Library prosa.analysis.facts.transform.edf_wc

Require Export prosa.analysis.facts.transform.edf_opt.
Require Export prosa.analysis.facts.transform.wc_correctness.
Require Export prosa.analysis.facts.behavior.deadlines.
Require Export prosa.analysis.facts.readiness.backlogged.

Optimality of Work-Conserving EDF on Ideal Uniprocessors

In this file, we establish the foundation needed to connect the EDF and work-conservation optimality theorems: if there is any work-conserving way to meet all deadlines (assuming an ideal uniprocessor schedule), then there is also an (ideal) EDF schedule that is work-conserving in which all deadlines are met.

Throughout this file, we assume ideal uniprocessor schedules.

Require Import prosa.model.processor.ideal.

Throughout this file, we assume the basic (i.e., Liu & Layland) readiness model.

Require Import prosa.model.readiness.basic.

Non-Idle Swaps

We start by showing that swapped, a function used in the inner-most level of edf_transform, maintains work conservation if the two instants being swapped are not idle.

Section NonIdleSwapWorkConservationLemmas.

For any given type of jobs...

Context {Job : JobType} `{JobCost Job} `{JobDeadline Job} `{JobArrival Job}.

... and any valid job arrival sequence.

Variable arr_seq: arrival_sequence Job.
Hypothesis H_arr_seq_valid: valid_arrival_sequence arr_seq.

...consider an ideal uniprocessor schedule...

Variable sched: schedule (ideal.processor_state Job).

...that is well-behaved (i.e., in which jobs execute only after having arrived and only if they are not yet complete, and in which all jobs come from the arrival sequence).

  Hypothesis H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched.
  Hypothesis H_completed_jobs_dont_execute: completed_jobs_dont_execute sched.
  Hypothesis H_from_arr_seq: jobs_come_from_arrival_sequence sched arr_seq.

Suppose we are given two specific times t1 and t2,...

Variables t1 t2 : instant.

...which we assume to be ordered (to avoid dealing with symmetric cases),...

Hypothesis H_well_ordered: t1 ≤ t2.

...and two jobs j1 and j2...

Variables j1 j2 : Job.

...such that j2 arrives before time t1,...

Hypothesis H_arrival_j2 : job_arrival j2 ≤ t1.

...j1 is scheduled at time t1, and...

Hypothesis H_t1_not_idle : scheduled_at sched j1 t1.

...j2 is scheduled at time t2.

Hypothesis H_t2_not_idle : scheduled_at sched j2 t2.

We let swap_sched denote the schedule in which the allocations at t1 and t2 have been swapped.

Let swap_sched := swapped sched t1 t2.

Now consider an arbitrary job j...

Variable j : Job.

...and an arbitrary instant t...

Variable t : instant.

...such that j arrives in arr_seq...

Hypothesis H_arrival_j : arrives_in arr_seq j.

...and is backlogged in swap_sched at instant t.

Hypothesis H_backlogged_j_t : backlogged swap_sched j t.

We proceed by case analysis. We first show that, if t equals t1, then swap_sched maintains work conservation. That is, there exists some job that's scheduled in swap_sched at instant t

  Lemma non_idle_swap_maintains_work_conservation_t1 :
    work_conserving arr_seq sched →
    t = t1 →
    ∃ j_other , scheduled_at swap_sched j_other t.

Similarly, if t equals t2 then then swap_sched maintains work conservation.

  Lemma non_idle_swap_maintains_work_conservation_t2 :
    work_conserving arr_seq sched →
    t = t2 →
    ∃ j_other , scheduled_at swap_sched j_other t.

If t is less than or equal to t1 then then then swap_sched maintains work conservation.

  Lemma non_idle_swap_maintains_work_conservation_LEQ_t1 :
    work_conserving arr_seq sched →
    t ≤ t1 →
    ∃ j_other , scheduled_at swap_sched j_other t.

Similarly, if t is greater than t2 then then then swap_sched maintains work conservation.

  Lemma non_idle_swap_maintains_work_conservation_GT_t2 :
    work_conserving arr_seq sched →
    t2 < t →
    ∃ j_other , scheduled_at swap_sched j_other t.

Lastly, we show that if t lies between t1 and t2 then work conservation is still maintained.

  Lemma non_idle_swap_maintains_work_conservation_BET_t1_t2 :
    work_conserving arr_seq sched →
    t1 < t ≤ t2 →
    ∃ j_other , scheduled_at swap_sched j_other t.

End NonIdleSwapWorkConservationLemmas.

Work-Conserving Swap Candidates

Now, we show that work conservation is maintained by the inner-most level of edf_transform, which is find_swap_candidate.

Section FSCWorkConservationLemmas.

For any given type of jobs...

Context {Job : JobType} `{JobCost Job} `{JobDeadline Job} `{JobArrival Job}.

...and any valid job arrival sequence,...

Variable arr_seq: arrival_sequence Job.
Hypothesis H_arr_seq_valid: valid_arrival_sequence arr_seq.

...consider an ideal uniprocessor schedule...

Variable sched: schedule (ideal.processor_state Job).

...that is well-behaved (i.e., in which jobs execute only after having arrived and only if they are not yet complete)...

Hypothesis H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched.
Hypothesis H_completed_jobs_dont_execute: completed_jobs_dont_execute sched.

...and in which all jobs come from the arrival sequence.

Hypothesis H_from_arr_seq: jobs_come_from_arrival_sequence sched arr_seq.

Suppose we are given a job j1...

Variable j1: Job.

...and a point in time t1...

Variable t1: instant.

...at which j1 is scheduled...

Hypothesis H_not_idle: scheduled_at sched j1 t1.

...and that is before j1's deadline.

Hypothesis H_deadline_not_missed: t1 < job_deadline j1.

We now show that, if t2 is a swap candidate returned by find_swap_candidate for t1, then swapping the processor allocations at the two instants maintains work conservation.

  Corollary fsc_swap_maintains_work_conservation :
    work_conserving arr_seq sched →
    work_conserving arr_seq (swapped sched t1 (edf_trans.find_swap_candidate sched t1 j1)).

End FSCWorkConservationLemmas.

Work-Conservation of the Point-Wise EDF Transformation

In the following section, we show that work conservation is maintained by the next level of edf_transform, which is make_edf_at.

Section MakeEDFWorkConservationLemmas.

For any given type of jobs...

Context {Job : JobType} `{JobCost Job} `{JobDeadline Job} `{JobArrival Job}.

... and any valid job arrival sequence, ...

Variable arr_seq: arrival_sequence Job.
Hypothesis H_arr_seq_valid: valid_arrival_sequence arr_seq.

... consider an ideal uniprocessor schedule ...

Variable sched: schedule (ideal.processor_state Job).

... in which all jobs come from the arrival sequence, ...

Hypothesis H_from_arr_seq: jobs_come_from_arrival_sequence sched arr_seq.

...that is well-behaved,...

Hypothesis H_jobs_must_arrive_to_execute: jobs_must_arrive_to_execute sched.
Hypothesis H_completed_jobs_dont_execute: completed_jobs_dont_execute sched.

...and in which no scheduled job misses a deadline.

Hypothesis H_no_deadline_misses: all_deadlines_met sched.

We analyze make_edf_at applied to an arbitrary point in time, which we denote t_edf in the following.

Variable t_edf: instant.

For brevity, let sched' denote the schedule obtained from make_edf_at applied to sched at time t_edf.

Let sched' := make_edf_at sched t_edf.

We show that, if a schedule is work-conserving, then applying make_edf_at to it at an arbitrary instant t_edf maintains work conservation.

Lemma mea_maintains_work_conservation :
work_conserving arr_seq sched → work_conserving arr_seq sched'.

End MakeEDFWorkConservationLemmas.

Work-Conserving EDF Prefix

On to the next layer, we prove that the transform_prefix function at the core of the EDF transformation maintains work conservation

Section EDFPrefixWorkConservationLemmas.

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}.

... and any valid job arrival sequence, ...

Variable arr_seq: arrival_sequence Job.
Hypothesis H_arr_seq_valid: valid_arrival_sequence arr_seq.

... consider an ideal uniprocessor schedule,...

Variable sched: schedule (ideal.processor_state Job).

... an arbitrary finite horizon, and ...

Variable horizon: instant.

...let sched_trans denote the schedule obtained by transforming sched up to the horizon.

Let sched_trans := edf_transform_prefix sched horizon.

Let schedule_behavior_premises define the premise that a schedule is: 1) well-behaved, 2) has all jobs coming from the arrival sequence arr_seq, and 3) in which no scheduled job misses its deadline

  Definition scheduled_behavior_premises (sched : schedule (processor_state Job)) :=
    jobs_must_arrive_to_execute sched
    ∧ completed_jobs_dont_execute sched
    ∧ jobs_come_from_arrival_sequence sched arr_seq
    ∧ all_deadlines_met sched.

For brevity, let P denote the predicate that a schedule satisfies scheduled_behavior_premises and is work-conserving.

Let P (sched : schedule (processor_state Job)) :=
scheduled_behavior_premises sched ∧ work_conserving arr_seq sched.

We show that if sched is work-conserving, then so is sched_trans.

Lemma edf_transform_prefix_maintains_work_conservation:
P sched → P sched_trans.

End EDFPrefixWorkConservationLemmas.

Work-Conservation of the EDF Transformation

Finally, having established that edf_transform_prefix maintains work conservation, we go ahead and prove that edf_transform maintains work conservation, too.

Section EDFTransformWorkConservationLemmas.

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}.

... and any valid job arrival sequence, ...

Variable arr_seq: arrival_sequence Job.
Hypothesis H_arr_seq_valid: valid_arrival_sequence arr_seq.

... consider a valid ideal uniprocessor schedule ...

Variable sched: schedule (ideal.processor_state Job).
Hypothesis H_sched_valid: valid_schedule sched arr_seq.

...and in which no scheduled job misses a deadline.

Hypothesis H_no_deadline_misses: all_deadlines_met sched.

We first note that sched satisfies scheduled_behavior_premises.

Lemma sched_satisfies_behavior_premises : scheduled_behavior_premises arr_seq sched.

We prove that, if the given schedule sched is work-conserving, then the schedule that results from transforming it into an EDF schedule is also work-conserving.

Lemma edf_transform_maintains_work_conservation :
work_conserving arr_seq sched → work_conserving arr_seq (edf_transform sched).

End EDFTransformWorkConservationLemmas.