Library prosa.analysis.facts.readiness.sequential

Require Export prosa.analysis.definitions.readiness.
Require Export prosa.analysis.definitions.work_bearing_readiness.
Require Export prosa.analysis.facts.behavior.completion.
Require Export prosa.analysis.facts.model.task_arrivals.

Throughout this file, we assume the sequential task readiness model, which means that a job is ready to execute only if all prior jobs of the same task have completed.

Require Export prosa.model.readiness.sequential.

In this section, we show some useful properties of the sequential task readiness model.

Section SequentialTasksReadiness.

Consider any type of job associated with any type of tasks ...

  Context {Job : JobType}.
  Context {Task : TaskType}.
  Context `{JobTask Job Task}.
  Context `{JobArrival Job}.
  Context `{JobCost Job}.

... and any kind of processor state.

Context {PState : ProcessorState Job}.

Consider any arrival sequence with consistent arrivals.

Variable arr_seq : arrival_sequence Job.
Hypothesis H_arrival_times_are_consistent : consistent_arrival_times arr_seq.

Recall that we assume sequential tasks.

#[local] Instance sequential_readiness_instance : JobReady Job PState :=
sequential_ready_instance arr_seq.

First, we observe that the sequential readiness model indeed lives up to its name.

Fact sequential_readiness_is_sequential :
sequential_readiness sequential_readiness_instance arr_seq.

Consider any valid schedule of arr_seq.

Variable sched : schedule PState.
Hypothesis H_valid_schedule : valid_schedule sched arr_seq.

Consider an FP policy that indicates a reflexive higher-or-equal priority relation.

Context {FP : FP_policy Task}.
Hypothesis H_priority_is_reflexive : reflexive_task_priorities FP.

We show that the sequential readiness model is non-clairvoyant.

Fact sequential_readiness_nonclairvoyance :
nonclairvoyant_readiness sequential_readiness_instance.

Next, we show that the sequential readiness model ensures that tasks are sequential. That is, that jobs of the same task execute in order of their arrival.

Lemma sequential_readiness_implies_sequential_tasks :
sequential_tasks arr_seq sched.

Finally, we show that the sequential readiness model is a work-bearing readiness model. That is, if a job j is pending but not ready at a time instant t, then there exists another (earlier) job of the same task that is pending and ready at time t.

Lemma sequential_readiness_implies_work_bearing_readiness :
work_bearing_readiness arr_seq sched.

End SequentialTasksReadiness.