# Library prosa.results.rta.prm.fifo.bounded_nps

Require Import prosa.analysis.facts.readiness.basic.
Require Export prosa.model.composite.valid_task_arrival_sequence.
Require Export prosa.analysis.facts.preemption.rtc_threshold.limited.
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.fifo.
Require Export prosa.analysis.abstract.restricted_supply.search_space.fifo_fixpoint.
Require Export prosa.analysis.facts.priority.fifo.
Require Export prosa.analysis.facts.priority.fifo_ahep_bound.
Require Export prosa.analysis.facts.model.sbf.periodic.

# RTA for FIFO Scheduling on Uniprocessors under the Periodic Resource Model

In the following, we derive a response-time analysis for FIFO schedulers, assuming a workload of sporadic real-time tasks, characterized by arbitrary arrival curves, executing upon a uniprocessor under the periodic resource model (see "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 RTAforFIFOModelwithArrivalCurves.

## 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, an arrival curve max_arrivals, and a run-to-completion threshold task_rtct, ...
Context {Task : TaskType} `{TaskCost Task} `{MaxArrivals Task}
`{TaskRunToCompletionThreshold Task}.
... and their associated jobs, where each job has a corresponding task job_task, an execution time job_cost, an arrival time job_arrival, and a list of preemption points job_preemptable.
Context {Job : JobType} `{JobTask Job Task} `{JobCost Job} `{JobArrival Job}
`{JobPreemptable Job}.

### 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.
We assume that tsk is described by a valid task run-to-completion threshold. That is, there exists a task parameter task_rtct such that task_rtct tsk is

- (1) no larger than tsk's WCET, and
- (2) for any job of task tsk, the job's run-to-completion threshold job_rtct is bounded by task_rtct tsk.

Hypothesis H_valid_run_to_completion_threshold :
valid_task_run_to_completion_threshold arr_seq tsk.

### 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 a 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.
We assume that the schedule complies with some valid preemption model. Under FIFO scheduling, the specific choice of preemption model does not matter, since the resulting schedule behaves as if it were non-preemptive: jobs are executed in arrival order without preemption.
Hypothesis H_valid_preemption_model : valid_preemption_model arr_seq sched.
We assume that the schedule respects the FIFO scheduling policy.
Hypothesis H_respects_policy :
respects_JLFP_policy_at_preemption_point arr_seq sched (FIFO Job).

### Periodic Resource Model

Assume that the processor supply follows the *periodic resource model* as defined in "Periodic Resource Model for Compositional Real-Time Guarantees" by Shin & Lee (RTSS 2003). Under this model, the supply is distributed periodically with a resource period Π and a resource allocation time γ: in every interval `[Π⋅k, Π⋅(k+1))`, the processor provides at least γ units of supply. Furthermore, let prm_sbf Π γ denote the corresponding SBF defined in the paper, which, as proven in the same paper and verified in prosa.analysis.facts.model.sbf.periodic, is a valid SBF.
Variable Π γ : duration.
Hypothesis H_periodic_resource_model : periodic_resource_model Π γ 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) prm_sbf Π γ L ≥ total_request_bound_function ts L, as defined below.
As the lemma busy_intervals_are_bounded_rs_jlfp shows, under FIFO 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
∧ prm_sbf Π γ 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 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 ts L A →
∃ (F : duration),
prm_sbf Π γ F ≥ total_request_bound_function ts (A + ε)
∧ A + R ≥ F.
Finally, using the abstract restricted-supply analysis, we establish that any R that satisfies the stated equation is a sound response-time bound for FIFO scheduling on a unit-speed uniprocessor under the periodic resource model.
Theorem uniprocessor_response_time_bound_fifo :
∀ (L : duration),
busy_window_recurrence_solution L →
∀ (R : duration),
rta_recurrence_solution L R →
task_response_time_bound arr_seq sched tsk R.
Proof.
move⇒ L [BW_POS BW_SOL] R SOL js ARRs TSKs.
have [ZERO|POS] := posnP (job_cost js); first by rewrite /job_response_time_bound /completed_by ZERO.
have VBSBF : valid_busy_sbf arr_seq sched tsk (prm_sbf Π γ).
{ by apply: valid_pred_sbf_switch_predicate; last apply prm_sbf_valid. }
have US : unit_supply_bound_function (prm_sbf Π γ) by exact: prm_sbf_unit.
have POStsk: 0 < task_cost tsk
by move: TSKs ⇒ /eqP <-; apply: leq_trans; [apply POS | apply H_valid_task_arrival_sequence].
eapply uniprocessor_response_time_bound_restricted_supply
with (L := L) (intra_IBF := fun A Δ ⇒ (\sum_(tsko <- ts) task_request_bound_function tsko (A + ε)) - task_cost tsk) ⇒ //.
- exact: instantiated_i_and_w_are_coherent_with_schedule.
- eapply busy_intervals_are_bounded_rs_jlfp with (blocking_bound := fun _ ⇒ 0)=> //.
+ exact: instantiated_i_and_w_are_coherent_with_schedule.
+ by apply: FIFO_implies_no_service_inversion.
- apply: valid_pred_sbf_switch_predicate; last first.
+ by apply prm_sbf_valid.
+ by move ⇒ ? ? ? ? [? ?]; split ⇒ //.
- move ⇒ t1 t2 Δ j ARR TSK BUSY LT NCOMPL A OFF.
move: (OFF _ _ BUSY) ⇒ EQA; subst A.
have [ZERO|POSj] := posnP (job_cost j).
{ by exfalso; rewrite /completed_by ZERO in NCOMPL. }
eapply leq_trans; first by eapply cumulative_intra_interference_split ⇒ //.
rewrite -[leqRHS]add0n leq_add //.
+ rewrite (leqRW (service_inversion_widen _ _ _ _ _ _ _ _ _)).
× apply: FIFO_implies_no_service_inversion ⇒ //.
by apply instantiated_busy_interval_equivalent_busy_interval ⇒ //.
× by done.
× by done.
+ unshelve apply: bound_on_hep_workload; (try apply H_fixed_point).
all: try apply H_L_positive.
all: try done.
apply instantiated_busy_interval_equivalent_busy_interval ⇒ //.
- apply: soln_abstract_response_time_recurrence ⇒ //.
+ by exact: prm_sbf_monotone.
+ by apply: non_pathological_max_arrivals =>//; apply H_valid_task_arrival_sequence.
Qed.
End RTAforFIFOModelwithArrivalCurves.