# Library prosa.results.rta.arm.fp.fully_nonpreemptive

Require Import prosa.analysis.facts.readiness.basic.
Require Export prosa.model.composite.valid_task_arrival_sequence.
Require Export prosa.analysis.facts.preemption.task.nonpreemptive.
Require Export prosa.analysis.facts.preemption.rtc_threshold.nonpreemptive.
Require Export prosa.analysis.abstract.restricted_supply.task_intra_interference_bound.
Require Export prosa.analysis.abstract.restricted_supply.bounded_bi.fp.
Require Export prosa.analysis.abstract.restricted_supply.search_space.fp.
Require Export prosa.analysis.facts.model.sbf.average.

# RTA for Fully Non-Preemptive FP Scheduling on Uniprocessors under the Average Resource Model

In the following, we derive a response-time analysis for non-preemptive FP scheduling, assuming a workload of sporadic real-time tasks, characterized by arbitrary arrival curves, executing upon a uniprocessor under the average resource model (inspired by the paper "Periodic Resource Model for Compositional Real-Time Guarantees" by Shin & Lee, RTSS 2003). To this end, we instantiate the sequential variant of abstract Restricted-Supply Analysis (aRSA) as provided in the prosa.analysis.abstract.restricted_supply module.
Section RTAforFullyNonPreemptiveFPModelwithArrivalCurves.

## 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,
- the task set and the task under analysis,
- the processor model,
- the sequence of job arrivals,
- the absence of self-suspensions,
- an arbitrary schedule of the task set, and finally,
- the resource-supply model.

### Tasks and Jobs

Consider tasks characterized by a WCET task_cost and an arrival curve max_arrivals, ...
Context {Task : TaskType} `{TaskCost Task} `{MaxArrivals Task}.
... and their associated jobs, where each job has a corresponding task job_task, an execution time job_cost, and an arrival time job_arrival.
Context {Job : JobType} `{JobTask Job Task} `{JobCost Job} `{JobArrival Job}.
Furthermore, assume that jobs and tasks are fully non-preemptive.
#[local] Existing Instance fully_nonpreemptive_job_model.
#[local] Existing Instance fully_nonpreemptive_task_model.
#[local] Existing Instance fully_nonpreemptive_rtc_threshold.

### The Task Set and the Task Under Analysis

Consider an arbitrary task set ts, and ...
Variable ts : seq Task.
... let tsk be any task in ts that is to be analyzed.
Variable tsk : Task.
Hypothesis H_tsk_in_ts : tsk \in ts.

### 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

Allow for any possible arrival sequence arr_seq consistent with the parameters of the task set ts. That is, arr_seq is a valid arrival sequence in which each job's cost is upper-bounded by its task's WCET, every job stems from a task in ts, and the number of arrivals respects each task's max_arrivals bound.
Variable arr_seq : arrival_sequence Job.
Hypothesis H_valid_task_arrival_sequence : valid_task_arrival_sequence ts arr_seq.

### Absence of Self-Suspensions

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.

### The Schedule

Consider an arbitrary fixed-priority policy FP that indicates a higher-or-equal priority relation among the tasks in ts, 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 a non-preemptive, work-conserving, valid uniprocessor schedule that corresponds to 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.
Hypothesis H_nonpreemptive_sched : nonpreemptive_schedule sched.
We assume that the schedule respects the given FP scheduling policy.
Hypothesis H_respects_policy :
respects_FP_policy_at_preemption_point arr_seq sched FP.
Furthermore, we require that the schedule ensures that jobs of the same task are executed in the order of their arrival.
Hypothesis H_sequential_tasks : sequential_tasks arr_seq sched.

### Average Resource Model

Assume that the processor supply follows the *average resource model*. Under this model, for any interval `[t1, t2)`, and given a resource period Π, a resource allocation time Θ, and a supply delay ν, the processor provides at least (t2 - t1 - ν) ⋅ Θ / Π units of supply. Intuitively, this means that on _average, the processor delivers Θ units of output every Π units of time, while the distribution of supply is not ideal and is subject to fluctuations bounded by ν. Furthermore, let arm_sbf Π Θ ν denote the SBF, which, as proven in prosa.analysis.facts.model.sbf.average, is a valid SBF.
Variables Π Θ ν : duration.
Hypothesis H_average_resource_model : average_resource_model Π Θ ν sched.

## 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) arm_sbf Π Θ ν L ≥ blocking_bound ts tsk + total_hep_request_bound_function_FP ts tsk L, as defined below.
As the lemma busy_intervals_are_bounded_rs_fp shows, under FP 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
∧ arm_sbf Π Θ ν L ≥ blocking_bound ts tsk + total_hep_request_bound_function_FP ts tsk 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 for task tsk if, for any given offset A in the search space (w.r.t. the busy-window bound L), the response-time bound "recurrence" (i.e., inequality) has a solution F not exceeding A + R.
Definition rta_recurrence_solution L R :=
∀ (A : duration),
is_in_search_space tsk L A →
∃ (F : duration),
arm_sbf Π Θ ν F ≥ blocking_bound ts tsk
+ (task_request_bound_function tsk (A + ε) - (task_cost tsk - ε))
+ total_ohep_request_bound_function_FP ts tsk F
∧ arm_sbf Π Θ ν (A + R) ≥ arm_sbf Π Θ ν F + (task_cost tsk - ε)
∧ A + R ≥ F.
Finally, using the sequential variant of abstract restricted-supply analysis, we establish that, given a bound on the maximum busy-window length L, any such R is indeed a sound response-time bound for task tsk under fully-non-preemptive fixed-priority scheduling on a unit-speed uniprocessor under the average resource model.
Theorem uniprocessor_response_time_bound_fully_non_preemptive_fp :
∀ (L : duration),
busy_window_recurrence_solution L →
∀ (R : duration),
rta_recurrence_solution L R →
task_response_time_bound arr_seq sched tsk R.
End RTAforFullyNonPreemptiveFPModelwithArrivalCurves.