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

Require Export prosa.model.schedule.priority_driven.
Require Export prosa.analysis.abstract.ideal.iw_instantiation.
Require Export prosa.analysis.facts.busy_interval.existence.
Require Export prosa.analysis.abstract.ideal.abstract_seq_rta.
Require Export prosa.analysis.facts.model.task_cost.

# Abstract RTA for FP-schedulers with Bounded Priority Inversion

In this module we instantiate the Abstract Response-Time analysis (aRTA) to FP-schedulers for ideal uniprocessor model of real-time tasks with arbitrary arrival models.
Given FP priority policy and an ideal uniprocessor 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 AbstractRTAforFPwithArrivalCurves.
Consider any type of tasks ...
Context {Task : TaskType}.
Context `{TaskCost Task}.
Context `{TaskRunToCompletionThreshold 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}.
Consider an FP policy that indicates a higher-or-equal priority relation, and assume that the relation is reflexive. Note that we do not relate the FP policy with the scheduler. However, we define functions for Interference and Interfering Workload that actively use the concept of priorities. We require the FP policy to be reflexive, so a job cannot cause lower-priority interference (i.e. priority inversion) to itself.
Context {FP : FP_policy Task}.
Hypothesis H_priority_is_reflexive : reflexive_task_priorities FP.
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 ideal uniprocessor schedule of this arrival sequence, ...
Variable sched : schedule (ideal.processor_state Job).
... allow for any work-bearing notion of job readiness, ...
Context `{!JobReady Job (ideal.processor_state Job)}.
Hypothesis H_job_ready : work_bearing_readiness arr_seq sched.
... and assume that the schedule is valid.
Hypothesis H_sched_valid : valid_schedule sched arr_seq.

## 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 we have sequential tasks, i.e., jobs from the same task execute in the order of their arrival.
Hypothesis H_sequential_tasks : sequential_tasks arr_seq sched.
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_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.
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.
For clarity, let's define some local names.
Let job_pending_at := pending sched.
Let job_scheduled_at := scheduled_at sched.
Let job_completed_by := completed_by sched.
Let job_backlogged_at := backlogged sched.
Let response_time_bounded_by := task_response_time_bound arr_seq sched.
We introduce task_rbf as an abbreviation of the task request bound function, which is defined as task_cost(tsk) × max_arrivals(tsk,Δ).
Let task_rbf := task_request_bound_function tsk.
Using the sum of individual request bound functions, we define the request bound function of all tasks with higher-or-equal priority (with respect to tsk).
Let total_hep_rbf := total_hep_request_bound_function_FP ts tsk.
Similarly, we define the request bound function of all tasks other than tsk with higher-or-equal priority (with respect to tsk).
Let total_ohep_rbf :=
total_ohep_request_bound_function_FP ts tsk.
Assume that there exists a constant priority_inversion_bound that bounds the length of any priority inversion experienced by any job of tsk. Since we analyze only task tsk, we ignore the lengths of priority inversions incurred by any other tasks.
Variable priority_inversion_bound : duration.
Hypothesis H_priority_inversion_is_bounded :
priority_inversion_is_bounded_by
arr_seq sched tsk (constant 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 = priority_inversion_bound + total_hep_rbf L.
To reduce the time complexity of the analysis, recall the notion of search space. Intuitively, this corresponds to all "interesting" arrival offsets that the job under analysis might have with regard to the beginning of its busy-window.
Definition is_in_search_space A := (A < L) && (task_rbf A != task_rbf (A + ε)).
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
+ (task_rbf (A + ε) - (task_cost tsk - task_rtct tsk))
+ total_ohep_rbf (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 FP, we define task_IBF as the sum of the priority interference bound and the higher-or-equal-priority workload.
Let task_IBF (R : duration) := priority_inversion_bound + total_ohep_rbf R.

## Filling Out Hypotheses Of Abstract RTA Theorem

In this section we prove that all preconditions necessary to use the abstract theorem are satisfied.
Recall that L is assumed to be a fixed point of the busy interval recurrence. Thanks to this fact, we can prove that every busy interval (according to the concrete definition) is bounded. In addition, we know that the conventional concept of busy interval and the one obtained from the abstract definition (with the interference and interfering workload) coincide. Thus, it follows that any busy interval (in the abstract sense) is bounded.
Lemma instantiated_busy_intervals_are_bounded :
busy_intervals_are_bounded_by arr_seq sched tsk L.
Next, we prove that task_IBF is indeed an interference bound.
Recall that in module abstract_seq_RTA hypothesis task_interference_is_bounded_by expects to receive a function that maps some task tsk, the relative arrival time of a job j of task tsk, and the length of the interval to the maximum amount of interference.
However, in this module we analyze only one task -- tsk, therefore it is “hard-coded” inside the interference bound function task_IBF. Moreover, in case of a model with fixed priorities, interference that some job j incurs from higher-or-equal priority jobs does not depend on the relative arrival time of job j. Therefore, in order for the task_IBF signature to match the required signature in module abstract_seq_RTA, we wrap the task_IBF function in a function that accepts, but simply ignores, the task and the relative arrival time.
Lemma instantiated_task_interference_is_bounded :
task_interference_is_bounded_by
arr_seq sched tsk (fun A R ⇒ task_IBF R).
Finally, we show that there exists a solution for the response-time recurrence.
Section SolutionOfResponseTimeRecurrenceExists.
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 Δ.
Let total_interference_bound A Δ :=
task_rbf (A + ε) - task_cost tsk + task_IBF 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 : duration),
A + F ≥ task_rbf (A + ε) - (task_cost tsk - task_rtct tsk) + task_IBF (A + F)
∧ R ≥ F + (task_cost tsk - task_rtct tsk).
End SolutionOfResponseTimeRecurrenceExists.

## 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_fp :
response_time_bounded_by tsk R.
End AbstractRTAforFPwithArrivalCurves.