Correctness of the work-conservation transformation

This file contains the main argument of the work-conservation proof, starting with an analysis of the individual functions that drive the work-conservation transformation of a given reference schedule and ending with the proofs of individual properties of the obtained work-conserving schedule. In order to discuss the correctness of the work-conservation transformation at a high level, we first need a set of lemmas about the inner parts of the procedure.

We assume ideal uni-processor schedules.

We assume the basic (i.e., Liu & Layland) readiness model under which any pending job is ready.

Consider any type of jobs with arrival times, costs, and deadlines...

...and an arbitrary arrival sequence.

We introduce the notion of work-conservation at a given time t. The definition is based on the concept of readiness of a job, and states that the presence of a ready job implies that the processor is not idle.

First, we prove some useful properties about the most fundamental operation of the work-conservation transformation: swapping two processor states t1 and fsc, with fsc being a valid swap candidate of t1.

Consider an ideal uniprocessor schedule...

...in which jobs must be ready to execute.

Consider an arbitrary time instant t1.

Let us define fsc as the result of the search for a swap candidate starting from t1...

...and sched' as the schedule resulting from the swap.

First, we show that, in any case, the result of the search will yield an instant that is in the future (or, in case of failure, equal to t1).

Also, we show that the search will not yield jobs that arrive later than the given reference time.

Next, we extend the previous lemma by stating that no job in the transformed schedule is scheduled before its arrival.

Finally we show that, in the transformed schedule, jobs are scheduled only if they are ready.

In the following section, we put our attention on the point-wise transformation performed at each point in time prior to the horizon.

Consider an ideal uniprocessor schedule...

...and take an arbitrary point in time...

...we define sched' as the resulting schedule after one point-wise transformation.

We start by proving that the point-wise transformation can only lead to higher service for a job at a given time. This is true because we swap only idle processor states with ones in which a job is scheduled.

Next, we show that any ready job in the transformed schedule must be ready also in the original one, since the transformation can only lead to higher service.

Since the search for a swap candidate is performed until the latest deadline among all the jobs arrived before the reference time, we need to show that the computed deadline is indeed the latest.

Next, we want to show that, if a job arriving from the arrival sequence is ready at some instant, then the point-wise transformation is guaranteed to find a job to swap with. We will proceed by doing a case analysis, and show that it is impossible that a swap candidate is not found.

We need to assume that, in the original schedule, all the deadlines of the jobs coming from the arrival sequence are met, in order to be sure that a ready job will be eventually scheduled.

We define max_dl as the maximum deadline for all jobs arrived before t.

Next, we define search_result as the result of the search for a swap candidate. In order to take the first result, it is sufficient to define the ordering function as a constant false.

First, we consider the case in which the procedure finds a job to swap with.

Assuming that the processor is idle at time t...

...let t_swap be a time instant found by the search procedure.

We show that, since the search only yields relevant processor states, a job is found.

Conversely, we prove that assuming that the search yields no result brings to a contradiction.

Consider a job that arrives in the arrival sequence, and assume that it is ready at time t in the transformed schedule.

Moreover, assume the search for a swap candidate yields nothing.

First, note that, since nothing was found, it means there is no relevant processor state between t and max_dl.

Since j is ready at time t, then it must be incomplete.

And since j is incomplete and meets its deadline, the deadline of j is in the future.

On the other hand, since we know that there is no relevant state between t and max_dl, then it must be the case that j is never scheduled in this period, and hence gets no service.

Combining the previous lemmas, we can deduce that j misses its deadline.

The fact that j misses its deadline contradicts the fact that all deadlines of jobs coming from the arrival sequence are met. We have a contradiction.

Next, we show that make_wc_at always manages to establish the work-conservation property at the given time. Using the above case analysis, we can conclude that the presence of a ready job always leads to a valid swap.

Next we prove that, given a schedule that respects the work-conservation property until t-1, applying the point-wise transformation at time t will extend the property until t.

We now show that the point-wise transformation does not introduce any new job that does not come from the arrival sequence.

We also show that the point-wise transformation does not schedule jobs in instants in which they are not ready.

Finally, we show that the point-wise transformation does not introduce deadline misses.

In the following section, we proceed by proving some useful properties respected by the partial schedule obtained by applying the work-conservation transformation up to an arbitrary horizon.

Consider an ideal uniprocessor schedule.

We start by proving that the transformation performed with two different horizons will yield two schedules that are identical until the earlier horizon.

Consider two horizons...

...and assume w.l.o.g. that they are ordered...

...we define two schedules, resulting from the transformation performed, respectively, until the first and the second horizon.

Then, we show that the two schedules are guaranteed to be equal until the earlier horizon.

Next, we show that repeating the point-wise transformation up to a given horizon does not introduce any deadline miss.

Assuming that all deadlines of jobs coming from the arrival sequence are met...

...let us define sched' as the schedule resulting from the full work-conservation transformation. Note that, if the schedule is sampled at time t, the transformation is performed until t+1.

Consider a job from the arrival sequence.

We show that, in the transformed schedule, the service of the job is always greater or equal than in the original one, at any given time.

Finally, it follows directly that the transformed schedule cannot introduce a deadline miss for any job from the arrival sequence.

Next, consider a given time, used as horizon for the transformation...

...and let us call sched' the schedule resulting from the transformation performed until h.

We prove that sched' will never introduce jobs not coming from the arrival sequence.

Similarly, we can show that sched' will only schedule jobs if they are ready.

Finally, we can leverage all the previous results to prove statements about the full work-conservation transformation.

We assume the basic (i.e., Liu & Layland) readiness model under which any pending job is ready.

Consider any type of jobs with arrival times, costs, and deadlines...

...an arbitrary valid arrival sequence...

...and an ideal uniprocessor schedule...

...in which jobs come from the arrival sequence, and must be ready to execute...

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

Let us call sched_wc the schedule obtained after applying the work-conservation transformation.

First, we show that any scheduled job still comes from the arrival sequence.

Similarly, jobs are only scheduled if they are ready.

Also, no deadline misses are introduced.

Finally, we can show that the transformation leads to a schedule in which the processor is not idle if a job is ready.

We can easily extend the previous lemma to obtain the definition of a work-conserving schedule.

Ultimately, we can show that the work-conservation transformation maintains all the properties of validity, does not introduce new deadline misses, and establishes the work-conservation property.