# Library prosa.results.rta.rs.edf.fully_preemptive

Require Import prosa.analysis.facts.readiness.basic.
Require Export prosa.analysis.facts.preemption.task.preemptive.
Require Export prosa.analysis.facts.preemption.rtc_threshold.preemptive.
Require Export prosa.analysis.abstract.restricted_supply.task_intra_interference_bound.
Require Export prosa.analysis.abstract.restricted_supply.bounded_bi.jlfp.
Require Export prosa.analysis.abstract.restricted_supply.search_space.edf.
Require Export prosa.analysis.facts.model.task_cost.
Require Export prosa.analysis.facts.priority.edf.
Require Export prosa.analysis.facts.blocking_bound.edf.
Require Export prosa.analysis.facts.workload.edf_athep_bound.

# RTA for Fully Preemptive EDF Scheduling on Restricted-Supply Uniprocessors

In the following, we derive a response-time analysis for EDF schedulers, assuming a workload of sporadic real-time tasks characterized by arbitrary arrival curves executing upon a uniprocessor with arbitrary supply restrictions. To this end, we instantiate the sequential variant of abstract Restricted-Supply Response-Time Analysis (aRSA) as provided in the prosa.analysis.abstract.restricted_supply module.
Section RTAforFullyPreemptiveEDFModelwithArrivalCurves.

## Defining the System Model

Before any formal claims can be stated, an initial setup is needed to define the system model under consideration. To this end, we next introduce and define the following notions using Prosa's standard definitions and behavioral semantics:

- tasks, jobs, and their parameters,
- processor model,
- the sequence of job arrivals,
- worst-case execution time (WCET) and the absence of self-suspensions,
- the set of tasks under analysis,
- the task under analysis,
- an arbitrary schedule of the task set, and finally,
- a supply-bound function.

### Tasks and Jobs

Consider any type of tasks, each characterized by a WCET task_cost, relative deadline task_deadline, and an arrival curve max_arrivals, ...
Context {Task : TaskType}.
Context `{TaskCost Task}.
Context `{TaskDeadline Task}.
Context `{MaxArrivals Task}.
... and any type of jobs associated with these tasks, where each job has a task job_task, a cost job_cost, and an arrival time job_arrival.
Context {Job : JobType}.
Context `{JobTask Job Task}.
Context `{JobCost Job}.
Context `{JobArrival Job}.
Furthermore, assume that jobs and tasks are fully preemptive.
#[local] Existing Instance fully_preemptive_job_model.
#[local] Existing Instance fully_preemptive_task_model.
#[local] Existing Instance fully_preemptive_rtc_threshold.

### Processor Model

Consider any kind of fully-supply-consuming, unit-supply processor state model.
Context `{PState : ProcessorState Job}.
Hypothesis H_uniprocessor_proc_model : uniprocessor_model PState.
Hypothesis H_unit_supply_proc_model : unit_supply_proc_model PState.
Hypothesis H_consumed_supply_proc_model : fully_consuming_proc_model PState.

### The Job Arrival Sequence

Consider any arrival sequence arr_seq with consistent, non-duplicate arrivals.
Variable arr_seq : arrival_sequence Job.
Hypothesis H_valid_arrival_sequence : valid_arrival_sequence arr_seq.

### Absence of Self-Suspensions and WCET Compliance

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 further require that a job's cost cannot exceed its task's stated WCET.
Hypothesis H_valid_job_cost : arrivals_have_valid_job_costs arr_seq.

### The Task Set

We consider an arbitrary task set ts ...
Variable ts : seq Task.
... and assume that all jobs stem from tasks in this task set.
Hypothesis H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts.
We assume that max_arrivals is a family of valid arrival curves that constrains the arrival sequence arr_seq, i.e., for any task tsk in ts, max_arrival tsk is (1) an arrival bound of tsk, and ...
Hypothesis H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts.
... (2) a monotonic function that equals 0 for the empty interval delta = 0.
Hypothesis H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals.

### The Task Under Analysis

Let tsk be any task in ts that is to be analyzed.
Variable tsk : Task.
Hypothesis H_tsk_in_ts : tsk \in ts.

### The Schedule

Consider any arbitrary, work-conserving, valid uniprocessor schedule of the given arrival sequence arr_seq (and hence the given task set ts).
Variable sched : schedule PState.
Hypothesis H_valid_schedule : valid_schedule sched arr_seq.
Hypothesis H_work_conserving : work_conserving arr_seq sched.
Assume that the schedule respects the EDF policy.
Hypothesis H_respects_policy :
respects_JLFP_policy_at_preemption_point arr_seq sched (EDF Job).

### Supply-Bound Function

Assume the minimum amount of supply that any job of task tsk receives is defined by a monotone unit-supply-bound function SBF.
Context {SBF : SupplyBoundFunction}.
Hypothesis H_SBF_monotone : sbf_is_monotone SBF.
Hypothesis H_unit_SBF : unit_supply_bound_function SBF.
We assume that SBF properly characterizes all busy intervals (w.r.t. task tsk) in sched. That is, (1) SBF 0 = 0 and (2) for any duration Δ, at least SBF Δ supply is available in any busy-interval prefix of length Δ.
Hypothesis H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF.

## Maximum Length of a Busy Interval

In order to apply aRSA, we require a bound on the maximum busy-window length. To this end, let L be any positive solution of the busy-interval "recurrence" (i.e., inequality) SBF L ≥ total_request_bound_function ts L, as defined below.
As the lemma busy_intervals_are_bounded_rs_jlfp shows, under EDF scheduling, this condition is sufficient to guarantee that the maximum busy-window length is at most L, i.e., the length of any busy interval is bounded by L.
Definition busy_window_recurrence_solution (L : duration) :=
L > 0
∧ SBF L ≥ total_request_bound_function ts L.

## Response-Time Bound

Having established all necessary preliminaries, it is finally time to state the claimed response-time bound R.
A value R is a response-time bound if, for any given offset A in the search space, the response-time bound recurrence has a solution F not exceeding A + R.
Definition rta_recurrence_solution L R :=
∀ (A : duration),
is_in_search_space ts tsk L A →
∃ (F : duration),
SBF F ≥ task_request_bound_function tsk (A + ε)
+ bound_on_athep_workload ts tsk A F
∧ A + R ≥ F.
Finally, using the sequential variant of abstract restricted-supply analysis, we establish that any such R is a sound response-time bound for the concrete model of fully-preemptive EDF scheduling with arbitrary supply restrictions.
Theorem uniprocessor_response_time_bound_fully_preemptive_edf :
∀ (L : duration),
busy_window_recurrence_solution L →
∀ (R : duration),
rta_recurrence_solution L R →
task_response_time_bound arr_seq sched tsk R.
End RTAforFullyPreemptiveEDFModelwithArrivalCurves.