Library rt.restructuring.analysis.edf.rta.nonpr_reg.concrete_models.nonpreemptive
Require Export rt.restructuring.analysis.edf.rta.nonpr_reg.response_time_bound.
Require Export rt.restructuring.analysis.basic_facts.preemption.task.nonpreemptive.
Require Export rt.restructuring.analysis.basic_facts.preemption.rtc_threshold.nonpreemptive.
Require Import rt.restructuring.model.priority.edf.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq path fintype bigop.
Require Export rt.restructuring.analysis.basic_facts.preemption.task.nonpreemptive.
Require Export rt.restructuring.analysis.basic_facts.preemption.rtc_threshold.nonpreemptive.
Require Import rt.restructuring.model.priority.edf.
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.
Throughout this file, we assume the fully non-preemptive task model.
RTA for Fully Non-Preemptive FP Model
In this module we prove the RTA theorem for the fully non-preemptive EDF model.
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 `{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.
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.
Consider an arbitrary task set ts, ...
... assume that all jobs come from this 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.
Next, consider any ideal non-preemptive uniprocessor schedule of this arrival sequence ...
Variable sched : schedule (ideal.processor_state Job).
Hypothesis H_nonpreemptive_sched : is_nonpreemptive_schedule sched.
Hypothesis H_jobs_come_from_arrival_sequence:
jobs_come_from_arrival_sequence sched arr_seq.
Hypothesis H_nonpreemptive_sched : is_nonpreemptive_schedule sched.
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.
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 nonpreemptive
segments).
Let's define some local names for clarity.
Let response_time_bounded_by :=
task_response_time_bound arr_seq sched.
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.
task_response_time_bound arr_seq sched.
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.
We introduce the abbreviation "rbf" for the task request bound function,
which is defined as task_cost(T) × max_arrivals(T,Δ) for a task T.
Next, we introduce task_rbf as an abbreviation
for the task request bound function of task tsk.
Using the sum of individual request bound functions, we define the request bound
function of all tasks (total request bound function).
We also define a bound for the priority inversion caused by jobs with lower priority.
Let blocking_bound :=
\max_(tsk_o <- ts | (tsk_o != tsk) && (D tsk_o > D tsk))
(task_cost tsk_o - ε).
\max_(tsk_o <- ts | (tsk_o != tsk) && (D tsk_o > D tsk))
(task_cost tsk_o - ε).
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 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: nat.
Hypothesis H_R_is_maximum:
∀ A,
is_in_search_space A →
∃ F,
A + F = blocking_bound + (task_rbf (A + ε) - (task_cost tsk - ε))
+ bound_on_total_hep_workload A (A + F) ∧
F + (task_cost tsk - ε) ≤ R.
Hypothesis H_R_is_maximum:
∀ A,
is_in_search_space A →
∃ F,
A + F = blocking_bound + (task_rbf (A + ε) - (task_cost tsk - ε))
+ bound_on_total_hep_workload A (A + F) ∧
F + (task_cost tsk - ε) ≤ R.
Now, we can leverage the results for the abstract model with bounded nonpreemptive segments
to establish a response-time bound for the more concrete model of fully nonpreemptive scheduling.
Theorem uniprocessor_response_time_bound_fully_nonpreemptive_edf:
response_time_bounded_by tsk R.
Proof.
case: (posnP (task_cost tsk)) ⇒ [ZERO|POS].
{ intros j ARR TSK.
have ZEROj: job_cost j = 0.
{ move: (H_job_cost_le_task_cost j ARR) ⇒ NEQ.
rewrite /job_cost_le_task_cost TSK ZERO in NEQ.
by apply/eqP; rewrite -leqn0.
}
by rewrite /job_response_time_bound /completed_by ZEROj.
}
eapply uniprocessor_response_time_bound_edf_with_bounded_nonpreemptive_segments with (L0 := L).
all: eauto 2 with basic_facts.
Qed.
End RTAforFullyNonPreemptiveEDFModelwithArrivalCurves.
response_time_bounded_by tsk R.
Proof.
case: (posnP (task_cost tsk)) ⇒ [ZERO|POS].
{ intros j ARR TSK.
have ZEROj: job_cost j = 0.
{ move: (H_job_cost_le_task_cost j ARR) ⇒ NEQ.
rewrite /job_cost_le_task_cost TSK ZERO in NEQ.
by apply/eqP; rewrite -leqn0.
}
by rewrite /job_response_time_bound /completed_by ZEROj.
}
eapply uniprocessor_response_time_bound_edf_with_bounded_nonpreemptive_segments with (L0 := L).
all: eauto 2 with basic_facts.
Qed.
End RTAforFullyNonPreemptiveEDFModelwithArrivalCurves.