_sequential_tasks, H_i_w_are_task_consistent
and H_task_interference_is_bounded_by imply H_job_interference_is_bounded, and the
fact that H_R_is_maximum_seq implies H_R_is_maximum.
In this section we show that there exists a bound for cumulative interference for any
job of task tsk, i.e., the hypothesis H_job_interference_is_bounded holds.
Variable j : Job.
Hypothesis H_j_arrives : arrives_in arr_seq j.
Hypothesis H_job_of_tsk : job_task j = tsk.
Hypothesis H_job_cost_positive : job_cost_positive j.
Hypothesis H_j_arrives : arrives_in arr_seq j.
Hypothesis H_job_of_tsk : job_task j = tsk.
Hypothesis H_job_cost_positive : job_cost_positive j.
Let's define A as a relative arrival time of job j (with respect to time t1).
Consider an arbitrary time x ...
... such that (t1 + x) is inside the busy interval...
... and job j is not completed by time (t1 + x).
In this section, we show that the cumulative interference of job j in the interval t1, t1 + x)
is bounded by the sum of the task workload in the interval [t1, t1 + A + ε) and the cumulative
interference of [j]'s task in the interval [t1, t1 + x). Note that the task workload is computed
only on the interval [t1, t1 + A + ε). Thanks to the hypothesis about sequential tasks, jobs of
task [tsk] that arrive after [t1 + A + ε] cannot interfere with j.
We start by proving a simpler analog of the lemma which states that at
any time instant t ∈ t1, t1 + x) the sum of [interference j t] and
[scheduled_at j t] is no larger than the sum of [the service received
by jobs of task tsk at time t] and [task_iterference tsk t].
Next we consider 4 cases.
Assume the processor is idle at time t.
In case when the processor is idle, one can show that
interference j t = 1, scheduled_at j t = 0. But since
interference doesn't come from a job of task tsk
task_interference tsk = 1. Which reduces to 1 ≤ 1.
Lemma interference_plus_sched_le_serv_of_task_plus_task_interference_idle:
interference j t + scheduled_at sched j t ≤
service_of_jobs_at (job_of_task tsk) (arrivals_between t1 (t1 + A + ε)) t +
task_interference_received_before tsk t2 t.
Proof.
move: (H_busy_interval) ⇒ [[/andP [BUS LT] _] _].
rewrite /cumul_task_interference /definitions.cumul_interference
/Sequential_Abstract_RTA.cumul_task_interference /task_interference_received_before
/task_scheduled_at /task_schedule.task_scheduled_at /service_of_jobs_at
/service_of_jobs.service_of_jobs_at/= scheduled_at_def.
rewrite !H_idle/=.
rewrite big1_eq addn0 add0n.
case INT: (interference j t); last by done.
simpl; rewrite lt0b.
apply/hasP; ∃ j; last by done.
by rewrite mem_filter; apply/andP; split;
[rewrite H_job_of_tsk | eapply arrived_between_implies_in_arrivals; eassumption].
Qed.
End Case1.
Section Case2.
interference j t + scheduled_at sched j t ≤
service_of_jobs_at (job_of_task tsk) (arrivals_between t1 (t1 + A + ε)) t +
task_interference_received_before tsk t2 t.
Proof.
move: (H_busy_interval) ⇒ [[/andP [BUS LT] _] _].
rewrite /cumul_task_interference /definitions.cumul_interference
/Sequential_Abstract_RTA.cumul_task_interference /task_interference_received_before
/task_scheduled_at /task_schedule.task_scheduled_at /service_of_jobs_at
/service_of_jobs.service_of_jobs_at/= scheduled_at_def.
rewrite !H_idle/=.
rewrite big1_eq addn0 add0n.
case INT: (interference j t); last by done.
simpl; rewrite lt0b.
apply/hasP; ∃ j; last by done.
by rewrite mem_filter; apply/andP; split;
[rewrite H_job_of_tsk | eapply arrived_between_implies_in_arrivals; eassumption].
Qed.
End Case1.
Section Case2.
Variable j' : Job.
Hypothesis H_sched : sched t = Some j'.
Hypothesis H_not_job_of_tsk : job_task j' != tsk.
Hypothesis H_sched : sched t = Some j'.
Hypothesis H_not_job_of_tsk : job_task j' != tsk.
If a job j' from another task is scheduled at time t,
then interference j t = 1, scheduled_at j t = 0. But
since interference doesn't come from a job of task tsk
task_interference tsk = 1. Which reduces to 1 ≤ 1.
Lemma interference_plus_sched_le_serv_of_task_plus_task_interference_task:
interference j t + scheduled_at sched j t ≤
service_of_jobs_at (job_of_task tsk) (arrivals_between t1 (t1 + A + ε)) t +
task_interference_received_before tsk t2 t.
Proof.
move: (H_busy_interval) ⇒ [[/andP [BUS LT] _] _].
rewrite /cumul_task_interference /definitions.cumul_interference
/Sequential_Abstract_RTA.cumul_task_interference /task_interference_received_before
/task_scheduled_at /task_schedule.task_scheduled_at /service_of_jobs_at
/service_of_jobs.service_of_jobs_at scheduled_at_def/=.
have ARRs: arrives_in arr_seq j'; first by apply H_jobs_come_from_arrival_sequence with t; rewrite scheduled_at_def; apply/eqP.
rewrite H_sched H_not_job_of_tsk; simpl.
rewrite (negbTE (option_inj_neq (neqprop_to_neqbool
(diseq (fun j ⇒ job_task j = tsk) _ _
(neqbool_to_neqprop H_not_job_of_tsk) H_job_of_tsk)))) addn0.
have ZERO: \sum_(i <- arrivals_between t1 (t1 + A + ε) | job_task i == tsk) (Some j' == Some i) = 0.
{ apply big1; move ⇒ j2 /eqP TSK2; apply/eqP; rewrite eqb0.
apply option_inj_neq, neqprop_to_neqbool, (diseq (fun j ⇒ job_task j = tsk) _ _
(neqbool_to_neqprop H_not_job_of_tsk) TSK2).
} rewrite ZERO ?addn0 add0n; simpl; clear ZERO.
case INT: (interference j t); last by done.
simpl; rewrite lt0b.
apply/hasP; ∃ j; last by done.
by rewrite mem_filter; apply/andP; split;
[rewrite H_job_of_tsk | eapply arrived_between_implies_in_arrivals; eassumption].
Qed.
End Case2.
Section Case3.
interference j t + scheduled_at sched j t ≤
service_of_jobs_at (job_of_task tsk) (arrivals_between t1 (t1 + A + ε)) t +
task_interference_received_before tsk t2 t.
Proof.
move: (H_busy_interval) ⇒ [[/andP [BUS LT] _] _].
rewrite /cumul_task_interference /definitions.cumul_interference
/Sequential_Abstract_RTA.cumul_task_interference /task_interference_received_before
/task_scheduled_at /task_schedule.task_scheduled_at /service_of_jobs_at
/service_of_jobs.service_of_jobs_at scheduled_at_def/=.
have ARRs: arrives_in arr_seq j'; first by apply H_jobs_come_from_arrival_sequence with t; rewrite scheduled_at_def; apply/eqP.
rewrite H_sched H_not_job_of_tsk; simpl.
rewrite (negbTE (option_inj_neq (neqprop_to_neqbool
(diseq (fun j ⇒ job_task j = tsk) _ _
(neqbool_to_neqprop H_not_job_of_tsk) H_job_of_tsk)))) addn0.
have ZERO: \sum_(i <- arrivals_between t1 (t1 + A + ε) | job_task i == tsk) (Some j' == Some i) = 0.
{ apply big1; move ⇒ j2 /eqP TSK2; apply/eqP; rewrite eqb0.
apply option_inj_neq, neqprop_to_neqbool, (diseq (fun j ⇒ job_task j = tsk) _ _
(neqbool_to_neqprop H_not_job_of_tsk) TSK2).
} rewrite ZERO ?addn0 add0n; simpl; clear ZERO.
case INT: (interference j t); last by done.
simpl; rewrite lt0b.
apply/hasP; ∃ j; last by done.
by rewrite mem_filter; apply/andP; split;
[rewrite H_job_of_tsk | eapply arrived_between_implies_in_arrivals; eassumption].
Qed.
End Case2.
Section Case3.
Variable j' : Job.
Hypothesis H_sched : sched t = Some j'.
Hypothesis H_not_job_of_tsk : job_task j' == tsk.
Hypothesis H_j_neq_j' : j != j'.
Hypothesis H_sched : sched t = Some j'.
Hypothesis H_not_job_of_tsk : job_task j' == tsk.
Hypothesis H_j_neq_j' : j != j'.
If a job j' (different from j) of task tsk is scheduled at time t, then
interference j t = 1, scheduled_at j t = 0. Moreover, since interference
comes from a job of the same task task_interference tsk = 0. However,
in this case service_of_jobs of tsk = 1. Which reduces to 1 ≤ 1.
Lemma interference_plus_sched_le_serv_of_task_plus_task_interference_job:
interference j t + scheduled_at sched j t ≤
service_of_jobs_at (job_of_task tsk) (arrivals_between t1 (t1 + A + ε)) t +
task_interference_received_before tsk t2 t.
Proof.
move: (H_busy_interval) ⇒ [[/andP [BUS LT] _] _].
rewrite /cumul_task_interference /definitions.cumul_interference
/Sequential_Abstract_RTA.cumul_task_interference /task_interference_received_before
/task_scheduled_at /task_schedule.task_scheduled_at /service_of_jobs_at
/service_of_jobs.service_of_jobs_at scheduled_at_def/=.
have ARRs: arrives_in arr_seq j'; first by apply H_jobs_come_from_arrival_sequence with t; rewrite scheduled_at_def; apply/eqP.
rewrite H_sched H_not_job_of_tsk addn0; simpl;
rewrite [Some j' == Some j](negbTE (option_inj_neq (neq_sym H_j_neq_j'))) addn0.
replace (interference j t) with true; last first.
{ have NEQT: t1 ≤ t < t2.
{ by move: H_t_in_interval ⇒ /andP [NEQ1 NEQ2]; apply/andP; split; last apply ltn_trans with (t1 + x). }
move: (H_work_conserving j t1 t2 t H_j_arrives H_job_of_tsk H_job_cost_positive H_busy_interval NEQT) ⇒ [Hn _].
apply/eqP;rewrite eq_sym eqb_id; apply/negPn/negP; intros CONTR; move: CONTR ⇒ /negP CONTR.
apply Hn in CONTR; rewrite scheduled_at_def in CONTR; simpl in CONTR.
by rewrite H_sched [Some j' == Some j](negbTE (option_inj_neq (neq_sym H_j_neq_j'))) in CONTR.
}
rewrite big_mkcond; apply/sum_seq_gt0P; ∃ j'; split; last first.
{ by move: H_not_job_of_tsk ⇒ /eqP TSK; rewrite /job_of_task TSK eq_refl eq_refl. }
{ intros. have ARR:= arrives_after_beginning_of_busy_interval j j' _ _ _ _ _ t1 t2 _ t.
feed_n 8 ARR; try (done || by move: H_t_in_interval ⇒ /andP [T1 T2]).
{ by move: H_not_job_of_tsk ⇒ /eqP TSK; rewrite TSK. }
{ move: H_sched ⇒ /eqP SCHEDt.
apply scheduled_implies_pending;
auto using ideal_proc_model_ensures_ideal_progress.
by rewrite scheduled_at_def. }
case_eq (job_arrival j' ≤ job_arrival j) ⇒ ARRNEQ.
{ move: ARR ⇒ /andP [РР _].
eapply arrived_between_implies_in_arrivals; eauto 2.
by apply/andP; split; last rewrite /A subnKC // addn1 ltnS. }
{ exfalso.
apply negbT in ARRNEQ; rewrite -ltnNge in ARRNEQ.
move: (H_sequential_tasks j j' t) ⇒ CONTR.
feed_n 3 CONTR; try done.
{ by rewrite /same_task eq_sym H_job_of_tsk. }
{ by move: H_sched ⇒ /eqP SCHEDt; rewrite scheduled_at_def. }
move: H_job_j_is_not_completed ⇒ /negP T; apply: T.
apply completion_monotonic with t; try done.
by apply ltnW; move: H_t_in_interval ⇒ /andP [_ NEQ]. }
}
Qed.
End Case3.
Section Case4.
interference j t + scheduled_at sched j t ≤
service_of_jobs_at (job_of_task tsk) (arrivals_between t1 (t1 + A + ε)) t +
task_interference_received_before tsk t2 t.
Proof.
move: (H_busy_interval) ⇒ [[/andP [BUS LT] _] _].
rewrite /cumul_task_interference /definitions.cumul_interference
/Sequential_Abstract_RTA.cumul_task_interference /task_interference_received_before
/task_scheduled_at /task_schedule.task_scheduled_at /service_of_jobs_at
/service_of_jobs.service_of_jobs_at scheduled_at_def/=.
have ARRs: arrives_in arr_seq j'; first by apply H_jobs_come_from_arrival_sequence with t; rewrite scheduled_at_def; apply/eqP.
rewrite H_sched H_not_job_of_tsk addn0; simpl;
rewrite [Some j' == Some j](negbTE (option_inj_neq (neq_sym H_j_neq_j'))) addn0.
replace (interference j t) with true; last first.
{ have NEQT: t1 ≤ t < t2.
{ by move: H_t_in_interval ⇒ /andP [NEQ1 NEQ2]; apply/andP; split; last apply ltn_trans with (t1 + x). }
move: (H_work_conserving j t1 t2 t H_j_arrives H_job_of_tsk H_job_cost_positive H_busy_interval NEQT) ⇒ [Hn _].
apply/eqP;rewrite eq_sym eqb_id; apply/negPn/negP; intros CONTR; move: CONTR ⇒ /negP CONTR.
apply Hn in CONTR; rewrite scheduled_at_def in CONTR; simpl in CONTR.
by rewrite H_sched [Some j' == Some j](negbTE (option_inj_neq (neq_sym H_j_neq_j'))) in CONTR.
}
rewrite big_mkcond; apply/sum_seq_gt0P; ∃ j'; split; last first.
{ by move: H_not_job_of_tsk ⇒ /eqP TSK; rewrite /job_of_task TSK eq_refl eq_refl. }
{ intros. have ARR:= arrives_after_beginning_of_busy_interval j j' _ _ _ _ _ t1 t2 _ t.
feed_n 8 ARR; try (done || by move: H_t_in_interval ⇒ /andP [T1 T2]).
{ by move: H_not_job_of_tsk ⇒ /eqP TSK; rewrite TSK. }
{ move: H_sched ⇒ /eqP SCHEDt.
apply scheduled_implies_pending;
auto using ideal_proc_model_ensures_ideal_progress.
by rewrite scheduled_at_def. }
case_eq (job_arrival j' ≤ job_arrival j) ⇒ ARRNEQ.
{ move: ARR ⇒ /andP [РР _].
eapply arrived_between_implies_in_arrivals; eauto 2.
by apply/andP; split; last rewrite /A subnKC // addn1 ltnS. }
{ exfalso.
apply negbT in ARRNEQ; rewrite -ltnNge in ARRNEQ.
move: (H_sequential_tasks j j' t) ⇒ CONTR.
feed_n 3 CONTR; try done.
{ by rewrite /same_task eq_sym H_job_of_tsk. }
{ by move: H_sched ⇒ /eqP SCHEDt; rewrite scheduled_at_def. }
move: H_job_j_is_not_completed ⇒ /negP T; apply: T.
apply completion_monotonic with t; try done.
by apply ltnW; move: H_t_in_interval ⇒ /andP [_ NEQ]. }
}
Qed.
End Case3.
Section Case4.
If job j is scheduled at time t, then interference = 0, scheduled_at = 1, but
note that service_of_jobs of tsk = 1, therefore inequality reduces to 1 ≤ 1.
Lemma interference_plus_sched_le_serv_of_task_plus_task_interference_j:
interference j t + scheduled_at sched j t ≤
service_of_jobs_at (job_of_task tsk) (arrivals_between t1 (t1 + A + ε)) t +
task_interference_received_before tsk t2 t.
Proof.
have j_is_in_arrivals_between: j \in arrivals_between t1 (t1 + A + ε).
{ eapply arrived_between_implies_in_arrivals; eauto 2.
move: (H_busy_interval) ⇒ [[/andP [GE _] [_ _]] _].
by apply/andP; split; last rewrite /A subnKC // addn1.
} intros.
rewrite /cumul_task_interference /definitions.cumul_interference
/Sequential_Abstract_RTA.cumul_task_interference /task_interference_received_before
/task_scheduled_at /task_schedule.task_scheduled_at /service_of_jobs_at
/service_of_jobs.service_of_jobs_at scheduled_at_def.
rewrite H_sched H_job_of_tsk neq_antirefl addn0; simpl.
move: (H_work_conserving j _ _ t H_j_arrives H_job_of_tsk H_job_cost_positive H_busy_interval) ⇒ WORK.
feed WORK.
{ move: H_t_in_interval ⇒ /andP [NEQ1 NEQ2].
by apply/andP; split; last apply ltn_trans with (t1 + x). }
move: WORK ⇒ [_ ZIJT].
feed ZIJT; first by rewrite scheduled_at_def H_sched; simpl.
move: ZIJT ⇒ /negP /eqP; rewrite eqb_negLR; simpl; move ⇒ /eqP ZIJT; rewrite ZIJT; simpl; rewrite add0n.
rewrite !eq_refl; simpl; rewrite big_mkcond //=; apply/sum_seq_gt0P.
by ∃ j; split; [apply j_is_in_arrivals_between | rewrite /job_of_task H_job_of_tsk H_sched !eq_refl].
Qed.
End Case4.
interference j t + scheduled_at sched j t ≤
service_of_jobs_at (job_of_task tsk) (arrivals_between t1 (t1 + A + ε)) t +
task_interference_received_before tsk t2 t.
Proof.
have j_is_in_arrivals_between: j \in arrivals_between t1 (t1 + A + ε).
{ eapply arrived_between_implies_in_arrivals; eauto 2.
move: (H_busy_interval) ⇒ [[/andP [GE _] [_ _]] _].
by apply/andP; split; last rewrite /A subnKC // addn1.
} intros.
rewrite /cumul_task_interference /definitions.cumul_interference
/Sequential_Abstract_RTA.cumul_task_interference /task_interference_received_before
/task_scheduled_at /task_schedule.task_scheduled_at /service_of_jobs_at
/service_of_jobs.service_of_jobs_at scheduled_at_def.
rewrite H_sched H_job_of_tsk neq_antirefl addn0; simpl.
move: (H_work_conserving j _ _ t H_j_arrives H_job_of_tsk H_job_cost_positive H_busy_interval) ⇒ WORK.
feed WORK.
{ move: H_t_in_interval ⇒ /andP [NEQ1 NEQ2].
by apply/andP; split; last apply ltn_trans with (t1 + x). }
move: WORK ⇒ [_ ZIJT].
feed ZIJT; first by rewrite scheduled_at_def H_sched; simpl.
move: ZIJT ⇒ /negP /eqP; rewrite eqb_negLR; simpl; move ⇒ /eqP ZIJT; rewrite ZIJT; simpl; rewrite add0n.
rewrite !eq_refl; simpl; rewrite big_mkcond //=; apply/sum_seq_gt0P.
by ∃ j; split; [apply j_is_in_arrivals_between | rewrite /job_of_task H_job_of_tsk H_sched !eq_refl].
Qed.
End Case4.
We use the above case analysis to prove that any time instant
t ∈ t1, t1 + x) the sum of [interference j t] and [scheduled_at j t]
is no larger than the sum of [the service received by jobs of task
tsk at time t] and [task_iterference tsk t].
Lemma interference_plus_sched_le_serv_of_task_plus_task_interference:
interference j t + scheduled_at sched j t
≤ service_of_jobs_at (job_of_task tsk) (arrivals_between t1 (t1 + A + ε)) t
+ task_interference_received_before tsk t2 t.
Proof.
move: (H_busy_interval) ⇒ [[/andP [BUS LT] _] _].
case SCHEDt: (sched t) ⇒ [j1 | ].
2: by apply interference_plus_sched_le_serv_of_task_plus_task_interference_idle.
have ARRs: arrives_in arr_seq j1;
first by apply H_jobs_come_from_arrival_sequence with t; rewrite scheduled_at_def; apply/eqP.
case_eq (job_task j1 == tsk) ⇒ TSK.
2: by eapply interference_plus_sched_le_serv_of_task_plus_task_interference_task; [eassumption| apply/negbT].
case EQ: (j == j1); [move: EQ ⇒ /eqP EQ; subst j1 | ].
1: by apply interference_plus_sched_le_serv_of_task_plus_task_interference_j.
eapply interference_plus_sched_le_serv_of_task_plus_task_interference_job;
auto; repeat split; eauto; apply/eqP; move: EQ ⇒ /eqP EQ; auto.
Qed.
End CaseAnalysis.
interference j t + scheduled_at sched j t
≤ service_of_jobs_at (job_of_task tsk) (arrivals_between t1 (t1 + A + ε)) t
+ task_interference_received_before tsk t2 t.
Proof.
move: (H_busy_interval) ⇒ [[/andP [BUS LT] _] _].
case SCHEDt: (sched t) ⇒ [j1 | ].
2: by apply interference_plus_sched_le_serv_of_task_plus_task_interference_idle.
have ARRs: arrives_in arr_seq j1;
first by apply H_jobs_come_from_arrival_sequence with t; rewrite scheduled_at_def; apply/eqP.
case_eq (job_task j1 == tsk) ⇒ TSK.
2: by eapply interference_plus_sched_le_serv_of_task_plus_task_interference_task; [eassumption| apply/negbT].
case EQ: (j == j1); [move: EQ ⇒ /eqP EQ; subst j1 | ].
1: by apply interference_plus_sched_le_serv_of_task_plus_task_interference_j.
eapply interference_plus_sched_le_serv_of_task_plus_task_interference_job;
auto; repeat split; eauto; apply/eqP; move: EQ ⇒ /eqP EQ; auto.
Qed.
End CaseAnalysis.
Next we prove cumulative version of the lemma above.
Lemma cumul_interference_plus_sched_le_serv_of_task_plus_cumul_task_interference:
cumul_interference j t1 (t1 + x)
≤ (task_service_of_jobs_in (arrivals_between t1 (t1 + A + ε)) t1 (t1 + x)
- service_during sched j t1 (t1 + x)) + cumul_task_interference t2 t1 (t1 + x).
Proof.
have j_is_in_arrivals_between: j \in arrivals_between t1 (t1 + A + ε).
{ eapply arrived_between_implies_in_arrivals; eauto 2.
move: (H_busy_interval) ⇒ [[/andP [GE _] [_ _]] _].
by apply/andP; split; last rewrite /A subnKC // addn1.
}
rewrite /cumul_interference /cumul_interference /task_service_of_jobs_in
/service_of_jobs.task_service_of_jobs_in
/service_of_jobs exchange_big //=.
rewrite -(leq_add2r (\sum_(t1 ≤ t < (t1 + x)) service_at sched j t)).
rewrite [X in _ ≤ X]addnC addnA subnKC; last first.
{ rewrite exchange_big //= (big_rem j) //=; auto using j_is_in_arrivals_between.
by rewrite /job_of_task H_job_of_tsk eq_refl leq_addr. }
rewrite -big_split -big_split //=.
rewrite big_nat_cond [X in _ ≤ X]big_nat_cond leq_sum //; move ⇒ t /andP [NEQ _].
rewrite -scheduled_at_def.
by apply interference_plus_sched_le_serv_of_task_plus_task_interference.
Qed.
cumul_interference j t1 (t1 + x)
≤ (task_service_of_jobs_in (arrivals_between t1 (t1 + A + ε)) t1 (t1 + x)
- service_during sched j t1 (t1 + x)) + cumul_task_interference t2 t1 (t1 + x).
Proof.
have j_is_in_arrivals_between: j \in arrivals_between t1 (t1 + A + ε).
{ eapply arrived_between_implies_in_arrivals; eauto 2.
move: (H_busy_interval) ⇒ [[/andP [GE _] [_ _]] _].
by apply/andP; split; last rewrite /A subnKC // addn1.
}
rewrite /cumul_interference /cumul_interference /task_service_of_jobs_in
/service_of_jobs.task_service_of_jobs_in
/service_of_jobs exchange_big //=.
rewrite -(leq_add2r (\sum_(t1 ≤ t < (t1 + x)) service_at sched j t)).
rewrite [X in _ ≤ X]addnC addnA subnKC; last first.
{ rewrite exchange_big //= (big_rem j) //=; auto using j_is_in_arrivals_between.
by rewrite /job_of_task H_job_of_tsk eq_refl leq_addr. }
rewrite -big_split -big_split //=.
rewrite big_nat_cond [X in _ ≤ X]big_nat_cond leq_sum //; move ⇒ t /andP [NEQ _].
rewrite -scheduled_at_def.
by apply interference_plus_sched_le_serv_of_task_plus_task_interference.
Qed.
On the other hand, the service terms in the inequality
above can be upper-bound by the workload terms.
Lemma serv_of_task_le_workload_of_task_plus:
task_service_of_jobs_in (arrivals_between t1 (t1 + A + ε)) t1 (t1 + x)
- service_during sched j t1 (t1 + x) + cumul_task_interference t2 t1 (t1 + x)
≤ (task_workload_between t1 (t1 + A + ε) - job_cost j)
+ cumul_task_interference t2 t1 (t1 + x).
Proof.
have j_is_in_arrivals_between: j \in arrivals_between t1 (t1 + A + ε).
{ eapply arrived_between_implies_in_arrivals; eauto 2.
move: (H_busy_interval) ⇒ [[/andP [GE _] [_ _]] _].
by apply/andP; split; last rewrite /A subnKC // addn1.
}
rewrite leq_add2r.
rewrite /task_workload /task_service_of_jobs_in
/service_of_jobs.task_service_of_jobs_in/service_of_jobs /workload_of_jobs /job_of_task.
rewrite (big_rem j) ?[X in _ ≤ X - _](big_rem j) //=; auto using j_is_in_arrivals_between.
rewrite /job_of_task H_job_of_tsk eq_refl.
rewrite addnC -addnBA; last by done.
rewrite [X in _ ≤ X - _]addnC -addnBA; last by done.
rewrite !subnn !addn0.
by apply service_of_jobs_le_workload; auto using ideal_proc_model_provides_unit_service.
Qed.
task_service_of_jobs_in (arrivals_between t1 (t1 + A + ε)) t1 (t1 + x)
- service_during sched j t1 (t1 + x) + cumul_task_interference t2 t1 (t1 + x)
≤ (task_workload_between t1 (t1 + A + ε) - job_cost j)
+ cumul_task_interference t2 t1 (t1 + x).
Proof.
have j_is_in_arrivals_between: j \in arrivals_between t1 (t1 + A + ε).
{ eapply arrived_between_implies_in_arrivals; eauto 2.
move: (H_busy_interval) ⇒ [[/andP [GE _] [_ _]] _].
by apply/andP; split; last rewrite /A subnKC // addn1.
}
rewrite leq_add2r.
rewrite /task_workload /task_service_of_jobs_in
/service_of_jobs.task_service_of_jobs_in/service_of_jobs /workload_of_jobs /job_of_task.
rewrite (big_rem j) ?[X in _ ≤ X - _](big_rem j) //=; auto using j_is_in_arrivals_between.
rewrite /job_of_task H_job_of_tsk eq_refl.
rewrite addnC -addnBA; last by done.
rewrite [X in _ ≤ X - _]addnC -addnBA; last by done.
rewrite !subnn !addn0.
by apply service_of_jobs_le_workload; auto using ideal_proc_model_provides_unit_service.
Qed.
Finally, we show that the cumulative interference of job j in the interval t1, t1 + x)
is bounded by the sum of the task workload in the interval [t1, t1 + A + ε) and
the cumulative interference of [j]'s task in the interval [t1, t1 + x).
Lemma cumulative_job_interference_le_task_interference_bound:
cumul_interference j t1 (t1 + x)
≤ (task_workload_between t1 (t1 + A + ε) - job_cost j)
+ cumul_task_interference t2 t1 (t1 + x).
Proof.
apply leq_trans with
(task_service_of_jobs_in (arrivals_between t1 (t1 + A + ε)) t1 (t1 + x)
- service_during sched j t1 (t1 + x)
+ cumul_task_interference t2 t1 (t1 + x));
[ apply cumul_interference_plus_sched_le_serv_of_task_plus_cumul_task_interference
| apply serv_of_task_le_workload_of_task_plus].
Qed.
End TaskInterferenceBoundsInterference.
cumul_interference j t1 (t1 + x)
≤ (task_workload_between t1 (t1 + A + ε) - job_cost j)
+ cumul_task_interference t2 t1 (t1 + x).
Proof.
apply leq_trans with
(task_service_of_jobs_in (arrivals_between t1 (t1 + A + ε)) t1 (t1 + x)
- service_during sched j t1 (t1 + x)
+ cumul_task_interference t2 t1 (t1 + x));
[ apply cumul_interference_plus_sched_le_serv_of_task_plus_cumul_task_interference
| apply serv_of_task_le_workload_of_task_plus].
Qed.
End TaskInterferenceBoundsInterference.
In order to obtain a more convenient bound of the cumulative interference, we need to
abandon the actual workload in favor of a bound which depends on task parameters only.
So, we show that actual workload of the task excluding workload of any job j is no
greater than bound of workload excluding the cost of job j's task.
Lemma task_rbf_excl_tsk_bounds_task_workload_excl_j:
task_workload_between t1 (t1 + A + ε) - job_cost j ≤ task_rbf (A + ε) - task_cost tsk.
Proof.
move: H_j_arrives H_job_of_tsk H_busy_interval ⇒ ARR TSK [[/andP [JAGET1 JALTT2] _] _].
apply leq_trans with
(task_cost tsk × number_of_task_arrivals arr_seq tsk t1 (t1 + A + ε) - task_cost tsk); last first.
{ rewrite leq_sub2r // leq_mul2l; apply/orP; right.
rewrite -addnA -{2}[(A+1)](addKn t1).
by apply H_is_arrival_curve; auto using leq_addr. }
have Fact6: j \in arrivals_between (t1 + A) (t1 + A + ε).
{ apply arrived_between_implies_in_arrivals; try done.
apply/andP; split; rewrite /A subnKC //.
by rewrite addn1 ltnSn //. }
have Fact4: j \in arrivals_at arr_seq (t1 + A).
{ by move: ARR ⇒ [t ARR]; rewrite subnKC //; feed (H_arrival_times_are_consistent j t); try (subst t). }
have Fact1: 1 ≤ number_of_task_arrivals arr_seq tsk (t1 + A) (t1 + A + ε).
{ rewrite /number_of_task_arrivals /task_arrivals_between /arrival_sequence.arrivals_between.
by rewrite -count_filter_fun -has_count; apply/hasP; ∃ j; last rewrite TSK.
}
rewrite (@num_arrivals_of_task_cat _ _ _ _ _ (t1 + A)); last by apply/andP; split; rewrite leq_addr //.
rewrite mulnDr.
have Step1: task_workload_between t1 (t1 + A + ε)
= task_workload_between t1 (t1 + A) + task_workload_between (t1 + A) (t1 + A + ε).
{ by apply workload_of_jobs_cat; apply/andP; split; rewrite leq_addr. } rewrite Step1; clear Step1.
rewrite -!addnBA; first last.
{ by rewrite /task_workload_between /workload.task_workload_between /task_workload
/workload_of_jobs /job_of_task (big_rem j) //= TSK eq_refl leq_addr. }
{ apply leq_trans with (task_cost tsk); first by done.
by rewrite -{1}[task_cost tsk]muln1 leq_mul2l; apply/orP; right. }
rewrite leq_add; [by done | by eapply task_workload_le_num_of_arrivals_times_cost; eauto | ].
rewrite /task_workload_between /workload.task_workload_between /task_workload /workload_of_jobs
/arrival_sequence.arrivals_between /number_of_task_arrivals /task_arrivals_between
/arrival_sequence.arrivals_between /job_of_task.
rewrite {1}addn1 big_nat1 addn1 big_nat1.
rewrite (big_rem j) //= TSK !eq_refl; simpl.
rewrite addnC -addnBA // subnn addn0.
rewrite (filter_size_rem _ j); [ | by done | by rewrite TSK].
rewrite mulnDr mulnC muln1 -addnBA // subnn addn0 mulnC.
apply sum_majorant_constant.
move ⇒ j' ARR' /eqP TSK2.
by rewrite -TSK2; apply H_valid_job_cost; ∃ (t1 + A); apply rem_in in ARR'.
Qed.
task_workload_between t1 (t1 + A + ε) - job_cost j ≤ task_rbf (A + ε) - task_cost tsk.
Proof.
move: H_j_arrives H_job_of_tsk H_busy_interval ⇒ ARR TSK [[/andP [JAGET1 JALTT2] _] _].
apply leq_trans with
(task_cost tsk × number_of_task_arrivals arr_seq tsk t1 (t1 + A + ε) - task_cost tsk); last first.
{ rewrite leq_sub2r // leq_mul2l; apply/orP; right.
rewrite -addnA -{2}[(A+1)](addKn t1).
by apply H_is_arrival_curve; auto using leq_addr. }
have Fact6: j \in arrivals_between (t1 + A) (t1 + A + ε).
{ apply arrived_between_implies_in_arrivals; try done.
apply/andP; split; rewrite /A subnKC //.
by rewrite addn1 ltnSn //. }
have Fact4: j \in arrivals_at arr_seq (t1 + A).
{ by move: ARR ⇒ [t ARR]; rewrite subnKC //; feed (H_arrival_times_are_consistent j t); try (subst t). }
have Fact1: 1 ≤ number_of_task_arrivals arr_seq tsk (t1 + A) (t1 + A + ε).
{ rewrite /number_of_task_arrivals /task_arrivals_between /arrival_sequence.arrivals_between.
by rewrite -count_filter_fun -has_count; apply/hasP; ∃ j; last rewrite TSK.
}
rewrite (@num_arrivals_of_task_cat _ _ _ _ _ (t1 + A)); last by apply/andP; split; rewrite leq_addr //.
rewrite mulnDr.
have Step1: task_workload_between t1 (t1 + A + ε)
= task_workload_between t1 (t1 + A) + task_workload_between (t1 + A) (t1 + A + ε).
{ by apply workload_of_jobs_cat; apply/andP; split; rewrite leq_addr. } rewrite Step1; clear Step1.
rewrite -!addnBA; first last.
{ by rewrite /task_workload_between /workload.task_workload_between /task_workload
/workload_of_jobs /job_of_task (big_rem j) //= TSK eq_refl leq_addr. }
{ apply leq_trans with (task_cost tsk); first by done.
by rewrite -{1}[task_cost tsk]muln1 leq_mul2l; apply/orP; right. }
rewrite leq_add; [by done | by eapply task_workload_le_num_of_arrivals_times_cost; eauto | ].
rewrite /task_workload_between /workload.task_workload_between /task_workload /workload_of_jobs
/arrival_sequence.arrivals_between /number_of_task_arrivals /task_arrivals_between
/arrival_sequence.arrivals_between /job_of_task.
rewrite {1}addn1 big_nat1 addn1 big_nat1.
rewrite (big_rem j) //= TSK !eq_refl; simpl.
rewrite addnC -addnBA // subnn addn0.
rewrite (filter_size_rem _ j); [ | by done | by rewrite TSK].
rewrite mulnDr mulnC muln1 -addnBA // subnn addn0 mulnC.
apply sum_majorant_constant.
move ⇒ j' ARR' /eqP TSK2.
by rewrite -TSK2; apply H_valid_job_cost; ∃ (t1 + A); apply rem_in in ARR'.
Qed.
Finally, we use the lemmas above to obtain the bound on
interference in terms of task_rbf and task_interference.
Lemma cumulative_job_interference_bound:
cumul_interference j t1 (t1 + x)
≤ (task_rbf (A + ε) - task_cost tsk) + cumul_task_interference t2 t1 (t1 + x).
Proof.
set (y := t1 + x) in ×.
have IN: j \in arrivals_between t1 (t1 + A + ε).
{ eapply arrived_between_implies_in_arrivals; eauto 2.
move: (H_busy_interval) ⇒ [[/andP [GE _] _] _].
by apply/andP; split; last rewrite /A subnKC // addn1.
}
apply leq_trans with (task_workload_between t1 (t1+A+ε) - job_cost j + cumul_task_interference t2 t1 y).
- by apply cumulative_job_interference_le_task_interference_bound.
- rewrite leq_add2r.
eapply task_rbf_excl_tsk_bounds_task_workload_excl_j; eauto 2.
Qed.
End BoundOfCumulativeJobInterference.
cumul_interference j t1 (t1 + x)
≤ (task_rbf (A + ε) - task_cost tsk) + cumul_task_interference t2 t1 (t1 + x).
Proof.
set (y := t1 + x) in ×.
have IN: j \in arrivals_between t1 (t1 + A + ε).
{ eapply arrived_between_implies_in_arrivals; eauto 2.
move: (H_busy_interval) ⇒ [[/andP [GE _] _] _].
by apply/andP; split; last rewrite /A subnKC // addn1.
}
apply leq_trans with (task_workload_between t1 (t1+A+ε) - job_cost j + cumul_task_interference t2 t1 y).
- by apply cumulative_job_interference_le_task_interference_bound.
- rewrite leq_add2r.
eapply task_rbf_excl_tsk_bounds_task_workload_excl_j; eauto 2.
Qed.
End BoundOfCumulativeJobInterference.
In this section, we prove that H_R_is_maximum_seq implies H_R_is_maximum.
Variable j : Job.
Hypothesis H_j_arrives : arrives_in arr_seq j.
Hypothesis H_job_of_tsk : job_task j = tsk.
Hypothesis H_j_arrives : arrives_in arr_seq j.
Hypothesis H_job_of_tsk : job_task j = tsk.
For simplicity, let's define a local name for the search space.
We prove that H_R_is_maximum holds.
Lemma max_in_seq_hypothesis_implies_max_in_nonseq_hypothesis:
∀ (A : duration),
is_in_search_space A →
∃ (F : duration),
A + F = task_run_to_completion_threshold tsk +
(task_rbf (A + ε) - task_cost tsk + task_interference_bound_function tsk A (A + F)) ∧
F + (task_cost tsk - task_run_to_completion_threshold tsk) ≤ R.
Proof.
move: H_valid_run_to_completion_threshold ⇒ [PRT1 PRT2].
intros A INSP.
clear H_sequential_tasks H_interference_and_workload_consistent_with_sequential_tasks.
move: (H_R_is_maximum_seq _ INSP) ⇒ [F [FIX LE]].
∃ F; split; last by done.
rewrite {1}FIX.
apply/eqP.
rewrite addnA eqn_add2r.
rewrite addnBA; last first.
{ apply leq_trans with (task_rbf 1).
eapply task_rbf_1_ge_task_cost; eauto 2.
eapply task_rbf_monotone; eauto 2.
by rewrite addn1.
}
by rewrite subnBA; auto; rewrite addnC.
Qed.
End MaxInSeqHypothesisImpMaxInNonseqHypothesis.
∀ (A : duration),
is_in_search_space A →
∃ (F : duration),
A + F = task_run_to_completion_threshold tsk +
(task_rbf (A + ε) - task_cost tsk + task_interference_bound_function tsk A (A + F)) ∧
F + (task_cost tsk - task_run_to_completion_threshold tsk) ≤ R.
Proof.
move: H_valid_run_to_completion_threshold ⇒ [PRT1 PRT2].
intros A INSP.
clear H_sequential_tasks H_interference_and_workload_consistent_with_sequential_tasks.
move: (H_R_is_maximum_seq _ INSP) ⇒ [F [FIX LE]].
∃ F; split; last by done.
rewrite {1}FIX.
apply/eqP.
rewrite addnA eqn_add2r.
rewrite addnBA; last first.
{ apply leq_trans with (task_rbf 1).
eapply task_rbf_1_ge_task_cost; eauto 2.
eapply task_rbf_monotone; eauto 2.
by rewrite addn1.
}
by rewrite subnBA; auto; rewrite addnC.
Qed.
End MaxInSeqHypothesisImpMaxInNonseqHypothesis.
Finally, we apply the uniprocessor_response_time_bound theorem, and using the
lemmas above, we prove that all the requirements are satisfied. So, R is a response
time bound.
Theorem uniprocessor_response_time_bound_seq:
response_time_bounded_by tsk R.
Proof.
intros j ARR TSK.
eapply uniprocessor_response_time_bound with
(interference_bound_function :=
fun tsk A R ⇒ task_rbf (A + ε) - task_cost tsk + task_interference_bound_function tsk A R)
(interfering_workload0 := interfering_workload); eauto 2.
{ clear ARR TSK H_R_is_maximum_seq R j.
intros t1 t2 R j BUSY NEQ ARR TSK COMPL.
move: (posnP (@job_cost _ H3 j)) ⇒ [ZERO|POS].
{ exfalso.
move: COMPL ⇒ /negP COMPL; apply: COMPL.
by rewrite /service.completed_by /completed_by ZERO.
}
set (A := job_arrival j - t1) in ×.
apply leq_trans with
(task_rbf (A + ε) - task_cost tsk + cumul_task_interference t2 t1 (t1 + R)).
- by eapply cumulative_job_interference_bound; eauto 2.
- by rewrite leq_add2l; apply H_task_interference_is_bounded.
}
{ by eapply max_in_seq_hypothesis_implies_max_in_nonseq_hypothesis; eauto. }
Qed.
End ResponseTimeBound.
End Sequential_Abstract_RTA.
response_time_bounded_by tsk R.
Proof.
intros j ARR TSK.
eapply uniprocessor_response_time_bound with
(interference_bound_function :=
fun tsk A R ⇒ task_rbf (A + ε) - task_cost tsk + task_interference_bound_function tsk A R)
(interfering_workload0 := interfering_workload); eauto 2.
{ clear ARR TSK H_R_is_maximum_seq R j.
intros t1 t2 R j BUSY NEQ ARR TSK COMPL.
move: (posnP (@job_cost _ H3 j)) ⇒ [ZERO|POS].
{ exfalso.
move: COMPL ⇒ /negP COMPL; apply: COMPL.
by rewrite /service.completed_by /completed_by ZERO.
}
set (A := job_arrival j - t1) in ×.
apply leq_trans with
(task_rbf (A + ε) - task_cost tsk + cumul_task_interference t2 t1 (t1 + R)).
- by eapply cumulative_job_interference_bound; eauto 2.
- by rewrite leq_add2l; apply H_task_interference_is_bounded.
}
{ by eapply max_in_seq_hypothesis_implies_max_in_nonseq_hypothesis; eauto. }
Qed.
End ResponseTimeBound.
End Sequential_Abstract_RTA.