Library prosa.model.processor.ideal

The Ideal Uniprocessor Model

In this module, we define a central piece of the Prosa model: the notion of an ideal uniprocessor state. The word "ideal" here refers to the complete absence of runtime overheads or any other complications. In an ideal uniprocessor schedule, there are only two possible cases: at a given time, either a specific job is scheduled and makes unit-progress, or the processor is idle. To model this, we simply reuse the standard option type from the Coq standard library.

Section State.

Consider any type of jobs.

Variable Job: JobType.

We connect the notion of an ideal processor state with the generic interface for the processor-state abstraction in Prosa by declaring a so-called instance of the ProcessorState typeclass.

  Program Definition processor_state : ProcessorState Job :=
    {|

We define the ideal "processor state" as an option Job, which means that it is either Some j (where j is a Job) or None (which we use to indicate an idle instant).

State := option Job;

As this is a uniprocessor model, cores are implicitly defined as the unit type containing a single element as a placeholder. We say that a given job j is scheduled in a given state s iff s is Some j.

scheduled_on j s (_ : unit) := s == Some j;

Any state of an ideal processor provides exactly one unit of supply.

supply_on s (_ : unit) := 1;

We say that a given job j receives service in a given state s iff s is Some j.

      service_on j s (_ : unit) := if s == Some j then 1 else 0;
    |}.
  Next Obligation.
    by move⇒ j s ?; rewrite /nat_of_bool; case: ifP ⇒ // ? /negP[].
  Qed.
  Next Obligation.
    by move⇒ j s ?; rewrite /nat_of_bool; case: ifP ⇒ // ? /negP[].
  Qed.

End State.

Idle Instants

In this section, we define the notion of idleness for ideal uniprocessor schedules.

Section IsIdle.

Consider any type of jobs...

Context {Job : JobType}.
Variable arr_seq : arrival_sequence Job.

... and any ideal uniprocessor schedule of such jobs.

Variable sched : schedule ((*ideal*)processor_state Job).

We say that the processor is idle at time t iff there is no job being scheduled.

Definition ideal_is_idle (t : instant) := sched t == None.

End IsIdle.