# Library prosa.results.rta.ideal.edf.fully_nonpreemptive

Require Import prosa.model.readiness.basic.
Require Export prosa.results.rta.ideal.edf.bounded_nps.
Require Export prosa.analysis.facts.preemption.task.nonpreemptive.
Require Export prosa.analysis.facts.preemption.rtc_threshold.nonpreemptive.
Require Export prosa.analysis.facts.readiness.basic.
Require Export prosa.model.task.preemption.fully_nonpreemptive.

# RTA for Fully Non-Preemptive EDF

In this module we prove the RTA theorem for the fully non-preemptive EDF model.

## Setup and Assumptions

Section RTAforFullyNonPreemptiveEDFModelwithArrivalCurves.
Consider any type of tasks ...
Context {Task : TaskType}.
Context `{TaskCost Task}.
Context `{TaskDeadline Task}.
... and any type of jobs associated with these tasks.
Context {Job : JobType}.
Context `{JobTask Job Task}.
Context `{JobArrival Job}.
Context `{JobCost Job}.
We assume the classic (i.e., Liu & Layland) model of readiness without jitter or self-suspensions, wherein pending jobs are always ready.
#[local] Existing Instance basic_ready_instance.
We assume that jobs and tasks are fully nonpreemptive.
#[local] Existing Instance fully_nonpreemptive_job_model.
#[local] Existing Instance fully_nonpreemptive_task_model.
#[local] Existing Instance fully_nonpreemptive_rtc_threshold.
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.
Consider an arbitrary task set ts, ...
Variable ts : list Task.
... assume that all jobs come from this 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.
Next, consider any valid ideal non-preemptive uniprocessor schedule of this arrival sequence ...
Variable sched : schedule (ideal.processor_state Job).
Hypothesis H_sched_valid : valid_schedule sched arr_seq.
Hypothesis H_nonpreemptive_sched : nonpreemptive_schedule 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.
Context {JLFP : JLFP_policy Job}.
Hypothesis H_policy_is_EDF : policy_is_EDF JLFP.
Hypothesis H_respects_policy :
respects_JLFP_policy_at_preemption_point arr_seq sched JLFP.

## Total Workload and Length of Busy Interval

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.
Let rbf := task_request_bound_function.
Next, we introduce task_rbf as an abbreviation for the task request bound function of task tsk.
Let task_rbf := rbf tsk.
Using the sum of individual request bound functions, we define the request bound function of all tasks (total request bound function).
Let total_rbf := total_request_bound_function ts.
We also define a bound for the priority inversion caused by jobs with lower priority.
Let blocking_bound A :=
\max_(tsk_o <- ts | (blocking_relevant tsk_o)
&& (task_deadline tsk_o > task_deadline tsk + A))
(task_cost tsk_o - ε).
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 = total_rbf L.

## Response-Time Bound

To reduce the time complexity of the analysis, recall the notion of search space.
Let is_in_search_space := bounded_nps.is_in_search_space ts tsk L.
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 A + (task_rbf (A + ε) - (task_cost tsk - ε))
+ bound_on_athep_workload ts tsk A (A + F)
∧ R ≥ F + (task_cost tsk - ε).
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 :
task_response_time_bound arr_seq sched tsk R.
End RTAforFullyNonPreemptiveEDFModelwithArrivalCurves.