Library rt.analysis.uni.basic.fp_rta_theory
Require Import rt.util.all.
Require Import rt.model.arrival.basic.job rt.model.arrival.basic.task rt.model.priority rt.model.arrival.basic.task_arrival
rt.model.arrival.basic.arrival_bounds.
Require Import rt.model.schedule.uni.schedule_of_task rt.model.schedule.uni.workload
rt.model.schedule.uni.schedulability rt.model.schedule.uni.response_time
rt.model.schedule.uni.service.
Require Import rt.model.schedule.uni.basic.busy_interval rt.model.schedule.uni.basic.platform.
Require Import rt.analysis.uni.basic.workload_bound_fp.
Module ResponseTimeAnalysisFP.
Import Job ScheduleOfTask SporadicTaskset Priority ResponseTime
TaskArrival ArrivalBounds WorkloadBoundFP Platform Schedulability
BusyInterval Workload Service.
(* In this section, we prove that any fixed point in the RTA for uniprocessor
FP scheduling is a response-time bound. *)
Section ResponseTimeBound.
Context {SporadicTask: eqType}.
Variable task_cost: SporadicTask → time.
Variable task_period: SporadicTask → time.
Variable task_deadline: SporadicTask → time.
Context {Job: eqType}.
Variable job_arrival: Job → time.
Variable job_cost: Job → time.
Variable job_deadline: Job → time.
Variable job_task: Job → SporadicTask.
(* Assume any job arrival sequence with consistent, duplicate-free arrivals... *)
Variable arr_seq: arrival_sequence Job.
Hypothesis H_arrival_times_are_consistent: arrival_times_are_consistent job_arrival arr_seq.
Hypothesis H_no_duplicate_arrivals: arrival_sequence_is_a_set arr_seq.
(* ... in which jobs arrive sporadically and have valid parameters. *)
Hypothesis H_sporadic_tasks:
sporadic_task_model task_period job_arrival job_task arr_seq.
Hypothesis H_valid_job_parameters:
∀ j,
arrives_in arr_seq j →
valid_sporadic_job task_cost task_deadline job_cost job_deadline job_task j.
(* Consider a task set ts where all tasks have valid parameters... *)
Variable ts: seq SporadicTask.
Hypothesis H_valid_task_parameters:
valid_sporadic_taskset task_cost task_period task_deadline ts.
(* ... and assume that all jobs in the arrival sequence come from the task set. *)
Hypothesis H_all_jobs_from_taskset:
∀ j, arrives_in arr_seq j → job_task j \in ts.
(* Next, consider any uniprocessor schedule of this arrival sequence...*)
Variable sched: schedule Job.
Hypothesis H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence sched arr_seq.
(* ... where jobs do not execute before their arrival times nor after completion. *)
Hypothesis H_jobs_must_arrive_to_execute:
jobs_must_arrive_to_execute job_arrival sched.
Hypothesis H_completed_jobs_dont_execute:
completed_jobs_dont_execute job_cost sched.
(* Consider an FP policy that indicates a higher-or-equal priority relation,
and assume that the relation is reflexive and transitive. *)
Variable higher_eq_priority: FP_policy SporadicTask.
Hypothesis H_priority_is_reflexive: FP_is_reflexive higher_eq_priority.
Hypothesis H_priority_is_transitive: FP_is_transitive higher_eq_priority.
(* Next, assume that the schedule is a work-conserving FP schedule. *)
Hypothesis H_work_conserving: work_conserving job_arrival job_cost arr_seq sched.
Hypothesis H_respects_fp_policy:
respects_FP_policy job_arrival job_cost job_task arr_seq sched higher_eq_priority.
(* Now we proceed with the analysis.
Let tsk be any task in ts that is to be analyzed. *)
Variable tsk: SporadicTask.
Hypothesis H_tsk_in_ts: tsk \in ts.
(* Recall the definition of response-time bound and the total workload bound W
for tasks with higher-or-equal priority (with respect to tsk). *)
Let response_time_bounded_by :=
is_response_time_bound_of_task job_arrival job_cost job_task arr_seq sched.
Let W := total_workload_bound_fp task_cost task_period higher_eq_priority ts tsk.
(* Let R be any positive fixed point of the response-time recurrence. *)
Variable R: time.
Hypothesis H_R_positive: R > 0.
Hypothesis H_response_time_is_fixed_point: R = W R.
(* Since R = W R bounds the workload of higher-or-equal priority
in any interval of length R, it follows from the busy-interval
lemmas that R bounds the response-time of job j.
(For more details, see model/uni/basic/busy_interval.v and
analysis/uni/basic/workload_bound_fp.v.) *)
Theorem uniprocessor_response_time_bound_fp:
response_time_bounded_by tsk R.
End ResponseTimeBound.
End ResponseTimeAnalysisFP.
Require Import rt.model.arrival.basic.job rt.model.arrival.basic.task rt.model.priority rt.model.arrival.basic.task_arrival
rt.model.arrival.basic.arrival_bounds.
Require Import rt.model.schedule.uni.schedule_of_task rt.model.schedule.uni.workload
rt.model.schedule.uni.schedulability rt.model.schedule.uni.response_time
rt.model.schedule.uni.service.
Require Import rt.model.schedule.uni.basic.busy_interval rt.model.schedule.uni.basic.platform.
Require Import rt.analysis.uni.basic.workload_bound_fp.
Module ResponseTimeAnalysisFP.
Import Job ScheduleOfTask SporadicTaskset Priority ResponseTime
TaskArrival ArrivalBounds WorkloadBoundFP Platform Schedulability
BusyInterval Workload Service.
(* In this section, we prove that any fixed point in the RTA for uniprocessor
FP scheduling is a response-time bound. *)
Section ResponseTimeBound.
Context {SporadicTask: eqType}.
Variable task_cost: SporadicTask → time.
Variable task_period: SporadicTask → time.
Variable task_deadline: SporadicTask → time.
Context {Job: eqType}.
Variable job_arrival: Job → time.
Variable job_cost: Job → time.
Variable job_deadline: Job → time.
Variable job_task: Job → SporadicTask.
(* Assume any job arrival sequence with consistent, duplicate-free arrivals... *)
Variable arr_seq: arrival_sequence Job.
Hypothesis H_arrival_times_are_consistent: arrival_times_are_consistent job_arrival arr_seq.
Hypothesis H_no_duplicate_arrivals: arrival_sequence_is_a_set arr_seq.
(* ... in which jobs arrive sporadically and have valid parameters. *)
Hypothesis H_sporadic_tasks:
sporadic_task_model task_period job_arrival job_task arr_seq.
Hypothesis H_valid_job_parameters:
∀ j,
arrives_in arr_seq j →
valid_sporadic_job task_cost task_deadline job_cost job_deadline job_task j.
(* Consider a task set ts where all tasks have valid parameters... *)
Variable ts: seq SporadicTask.
Hypothesis H_valid_task_parameters:
valid_sporadic_taskset task_cost task_period task_deadline ts.
(* ... and assume that all jobs in the arrival sequence come from the task set. *)
Hypothesis H_all_jobs_from_taskset:
∀ j, arrives_in arr_seq j → job_task j \in ts.
(* Next, consider any uniprocessor schedule of this arrival sequence...*)
Variable sched: schedule Job.
Hypothesis H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence sched arr_seq.
(* ... where jobs do not execute before their arrival times nor after completion. *)
Hypothesis H_jobs_must_arrive_to_execute:
jobs_must_arrive_to_execute job_arrival sched.
Hypothesis H_completed_jobs_dont_execute:
completed_jobs_dont_execute job_cost sched.
(* Consider an FP policy that indicates a higher-or-equal priority relation,
and assume that the relation is reflexive and transitive. *)
Variable higher_eq_priority: FP_policy SporadicTask.
Hypothesis H_priority_is_reflexive: FP_is_reflexive higher_eq_priority.
Hypothesis H_priority_is_transitive: FP_is_transitive higher_eq_priority.
(* Next, assume that the schedule is a work-conserving FP schedule. *)
Hypothesis H_work_conserving: work_conserving job_arrival job_cost arr_seq sched.
Hypothesis H_respects_fp_policy:
respects_FP_policy job_arrival job_cost job_task arr_seq sched higher_eq_priority.
(* Now we proceed with the analysis.
Let tsk be any task in ts that is to be analyzed. *)
Variable tsk: SporadicTask.
Hypothesis H_tsk_in_ts: tsk \in ts.
(* Recall the definition of response-time bound and the total workload bound W
for tasks with higher-or-equal priority (with respect to tsk). *)
Let response_time_bounded_by :=
is_response_time_bound_of_task job_arrival job_cost job_task arr_seq sched.
Let W := total_workload_bound_fp task_cost task_period higher_eq_priority ts tsk.
(* Let R be any positive fixed point of the response-time recurrence. *)
Variable R: time.
Hypothesis H_R_positive: R > 0.
Hypothesis H_response_time_is_fixed_point: R = W R.
(* Since R = W R bounds the workload of higher-or-equal priority
in any interval of length R, it follows from the busy-interval
lemmas that R bounds the response-time of job j.
(For more details, see model/uni/basic/busy_interval.v and
analysis/uni/basic/workload_bound_fp.v.) *)
Theorem uniprocessor_response_time_bound_fp:
response_time_bounded_by tsk R.
End ResponseTimeBound.
End ResponseTimeAnalysisFP.