# Library prosa.results.rta.prm.fp.limited_preemptive

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.fp.
Require Export prosa.analysis.abstract.restricted_supply.search_space.fp.
Require Export prosa.analysis.facts.model.sbf.periodic.

# RTA for FP Scheduling with Fixed Preemption Points on Uniprocessors under the Periodic Resource Model

In the following, we derive a response-time analysis for FP schedulers, assuming a workload of sporadic real-time tasks with fixed preemption points, 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 RTAforLimitedPreemptiveFPModelwithArrivalCurves.

## 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 list of preemption points task_preemption_points, ...
Context {Task : TaskType} `{TaskCost Task} `{MaxArrivals Task} `{TaskPreemptionPoints 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}
`{JobPreemptionPoints Job}.
We assume that jobs are limited-preemptive.
#[local] Existing Instance limited_preemptive_job_model.

### 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 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.

### 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 a model with fixed preemption points. I.e., each task is divided into a number of non-preemptive segments by inserting statically predefined preemption points.
Hypothesis H_valid_model_with_fixed_preemption_points :
valid_fixed_preemption_points_model arr_seq ts.

### 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.
Context {RM : JobReady Job PState}.
Hypothesis H_basic_readiness : basic_readiness RM.

### 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 work-conserving, valid uniprocessor schedule with limited preemptions 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_schedule_with_limited_preemptions :
schedule_respects_preemption_model arr_seq 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.

### 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 ≥ 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
∧ prm_sbf Π γ 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),
prm_sbf Π γ F ≥ blocking_bound ts tsk
+ (task_request_bound_function tsk (A + ε) - (task_last_nonpr_segment tsk - ε))
+ total_ohep_request_bound_function_FP ts tsk F
∧ prm_sbf Π γ (A + R) ≥ prm_sbf Π γ F + (task_last_nonpr_segment 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 FP scheduling with limited preemptions on a unit-speed uniprocessor under the periodic resource model.
Theorem uniprocessor_response_time_bound_limited_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 RTAforLimitedPreemptiveFPModelwithArrivalCurves.