# Library prosa.results.fixed_priority.rta.bounded_pi

Require Export prosa.model.schedule.priority_driven.

Require Export prosa.analysis.abstract.ideal_jlfp_rta.

Require Export prosa.analysis.facts.busy_interval.busy_interval.

Require Export prosa.analysis.abstract.ideal_jlfp_rta.

Require Export prosa.analysis.facts.busy_interval.busy_interval.

# 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 uni-processor model of real-time tasks with arbitrary arrival models.
Consider any type of tasks ...

... 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}.

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.

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.

Next, consider any ideal uni-processor schedule of this arrival sequence, ...

... allow for any work-bearing notion of job readiness, ...

Context `{@JobReady Job (ideal.processor_state Job) Cost Arrival}.

Hypothesis H_job_ready : work_bearing_readiness arr_seq sched.

Hypothesis H_job_ready : work_bearing_readiness arr_seq sched.

... and assume that the schedule is valid.

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.

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.
Assume we have sequential tasks, i.e, jobs from the
same task execute in the order of their arrival.

Assume that a job cost cannot be larger than a task cost.

Consider an arbitrary task set ts.

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

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.

Consider a valid preemption model...

...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.

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.

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).

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).

Similarly, we define the request bound function of all tasks other
than tsk with higher-or-equal priority (with respect to 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_constant

arr_seq sched tsk priority_inversion_bound.

Hypothesis H_priority_inversion_is_bounded:

priority_inversion_is_bounded_by_constant

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 = priority_inversion_bound + total_hep_rbf L.

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.

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).

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).

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.

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.

Let interfering_workload (j : Job) (t : instant) :=

ideal_jlfp_rta.interfering_workload arr_seq sched j t.

ideal_jlfp_rta.interfering_workload arr_seq sched j t.

Finally, we define the interference bound function
(IBF_other). IBF_other 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 IBF_other
as the sum of the priority interference bound and the
higher-or-equal-priority workload.

## 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 interference interfering_workload L.

busy_intervals_are_bounded_by arr_seq sched tsk interference interfering_workload L.

Next, we prove that IBF_other 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 t, the relative arrival time of a job j of task t,
and the length of the interval to the maximum amount of interference (for more details see
files limited.abstract_RTA.definitions and limited.abstract_RTA.abstract_seq_rta).
However, in this module we analyze only one task -- tsk, therefore it is “hard-coded”
inside the interference bound function IBF_other. 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 IBF_other signature to
match the required signature in module abstract_seq_RTA, we wrap the IBF_other 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 interference interfering_workload (fun t A R ⇒ IBF_other R).

task_interference_is_bounded_by

arr_seq sched tsk interference interfering_workload (fun t A R ⇒ IBF_other R).

Finally, we show that there exists a solution for the response-time recurrence.

Variable j : Job.

Hypothesis H_j_arrives : arrives_in arr_seq j.

Hypothesis H_job_of_tsk : job_of_task tsk j.

Hypothesis H_job_cost_positive: job_cost_positive j.

Hypothesis H_j_arrives : arrives_in arr_seq j.

Hypothesis H_job_of_tsk : job_of_task tsk j.

Hypothesis H_job_cost_positive: job_cost_positive j.

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 + IBF_other 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 tsk L total_interference_bound A.

Hypothesis H_A_is_in_abstract_search_space :

search_space.is_in_search_space tsk L total_interference_bound A.

We prove that A is also in the concrete search space.

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) + IBF_other (A + F) ∧

R ≥ F + (task_cost tsk - task_rtct tsk).

End SolutionOfResponseTimeRecurrenceExists.

End FillingOutHypothesesOfAbstractRTATheorem.

∃ (F : duration),

A + F ≥ task_rbf (A + ε) - (task_cost tsk - task_rtct tsk) + IBF_other (A + F) ∧

R ≥ F + (task_cost tsk - task_rtct tsk).

End SolutionOfResponseTimeRecurrenceExists.

End FillingOutHypothesesOfAbstractRTATheorem.