Newer
Older
Add LoadPath "../../" as rt.
Require Import rt.model.jitter.task rt.model.jitter.job rt.model.jitter.schedule
rt.model.jitter.priority rt.model.jitter.workload.
Require Import ssreflect ssrbool eqtype ssrnat seq fintype bigop.
Import ScheduleOfSporadicTaskWithJitter Priority Workload.
(* We import some of the basic definitions, but we need to re-define almost everything
since the definition of backlogged (and thus the definition of interference)
changes with jitter. *)
Require Import rt.model.basic.interference.
Export Interference.
Section InterferenceDefs.
Context {sporadic_task: eqType}.
Context {Job: eqType}.
Variable job_task: Job -> sporadic_task.
Variable job_jitter: Job -> time.
(* Assume any job arrival sequence...*)
Context {arr_seq: arrival_sequence Job}.
Context {num_cpus: nat}.
Variable sched: schedule num_cpus arr_seq.
(* Consider any job j that incurs interference. *)
Variable j: JobIn arr_seq.
(* Recall the definition of backlogged (pending and not scheduled). *)
Let job_is_backlogged (t: time) :=
backlogged job_cost job_jitter sched j t.
Section TotalInterference.
(* First, we define the total interference incurred by job j during [t1, t2)
as the cumulative time in which j is backlogged in this interval. *)
Definition total_interference (t1 t2: time) :=
\sum_(t1 <= t < t2) job_is_backlogged t.
End TotalInterference.
Section JobInterference.
(* Let job_other be a job that interferes with j. *)
Variable job_other: JobIn arr_seq.
(* We define the total interference caused by job_other during [t1, t2) as the
cumulative service received by job_other while j is backlogged. *)
Definition job_interference (t1 t2: time) :=
\sum_(t1 <= t < t2)
\sum_(cpu < num_cpus)
(job_is_backlogged t &&
scheduled_on sched job_other cpu t).
End JobInterference.
Section TaskInterference.
(* In order to define task interference, consider any interfering task tsk_other. *)
Variable tsk_other: sporadic_task.
(* We define the total interference caused by tsk during [t1, t2) as
the cumulative service received by tsk while j is backlogged. *)
Definition task_interference (t1 t2: time) :=
\sum_(t1 <= t < t2)
\sum_(cpu < num_cpus)
(job_is_backlogged t &&
task_scheduled_on job_task sched tsk_other cpu t).
End TaskInterference.
Section TaskInterferenceJobList.
Variable tsk_other: sporadic_task.
Definition task_interference_joblist (t1 t2: time) :=
\sum_(j <- jobs_scheduled_between sched t1 t2 | job_task j == tsk_other)
job_interference j t1 t2.
End TaskInterferenceJobList.
Section BasicLemmas.
(* Interference cannot be larger than the considered time window. *)
Lemma total_interference_le_delta :
forall t1 t2,
total_interference t1 t2 <= t2 - t1.
Proof.
unfold total_interference; intros t1 t2.
apply leq_trans with (n := \sum_(t1 <= t < t2) 1);
first by apply leq_sum; ins; apply leq_b1.
by rewrite big_const_nat iter_addn mul1n addn0 leqnn.
Qed.
Lemma job_interference_le_service :
forall j_other t1 t2,
job_interference j_other t1 t2 <= service_during sched j_other t1 t2.
Proof.
intros j_other t1 t2; unfold job_interference, service_during.
apply leq_sum; intros t _.
unfold service_at; rewrite [\sum_(_ < _ | scheduled_on _ _ _ _)_]big_mkcond.
apply leq_sum; intros cpu _.
destruct (job_is_backlogged t); [rewrite andTb | by rewrite andFb].
by destruct (scheduled_on sched j_other cpu t).
Qed.
Lemma task_interference_le_workload :
forall tsk t1 t2,
task_interference tsk t1 t2 <= workload job_task sched tsk t1 t2.
Proof.
unfold task_interference, workload; intros tsk t1 t2.
apply leq_sum; intros t _.
apply leq_sum; intros cpu _.
destruct (job_is_backlogged t); [rewrite andTb | by rewrite andFb].
unfold task_scheduled_on, service_of_task.
by destruct (sched cpu t).
Qed.
End BasicLemmas.
Section InterferenceSequentialJobs.
(* If jobs are sequential, ... *)
Hypothesis H_sequential_jobs: sequential_jobs sched.
(* ... then the interference incurred by a job in an interval
of length delta is at most delta. *)
Lemma job_interference_le_delta :
forall j_other t1 delta,
job_interference j_other t1 (t1 + delta) <= delta.
rename H_sequential_jobs into SEQ.
unfold job_interference, sequential_jobs in *.
intros j_other t1 delta.
apply leq_trans with (n := \sum_(t1 <= t < t1 + delta) 1);
last by rewrite big_const_nat iter_addn mul1n addn0 addKn leqnn.
apply leq_sum; intros t _.
destruct ([exists cpu, scheduled_on sched j_other cpu t]) eqn:EX.
move: EX => /existsP [cpu SCHED].
rewrite (bigD1 cpu) // /=.
rewrite big_mkcond (eq_bigr (fun x => 0)) /=;
first by simpl_sum_const; rewrite leq_b1.
intros cpu' _; des_if_goal; last by done.
destruct (scheduled_on sched j_other cpu' t) eqn:SCHED'; last by rewrite andbF.
move: SCHED SCHED' => /eqP SCHED /eqP SCHED'.
by specialize (SEQ j_other t cpu cpu' SCHED SCHED'); rewrite SEQ in Heq.
apply negbT in EX; rewrite negb_exists in EX.
move: EX => /forallP EX.
rewrite (eq_bigr (fun x => 0)); first by simpl_sum_const.
by intros cpu _; specialize (EX cpu); apply negbTE in EX; rewrite EX andbF.
End InterferenceSequentialJobs.
(* The sequential per-job interference bounds the actual interference. *)
Section BoundUsingPerJobInterference.
Lemma interference_le_interference_joblist :
forall tsk t1 t2,
task_interference tsk t1 t2 <= task_interference_joblist tsk t1 t2.
Proof.
intros tsk t1 t2.
unfold task_interference, task_interference_joblist, job_interference, job_is_backlogged.
rewrite [\sum_(_ <- _ sched _ _ | _) _]exchange_big /=.
rewrite big_nat_cond [\sum_(_ <= _ < _ | true) _]big_nat_cond.
apply leq_sum; move => t /andP [LEt _].
rewrite exchange_big /=.
apply leq_sum; intros cpu _.
destruct (backlogged job_cost job_jitter sched j t) eqn:BACK;
last by rewrite andFb (eq_bigr (fun x => 0));
first by rewrite big_const_seq iter_addn mul0n addn0.
rewrite andTb.
destruct (task_scheduled_on job_task sched tsk cpu t) eqn:SCHED; last by done.
unfold scheduled_on, task_scheduled_on in *.
destruct (sched cpu t) as [j' |] eqn:SOME; last by done.
rewrite big_mkcond /= (bigD1_seq j') /=; last by apply undup_uniq.
by rewrite SCHED eq_refl.
}
{
unfold jobs_scheduled_between.
rewrite mem_undup; apply mem_bigcat_nat with (j := t);
first by done.
apply mem_bigcat_ord with (j := cpu); first by apply ltn_ord.
by unfold make_sequence; rewrite SOME mem_seq1 eq_refl.
}
Qed.
End BoundUsingPerJobInterference.
End InterferenceDefs.
End Interference.