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.
We assume the classic (i.e., Liu & Layland) model of readiness without jitter or self-suspensions, wherein pending jobs are always ready.

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
edf_sched : schedule (ideal.processor_state Job),
valid_schedule edf_sched arr_seq
EDF_schedule edf_sched.
Proof.
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_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
edf_wc_sched : schedule (ideal.processor_state Job),
valid_schedule edf_wc_sched arr_seq
work_conserving arr_seq edf_wc_sched
EDF_schedule edf_wc_sched.
Proof.
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_DL_ARR_MET : all_deadlines_of_arrivals_met arr_seq wc_sched
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.
- 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 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. to 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
priority_compliant_sched : schedule (ideal.processor_state Job),
valid_schedule priority_compliant_sched arr_seq
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 ⇒ //.
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 ...
... in which no deadlines are missed, ...
...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
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.