# Library prosa.results.fixed_priority.rta.limited_preemptive

From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq path fintype bigop.

Require Export prosa.results.fixed_priority.rta.bounded_nps.

Require Export prosa.analysis.facts.preemption.rtc_threshold.limited.

Require Export prosa.analysis.facts.readiness.sequential.

Require Export prosa.model.task.preemption.limited_preemptive.

Require Export prosa.results.fixed_priority.rta.bounded_nps.

Require Export prosa.analysis.facts.preemption.rtc_threshold.limited.

Require Export prosa.analysis.facts.readiness.sequential.

Require Export prosa.model.task.preemption.limited_preemptive.

# RTA for FP-schedulers with Fixed Preemption Points

In this module we prove the RTA theorem for FP-schedulers with fixed preemption points.## Setup and Assumptions

We assume ideal uni-processor schedules.

#[local] Existing Instance ideal.processor_state.

Consider any type of tasks ...

... and any type of jobs associated with these tasks.

Context {Job : JobType}.

Context `{JobTask Job Task}.

Context `{JobArrival Job}.

Context `{JobCost Job}.

Context `{JobTask Job Task}.

Context `{JobArrival Job}.

Context `{JobCost Job}.

We assume that jobs are limited-preemptive.

#[local] Existing Instance limited_preemptive_job_model.

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.

Hypothesis H_valid_arrival_sequence : valid_arrival_sequence arr_seq.

Consider an arbitrary task set ts, ...

... assume that all jobs come from the task set, ...

... and the cost of a job cannot be larger than the task cost.

First, we assume we have the model with fixed preemption points.
I.e., each task is divided into a number of non-preemptive segments
by inserting statically predefined preemption points.

Context `{JobPreemptionPoints Job}

`{TaskPreemptionPoints Task}.

Hypothesis H_valid_model_with_fixed_preemption_points:

valid_fixed_preemption_points_model arr_seq ts.

`{TaskPreemptionPoints Task}.

Hypothesis H_valid_model_with_fixed_preemption_points:

valid_fixed_preemption_points_model arr_seq ts.

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.

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.

Recall that we assume sequential readiness.

Next, consider any valid ideal uni-processor schedule with limited preemptions of this arrival sequence ...

Variable sched : schedule (ideal.processor_state Job).

Hypothesis H_sched_valid : valid_schedule sched arr_seq.

Hypothesis H_schedule_respects_preemption_model:

schedule_respects_preemption_model arr_seq sched.

Hypothesis H_sched_valid : valid_schedule sched arr_seq.

Hypothesis H_schedule_respects_preemption_model:

schedule_respects_preemption_model arr_seq sched.

Consider an FP policy that indicates a higher-or-equal priority relation,
and assume that the relation is reflexive and transitive.

Context {FP : FP_policy Task}.

Hypothesis H_priority_is_reflexive : reflexive_priorities.

Hypothesis H_priority_is_transitive : transitive_priorities.

Hypothesis H_priority_is_reflexive : reflexive_priorities.

Hypothesis H_priority_is_transitive : transitive_priorities.

Next, we assume that the schedule is a work-conserving schedule...

... and the schedule respects the scheduling policy.

## Total Workload and Length of Busy Interval

Using the sum of individual request bound functions, we define
the request bound function of all tasks with higher priority
...

... and the request bound function of all tasks with higher
priority other than task tsk.

Next, we define a bound for the priority inversion caused by tasks of lower priority.

Let blocking_bound :=

\max_(tsk_other <- ts | ~~ hep_task tsk_other tsk)

(task_max_nonpreemptive_segment tsk_other - ε).

\max_(tsk_other <- ts | ~~ hep_task tsk_other tsk)

(task_max_nonpreemptive_segment tsk_other - ε).

Let L be any positive fixed point of the busy interval recurrence, determined by
the sum of blocking and higher-or-equal-priority workload.

Variable L : duration.

Hypothesis H_L_positive : L > 0.

Hypothesis H_fixed_point : L = blocking_bound + total_hep_rbf L.

Hypothesis H_L_positive : L > 0.

Hypothesis H_fixed_point : L = blocking_bound + total_hep_rbf L.

## Response-Time Bound

Next, consider any value R, and assume that for any given
arrival A from 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 : duration),

is_in_search_space A →

∃ (F : duration),

A + F ≥ blocking_bound

+ (task_rbf (A + ε) - (task_last_nonpr_segment tsk - ε))

+ total_ohep_rbf (A + F) ∧

R ≥ F + (task_last_nonpr_segment tsk - ε).

Hypothesis H_R_is_maximum:

∀ (A : duration),

is_in_search_space A →

∃ (F : duration),

A + F ≥ blocking_bound

+ (task_rbf (A + ε) - (task_last_nonpr_segment tsk - ε))

+ total_ohep_rbf (A + F) ∧

R ≥ F + (task_last_nonpr_segment tsk - ε).

Now, we can reuse the results for the abstract model with
bounded non-preemptive segments to establish a response-time
bound for the more concrete model of fixed preemption points.