# Library prosa.results.optimality.edf

Require Import prosa.model.readiness.basic.
Require Import prosa.model.preemption.fully_preemptive.
Require Export prosa.analysis.facts.transform.edf_opt.
Require Export prosa.analysis.facts.transform.edf_wc.
Require Export prosa.analysis.facts.edf_definitions.
Require prosa.implementation.priority.edf.

# Optimality of EDF on Ideal Uniprocessors

This module provides the famous EDF optimality theorem: if there is any way to meet all deadlines (assuming an ideal uniprocessor schedule), then there is also an (ideal) EDF schedule in which all deadlines are met.

## Optimality Theorem

Section Optimality.
Consider any given type of jobs, each characterized by execution costs, an arrival time, and an absolute deadline,...
Context {Job : JobType} `{JobCost Job} `{JobDeadline Job} `{JobArrival Job}.
... and any valid arrival sequence of such jobs.
Variable arr_seq : arrival_sequence Job.
Hypothesis H_arr_seq_valid : valid_arrival_sequence arr_seq.
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.
We assume that jobs are fully preemptive.
#[local] Existing Instance fully_preemptive_job_model.
Under these assumptions, EDF is optimal in the sense that, if there exists any schedule in which all jobs of arr_seq meet their deadline, then there also exists an EDF schedule in which all deadlines are met.
Theorem EDF_optimality :
(∃ any_sched : schedule (ideal.processor_state Job),
valid_schedule any_sched arr_seq
∧ all_deadlines_of_arrivals_met arr_seq any_sched) →
∃ edf_sched : schedule (ideal.processor_state Job),
valid_schedule edf_sched arr_seq
∧ all_deadlines_of_arrivals_met arr_seq edf_sched
∧ EDF_schedule edf_sched.
Proof.
move⇒ [sched [[COME READY] DL_ARR_MET]].
have ARR := jobs_must_arrive_to_be_ready sched READY.
have COMP := completed_jobs_are_not_ready sched READY.
move: (all_deadlines_met_in_valid_schedule _ _ COME DL_ARR_MET) ⇒ DL_MET.
set sched' := edf_transform sched.
∃ sched'. split; last split.
- by apply edf_schedule_is_valid.
- by apply edf_schedule_meets_all_deadlines_wrt_arrivals.
- by apply edf_transform_ensures_edf.
Qed.
Moreover, we note that, since EDF maintains work conservation, if there exists a schedule in which all jobs of arr_seq meet their deadline, then there also exists a work-conserving EDF in which all deadlines are met.
Theorem EDF_WC_optimality :
(∃ any_sched : schedule (ideal.processor_state Job),
valid_schedule any_sched arr_seq
∧ all_deadlines_of_arrivals_met arr_seq any_sched) →
∃ edf_wc_sched : schedule (ideal.processor_state Job),
valid_schedule edf_wc_sched arr_seq
∧ all_deadlines_of_arrivals_met arr_seq edf_wc_sched
∧ work_conserving arr_seq edf_wc_sched
∧ EDF_schedule edf_wc_sched.
Proof.
move⇒ [sched [[COME READY] DL_ARR_MET]].
move: (all_deadlines_met_in_valid_schedule _ _ COME DL_ARR_MET) ⇒ DL_MET.
set wc_sched := wc_transform arr_seq sched.
have wc_COME : jobs_come_from_arrival_sequence wc_sched arr_seq
by apply wc_jobs_come_from_arrival_sequence.
have wc_READY : jobs_must_be_ready_to_execute wc_sched
by apply wc_jobs_must_be_ready_to_execute.
have wc_ARR := jobs_must_arrive_to_be_ready wc_sched wc_READY.
have wc_COMP := completed_jobs_are_not_ready wc_sched wc_READY.
have wc_DL_ARR_MET : all_deadlines_of_arrivals_met arr_seq wc_sched
by apply wc_all_deadlines_of_arrivals_met; apply DL_ARR_MET.
move: (all_deadlines_met_in_valid_schedule _ _ wc_COME wc_DL_ARR_MET) ⇒ wc_DL_MET.
set sched' := edf_transform wc_sched.
∃ sched'. split; last split; last split.
- by apply edf_schedule_is_valid.
- by apply edf_schedule_meets_all_deadlines_wrt_arrivals.
- apply edf_transform_maintains_work_conservation; by [ | apply wc_is_work_conserving ].
- by apply edf_transform_ensures_edf.
Qed.
Remark: EDF_optimality is of course an immediate corollary of EDF_WC_optimality. We nonetheless have two separate proofs for historic reasons as EDF_optimality predates EDF_WC_optimality (in Prosa).
Finally, we state the optimality theorem also in terms of a policy-compliant schedule, which is a more generic notion used in Prosa for scheduling policies (rather than the simpler, but ad-hoc EDF_schedule predicate used above).
Given that we're considering the EDF priority policy and a fully preemptive job model, satisfying the EDF_schedule predicate is equivalent to satisfying the respects_policy_at_preemption_point w.r.t. the EDF policy predicate. The optimality of priority-compliant schedules that are work-conserving follows hence directly from the above EDF_WC_optimality theorem.
Corollary EDF_priority_compliant_WC_optimality :
(∃ any_sched : schedule (ideal.processor_state Job),
valid_schedule any_sched arr_seq
∧ all_deadlines_of_arrivals_met arr_seq any_sched) →
∃ priority_compliant_sched : schedule (ideal.processor_state Job),
valid_schedule priority_compliant_sched arr_seq
∧ all_deadlines_of_arrivals_met arr_seq priority_compliant_sched
∧ work_conserving arr_seq priority_compliant_sched
∧ respects_JLFP_policy_at_preemption_point arr_seq priority_compliant_sched (EDF Job).
Proof.
move /EDF_WC_optimality ⇒ [edf_sched [[ARR READY] [DL_MET [WC EDF]]]].
∃ edf_sched.
apply (EDF_schedule_equiv arr_seq) in EDF ⇒ //.
exact: (completed_jobs_are_not_ready edf_sched READY).
Qed.
End Optimality.

## Weak Optimality Theorem

We further state a weaker notion of the above optimality result that avoids a dependency on a given arrival sequence. Specifically, it establishes that, given a reference schedule without deadline misses, there exists an EDF schedule of the same jobs in which no deadlines are missed.
Section WeakOptimality.
For any given type of jobs, each characterized by execution costs, an arrival time, and an absolute deadline,...
Context {Job : JobType} `{JobCost Job} `{JobDeadline Job} `{JobArrival Job}.
... if we have a well-behaved reference schedule ...
Variable any_sched : schedule (ideal.processor_state Job).
Hypothesis H_must_arrive : jobs_must_arrive_to_execute any_sched.
Hypothesis H_completed_dont_execute : completed_jobs_dont_execute any_sched.
... in which no deadlines are missed, ...
Hypothesis H_all_deadlines_met : all_deadlines_met any_sched.
...then there also exists an EDF schedule in which no deadlines are missed (and in which exactly the same set of jobs is scheduled, as ensured by the last clause).
Theorem weak_EDF_optimality :
∃ edf_sched : schedule (ideal.processor_state Job),
jobs_must_arrive_to_execute edf_sched
∧ completed_jobs_dont_execute edf_sched
∧ all_deadlines_met edf_sched
∧ EDF_schedule edf_sched
∧ ∀ j, (∃ t, scheduled_at any_sched j t)
↔ (∃ t', scheduled_at edf_sched j t').
Proof.
set sched' := edf_transform any_sched.
∃ sched'. repeat split.
- by apply edf_transform_jobs_must_arrive.
- by apply edf_transform_completed_jobs_dont_execute.
- by apply edf_transform_deadlines_met.
- by apply edf_transform_ensures_edf.
- by move⇒ [t SCHED_j]; apply edf_transform_job_scheduled' with (t := t).
- by move⇒ [t SCHED_j]; apply edf_transform_job_scheduled with (t := t).
Qed.
End WeakOptimality.