# Library prosa.results.rta.ideal.edf.bounded_pi

Require Export prosa.analysis.facts.priority.edf.
Require Export prosa.analysis.definitions.schedulability.
Require Import prosa.model.readiness.basic.
Require Import prosa.model.priority.edf.
Require Import prosa.model.task.absolute_deadline.
Require Import prosa.analysis.abstract.ideal.cumulative_bounds.
Require Import prosa.analysis.facts.busy_interval.carry_in.
Require Import prosa.analysis.facts.readiness.basic.
Require Export prosa.analysis.abstract.ideal.abstract_seq_rta.
Require Export prosa.analysis.facts.model.task_cost.
Require Export prosa.analysis.facts.workload.edf_athep_bound.

# Abstract RTA for EDF-schedulers with Bounded Priority Inversion

In this module we instantiate the Abstract Response-Time analysis (aRTA) to EDF-schedulers for ideal uni-processor model of real-time tasks with arbitrary arrival models.
Given EDF priority policy and an ideal uni-processor scheduler model, we can explicitly specify interference, interfering_workload, and interference_bound_function. In this setting, we can define natural notions of service, workload, busy interval, etc. The important feature of this instantiation is that we can induce the meaningful notion of priority inversion. However, we do not specify the exact cause of priority inversion (as there may be different reasons for this, like execution of a non-preemptive segment or blocking due to resource locking). We only assume that a priority inversion is bounded.
Section AbstractRTAforEDFwithArrivalCurves.
Consider any type of tasks ...
Context {Task : TaskType}.
Context `{TaskCost Task}.
Context `{TaskDeadline Task}.
Context `{TaskRunToCompletionThreshold Task}.
Context `{TaskMaxNonpreemptiveSegment Task}.
... and any type of jobs associated with these tasks.
Context {Job : JobType}.
Context `{JobTask Job Task}.
Context {Arrival : JobArrival Job}.
Context {Cost : JobCost Job}.
Context `{JobPreemptable Job}.
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.
For clarity, let's denote the relative deadline of a task as D.
Let D tsk := task_deadline tsk.
Consider the EDF policy that indicates a higher-or-equal priority relation. Note that we do not relate the EDF policy with the scheduler. However, we define functions for Interference and Interfering Workload that actively use the concept of priorities.
Let EDF := EDF Job.
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.
Next, consider any valid ideal uni-processor schedule of this arrival sequence that follows the scheduling policy.
Variable sched : schedule (ideal.processor_state Job).
Hypothesis H_sched_valid : valid_schedule sched arr_seq.
Hypothesis H_respects_policy : respects_JLFP_policy_at_preemption_point arr_seq sched EDF.
To use the theorem uniprocessor_response_time_bound_seq from the Abstract RTA module, we need to specify functions of interference, interfering workload, and task_IBF. Next, we define interference and interfering workload; we return to task_IBF later.

## Instantiation of Interference

We say that job j incurs interference at time t iff it cannot execute due to a higher-or-equal-priority job being scheduled, or if it incurs a priority inversion.
#[local] Instance ideal_jlfp_interference : Interference Job :=
ideal_jlfp_interference arr_seq sched.

## Instantiation of Interfering Workload

The interfering workload, in turn, is defined as the sum of the priority inversion function and interfering workload of jobs with higher or equal priority.
#[local] Instance ideal_jlfp_interfering_workload : InterferingWorkload Job :=
ideal_jlfp_interfering_workload arr_seq sched.
Note that we differentiate between abstract and classical notions of work-conserving schedule.
Let work_conserving_ab := definitions.work_conserving arr_seq sched.
Let work_conserving_cl := work_conserving.work_conserving arr_seq sched.
We assume that the schedule is a work-conserving schedule in the classical sense, and later prove that the hypothesis about abstract work-conservation also holds.
Hypothesis H_work_conserving : work_conserving_cl.
Assume that a job cost cannot be larger than a task cost.
Hypothesis H_valid_job_cost :
arrivals_have_valid_job_costs arr_seq.
Consider an arbitrary task set ts.
Variable ts : list Task.
Next, we assume that all jobs come from the task set.
Hypothesis H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts.
Let max_arrivals be a family of valid arrival curves, i.e., for any task tsk in ts, max_arrivals 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.
Let tsk be any task in ts that is to be analyzed.
Variable tsk : Task.
Hypothesis H_tsk_in_ts : tsk \in ts.
Consider a valid preemption model ...
Hypothesis H_valid_preemption_model :
valid_preemption_model arr_seq sched.
... and a valid task run-to-completion threshold function. That is, task_rtct tsk is (1) no bigger than tsk's cost, (2) for any job of task tsk job_rtct is bounded by task_rtct.
Hypothesis H_valid_run_to_completion_threshold :
valid_task_run_to_completion_threshold arr_seq tsk.
We introduce rbf as an abbreviation of the task request bound function, which is defined as task_cost(T) × max_arrivals(T,Δ) for some task T.
Let rbf := task_request_bound_function.
Next, we introduce task_rbf as an abbreviation of the task request bound function of task tsk.
Let task_rbf := rbf tsk.
Using the sum of individual request bound functions, we define the request bound function of all tasks (total request bound function).
Let total_rbf := total_request_bound_function ts.
Assume that there exists a bound on the length of any priority inversion experienced by any job of task tsk. Since we analyze only task tsk, we ignore the lengths of priority inversions incurred by any other tasks.
Variable priority_inversion_bound : duration → duration.
Hypothesis H_priority_inversion_is_bounded :
priority_inversion_is_bounded_by
arr_seq sched tsk priority_inversion_bound.
Let L be any positive fixed point of the busy interval recurrence.
Variable L : duration.
Hypothesis H_L_positive : L > 0.
Hypothesis H_fixed_point : L = total_rbf L.
To reduce the time complexity of the analysis, we introduce the notion of search space for EDF. Intuitively, this corresponds to all "interesting" arrival offsets that the job under analysis might have with regard to the beginning of its busy-window.
In the case of the search space for EDF, we consider three conditions. First, we ask whether task_rbf A ≠ task_rbf (A + ε).
Definition task_rbf_changes_at (A : duration) := task_rbf A != task_rbf (A + ε).
Second, we ask whether there exists a task tsko from ts such that tsko ≠ tsk and rbf(tsko, A + D tsk - D tsko) ≠ rbf(tsko, A + ε + D tsk - D tsko). Note that we use a slightly uncommon notation has (λ tsko ⇒ P tskₒ) ts, which can be interpreted as follows: the task set ts contains a task tsko such that a predicate P holds for tsko.
Definition bound_on_total_hep_workload_changes_at A :=
has (fun tsko ⇒
(tsk != tsko)
&& (rbf tsko (A + D tsk - D tsko)
!= rbf tsko ((A + ε) + D tsk - D tsko))) ts.
Third, we ask whether priority_inversion_bound (A - ε) ≠ priority_inversion_bound A.
Definition priority_inversion_changes_at (A : duration) :=
priority_inversion_bound (A - ε) != priority_inversion_bound A.
The final search space for EDF is a set of offsets that are less than L and where priority_inversion_bound, task_rbf, or bound_on_total_hep_workload changes in value.
Definition is_in_search_space (A : duration) :=
(A < L) && (priority_inversion_changes_at A
|| task_rbf_changes_at A
|| bound_on_total_hep_workload_changes_at A).
Let R be a value that upper-bounds the solution of each response-time recurrence, i.e., for any relative arrival time A in the search space, there exists a corresponding solution F such that R ≥ F + (task cost - task lock-in service).
Variable R : duration.
Hypothesis H_R_is_maximum :
∀ (A : duration),
is_in_search_space A →
∃ (F : duration),
A + F ≥ priority_inversion_bound A
+ (task_rbf (A + ε) - (task_cost tsk - task_rtct tsk))
+ bound_on_athep_workload ts tsk A (A + F)
∧ R ≥ F + (task_cost tsk - task_rtct tsk).
Finally, we define the interference bound function (task_IBF). task_IBF bounds the interference if tasks are sequential. Since tasks are sequential, we exclude interference from other jobs of the same task. For EDF, we define task_IBF as the sum of the priority interference bound and the higher-or-equal-priority workload.
Let task_IBF (A R : duration) :=
priority_inversion_bound A + bound_on_athep_workload ts tsk A R.

## Filling Out Hypothesis Of Abstract RTA Theorem

In this section we prove that all hypotheses necessary to use the abstract theorem are satisfied.
First, we prove that task_IBF is indeed a valid bound on the cumulative task interference.
Lemma instantiated_task_interference_is_bounded :
task_interference_is_bounded_by arr_seq sched tsk task_IBF.
Finally, we show that there exists a solution for the response-time recurrence.
Section SolutionOfResponseTimeReccurenceExists.
To rule out pathological cases with the concrete search space, we assume that the task cost is positive and the arrival curve is non-pathological.
Hypothesis H_task_cost_pos : 0 < task_cost tsk.
Hypothesis H_arrival_curve_pos : 0 < max_arrivals tsk ε.
Given any job j of task tsk that arrives exactly A units after the beginning of the busy interval, the bound of the total interference incurred by j within an interval of length Δ is equal to task_rbf (A + ε) - task_cost tsk + task_IBF(A, Δ).
Let total_interference_bound (A Δ : duration) :=
task_rbf (A + ε) - task_cost tsk + task_IBF A Δ.
Next, consider any A from the search space (in the abstract sense).
Variable A : duration.
Hypothesis H_A_is_in_abstract_search_space :
search_space.is_in_search_space L total_interference_bound A.
We prove that A is also in the concrete search space.
Lemma A_is_in_concrete_search_space :
is_in_search_space A.
Then, there exists a solution for the response-time recurrence (in the abstract sense).
Corollary correct_search_space :
∃ F,
A + F ≥ task_rbf (A + ε) - (task_cost tsk - task_rtct tsk) + task_IBF A (A + F)
∧ R ≥ F + (task_cost tsk - task_rtct tsk).
End SolutionOfResponseTimeReccurenceExists.

## Final Theorem

Based on the properties established above, we apply the abstract analysis framework to infer that R is a response-time bound for tsk.
Theorem uniprocessor_response_time_bound_edf :
task_response_time_bound arr_seq sched tsk R.
End AbstractRTAforEDFwithArrivalCurves.