# Library prosa.results.rta.ideal.fp.bounded_nps

Require Export prosa.analysis.facts.busy_interval.pi_bound.
Require Export prosa.results.rta.ideal.fp.bounded_pi.
Require Export prosa.model.schedule.work_conserving.
Require Export prosa.analysis.definitions.busy_interval.classical.
Require Export prosa.analysis.definitions.blocking_bound.fp.

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

In this section we instantiate the Abstract RTA for FP-schedulers with Bounded Priority Inversion to FP-schedulers for ideal uniprocessor model of real-time tasks with arbitrary arrival models and bounded non-preemptive segments.
Recall that Abstract RTA for FP-schedulers with Bounded Priority Inversion does not specify the cause of priority inversion. In this section, we prove that the priority inversion caused by execution of non-preemptive segments is bounded. Thus the Abstract RTA for FP-schedulers is applicable to this instantiation.
Section RTAforFPwithBoundedNonpreemptiveSegmentsWithArrivalCurves.
Consider any type of tasks ...
Context {Task : TaskType}.
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}.
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_task_priorities FP.
Hypothesis H_priority_is_transitive : transitive_task_priorities FP.
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.
Next, consider any ideal uniprocessor schedule of this arrival sequence, ...
Variable sched : schedule (ideal.processor_state Job).
... 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.
... and assume that the schedule is valid.
Hypothesis H_sched_valid : valid_schedule sched arr_seq.
In addition, we assume the existence of a function mapping jobs to their preemption points ...
Context `{JobPreemptable Job}.
... 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.
Next, we assume that the schedule is a work-conserving schedule...
Hypothesis H_work_conserving : work_conserving arr_seq sched.
... and the schedule respects the scheduling policy.
Hypothesis H_respects_policy : respects_FP_policy_at_preemption_point arr_seq sched FP.
Assume we have sequential tasks, i.e., jobs from the same task execute in the order of their arrival.
Hypothesis H_sequential_tasks : sequential_tasks arr_seq sched.
Consider an arbitrary task set ts, ...
Variable ts : list Task.
... assume that all jobs come from the task set, ...
Hypothesis H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts.
... and the cost of a job cannot be larger than the task cost.
Hypothesis H_valid_job_cost :
arrivals_have_valid_job_costs arr_seq.
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.
Let tsk be any task in ts that is to be analyzed.
Variable tsk : Task.
Hypothesis H_tsk_in_ts : tsk \in ts.
Consider a valid preemption model ...
Hypothesis H_valid_preemption_model :
valid_preemption_model arr_seq sched.
... 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.
Hypothesis H_valid_run_to_completion_threshold :
valid_task_run_to_completion_threshold arr_seq tsk.
Let's define some local names for clarity.
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.

## 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 (i.e., the blocking term).
Lemma priority_inversion_is_bounded_by_blocking :
∀ j t1 t2,
arrives_in arr_seq j →
job_of_task tsk j →
busy_interval_prefix arr_seq sched j t1 t2 →
max_lp_nonpreemptive_segment arr_seq j t1 ≤ blocking_bound ts tsk.
Proof.
move⇒ j t1 t2 ARR TSK BUSY; move: TSK ⇒ /eqP TSK.
rewrite /blocking_bound /max_lp_nonpreemptive_segment.
apply: leq_trans; first exact: max_np_job_segment_bounded_by_max_np_task_segment.
apply: (@leq_trans (\max_(j_lp <- arrivals_between arr_seq 0 t1
| (~~ hep_task (job_task j_lp) tsk) && (0 < job_cost j_lp))
(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 /bigmax_leq_seqP ⇒ j' JINB /andP[NOTHEP POS].
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 the blocking_bound.
Lemma priority_inversion_is_bounded :
priority_inversion_is_bounded_by
arr_seq sched tsk (constant (blocking_bound ts tsk)).
Proof.
have PIB: priority_inversion_is_bounded_by arr_seq sched tsk (fun⇒ blocking_bound ts tsk); last by done.
apply: priority_inversion_is_bounded ⇒ //.
by exact: priority_inversion_is_bounded_by_blocking.
Qed.

## 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.
Section ResponseTimeBound.
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 ts tsk + total_hep_rbf L.
To reduce the time complexity of the analysis, recall the notion of search space.
Let is_in_search_space := is_in_search_space tsk L.
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 ts tsk
+ (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 non-preemptive 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 non-preemptive 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.