Library prosa.analysis.facts.transform.edf_opt

This file contains the main argument of the EDF optimality proof, starting with an analysis of the individual functions that drive the piece-wise transformation of a given reference schedule in an EDF schedule, and ending with proofs of individual properties of the obtained EDF schedule. Throughout this file, we assume ideal uniprocessor schedules.

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

We start by analyzing the helper function find_swap_candidate, which is a problem-specific wrapper around search_arg.

For any given type of jobs...

...consider an ideal uniprocessor schedule...

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

Suppose we are given a job j1...

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

...at which j1 is scheduled...

...and that is before its deadline.

First, we observe that under these assumptions the processor state at time t1 is "relevant" according to the notion of relevance underlying the EDF transformation, namely relevant_pstate.

Since t1 is relevant, we conclude that a search for a relevant state succeeds (if nothing else, it finds t1).

For rewriting purposes, we observe that the search_arg operation within find_swap_candidate yields the final result of find_swap_candidate.

There is a job that is scheduled at the time that find_swap_candidate returns, and that job arrives no later than at time t1.

Since we are considering a uniprocessor model, only one job is scheduled at a time. Hence once we know that a job is scheduled at the time that find_swap_candidate returns, we can conclude that it arrives not later than at time t1.

We observe that find_swap_candidate returns a value within a known finite interval.

For convenience, since we often only require the lower bound on the interval, we re-state it as a corollary.

The following lemma is a key step of the overall proof: the job scheduled at the time found by find_swap_candidate has the property that it has a deadline that is no later than that of any other job in the window given by time t1 and the deadline of the job scheduled at time t1.

As a special case of the above lemma, we observe that the job scheduled at the time given by find_swap_candidate in particular has a deadline no later than the job scheduled at time t1.

In the next section, we analyze properties of make_edf_at, which we abbreviate as "mea" in the following.

For any given type of jobs...

...consider an ideal uniprocessor schedule...

...that is well-behaved...

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

Since we will require this fact repeatedly, we briefly observe that, since no scheduled job misses its deadline, if a job is scheduled at some time t, then its deadline is later than t.

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

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

First, we observe that in sched' jobs still don't execute past completion.

Importantly, make_edf_at does not introduce any deadline misses, which is a crucial step in the EDF optimality argument.

As a result, we may conclude that any job scheduled at a time t has a deadline later than t.

Next comes a big step in the optimality proof: we observe that make_edf_at indeed ensures that EDF_at holds at time t_edf in sched'. As this is a larger argument, we proceed by case analysis and first establish a couple of helper lemmas in the following section.

Let j_orig denote the job scheduled in sched at time t_edf, let j_edf denote the job scheduled in sched' at time t_edf, and let j' denote any job scheduled in sched' at some time t' after t_edf...

... and that arrives before time t_edf.

We begin by observing three simple facts that will be used repeatedly in the case analysis. First, the deadline of j_orig is later than t_edf.

Second, by the definition of sched', j_edf is scheduled in sched at the time returned by find_swap_candidate.

Third, the deadline of j_edf is no later than the deadline of j_orig.

With the setup in place, we are now ready to begin the case analysis. First, we consider the simpler case where t' is no earlier than the deadline of j_orig. This case is simpler because t' being no earlier than j_orig's deadline implies that j' has deadline no earlier than j_orig (since no scheduled job in sched misses a deadline), which in turn has a deadline no earlier than j_edf.

Next, we consider the more difficult case, where t' is before the deadline of j_orig.

Finally, putting the preceding cases together, we obtain the result that make_edf_at establishes EDF_at at time t_edf.

We observe that make_edf_at maintains the property that jobs must arrive to execute.

We connect the fact that a job is scheduled in sched' to the fact that it must be scheduled somewhere in sched, too, since make_edf_at does not introduce any new jobs.

Conversely, if a job is scheduled in sched, it is also scheduled somewhere in sched' since make_edf_at does not lose any jobs.

Next, we observe that if all jobs in sched come from a given arrival sequence, then that's still the case in sched', too.

For given arrival sequence,...

...if all jobs in sched come from the arrival sequence,...

...then all jobs in sched' do, too.

For the final claim, assume that EDF_at already holds everywhere prior to time t_edf, i.e., that sched consists of an EDF prefix.

We establish a key property of make_edf_at: not only does it ensure EDF_at at time t_edf, it also maintains the fact that the schedule has an EDF prefix prior to time t_edf. In other words, it grows the EDF prefix by one time unit.

In the following section, we establish properties of edf_transform_prefix.

For any given type of jobs...

...consider an ideal uniprocessor schedule...

...that is well-behaved...

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

Consider any point in time, denoted horizon, and...

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

To start, we observe that sched' is still well-behaved and without deadline misses.

Because it is needed frequently, we extract the second clause of the above conjunction as a corollary.

We similarly observe that the absence of deadline misses implies that any scheduled job must have a deadline at a time later then when it is scheduled.

Since no jobs are lost or added to the schedule by edf_transform_prefix, we if a job is scheduled in the transformed schedule, then it is also scheduled at some point in the original schedule.

Conversely, if a job is scheduled in the original schedule, it is also scheduled at some point in the transformed schedule.

Next, we note that edf_transform_prefix maintains the property that all jobs stem from a given arrival sequence.

For any arrival sequence,...

...if all jobs in the original schedule come from the arrival sequence,...

...then all jobs in the transformed schedule still come from the same arrival sequence.

We establish the key property of edf_transform_prefix: that it indeed ensures that the resulting schedule ensures the EDF invariant up to the given horizon.

Finally, we observe that edf_transform_prefix is prefix-stable, which allows us to replace an earlier horizon with a later horizon. Note: this is in a separate section because we need edf_prefix_jobs_must_arrive generalized for any schedule.

For any given type of jobs...

...consider an ideal uniprocessor schedule...

...that is well-behaved...

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

In the following section, we finally establish properties of the overall EDF transformationedf_transform.

For any given type of jobs...

...consider an ideal uniprocessor schedule...

...that is well-behaved...

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

In the following, let sched_edf denote the EDF schedule obtained by transforming the given reference schedule.

We begin with a simple lemma relating sched_edf to its definition that allows us to easily look at any finite prefix of the EDF-transformed scheduled.

From this, we move on to the defining property of the transformation: the resulting schedule is actually an EDF schedule.

Next, we observe that completed jobs still don't execute in the resulting EDF schedule. This observation is needed to establish that the resulting EDF schedule is valid.

Similarly, we observe that no job is scheduled prior to its arrival.

We next establish the second key property: in the transformed EDF schedule, no scheduled job misses a deadline.

We observe that no new jobs are introduced: any job scheduled in the EDF schedule were also present in the reference schedule.

Conversely, we observe that no jobs are lost: any job scheduled in the reference schedule is also present in the EDF schedule.

Next, we note that edf_transform maintains the property that all jobs stem from a given arrival sequence.

For any arrival sequence,...

...if all jobs in the original schedule come from the arrival sequence,...

...then all jobs in the transformed EDF schedule still come from the same arrival sequence.

Finally, we state the theorems that jointly make up the EDF optimality claim.

For any given type of jobs...

... consider an arbitrary valid job arrival sequence ...

... and an ideal uniprocessor schedule...

... that corresponds to the given arrival sequence.

In the following, let equivalent_edf_schedule denote the schedule that results from the EDF transformation.

Suppose no job scheduled in the given reference schedule misses a deadline.

Then the resulting EDF schedule is a valid schedule for the given arrival sequence...

...and no scheduled job misses its deadline.

Next, we strengthen the above "no deadline misses" claim by relating it not just to all scheduled jobs, but to all jobs in the given arrival sequence.

Suppose no job that's part of the arrival sequence misses a deadline in the given reference schedule.

Then no job that's part of the arrival sequence misses a deadline in the EDF schedule, either.