Library prosa.model.priority.classes

Require Export prosa.model.task.concept.
Require Export prosa.util.rel.
Require Export prosa.util.list.

From mathcomp Require Export seq.

The FP, JLFP, and JLDP Priority Classes

In this module, we define the three well-known classes of priority relations: (1) fixed-priority (FP) policies, (2) job-level fixed-priority (JLFP) polices, and (3) job-level dynamic-priority (JLDP) policies, where (2) is a subset of (3), and (1) a subset of (2).
As a convention, we use "hep" to mean "higher or equal priority."
We define an FP policy as a relation among tasks, ...
Class FP_policy (Task: TaskType) := hep_task : rel Task.

... a JLFP policy as a relation among jobs, and ...
Class JLFP_policy (Job: JobType) := hep_job : rel Job.

... a JLDP policy as a relation among jobs that may vary over time.
NB: The preceding definitions currently make it difficult to express priority policies in which the priority of a job at a given time varies depending on the preceding schedule prefix (e.g., least-laxity first). That is, there is room for an even more general notion of a schedule-dependent JLDP policy, where the priority relation among jobs may vary depending both on time and the schedule prefix prior to a given time. This is left to future work.

Automatic FP ➔ JLFP ➔ JLDP Conversion

Since there are natural interpretations of FP and JLFP policies as JLFP and JLDP policies, respectively, we define conversions that express these generalizations. In practice, this means that Coq will be able to automatically satisfy a JLDP assumption if a JLFP or FP policy is in scope.
First, any FP policy can be interpreted as an JLFP policy by comparing jobs according to the priorities of their respective tasks.
Instance FP_to_JLFP (Job: JobType) (Task: TaskType)
         `{JobTask Job Task} `{FP_policy Task} : JLFP_policy Job :=
  fun j1 j2hep_task (job_task j1) (job_task j2).

Second, any JLFP policy implies a JLDP policy that simply ignores the time parameter.
Instance JLFP_to_JLDP (Job: JobType) `{JLFP_policy Job} : JLDP_policy Job :=
  fun _ j1 j2hep_job j1 j2.

Properties of Priority Policies

In the following section, we define key properties of common priority policies that proofs often depend on.

Section Priorities.

Consider any type of tasks ...
  Context {Task : TaskType}.
  Context `{TaskCost Task}.

... and any type of jobs associated with these tasks, ...
  Context {Job : JobType}.
  Context `{JobTask Job Task}.

.. and assume that jobs have a cost and an arrival time.
  Context `{JobArrival Job}.
  Context `{JobCost Job}.

In the following section, we define properties of JLDP policies, and by extension also properties of FP and JLFP policies.
  Section JLDP.

Consider any JLDP policy.
    Context `{JLDP_policy Job}.

We define what it means for a JLDP policy to be reflexive, transitive, and total. Note that these definitions, although phrased in terms of a given JLDP policy, can also be used for JLFP and FP policies due to the above-defined conversion instances.
A JLDP policy is reflexive if the relation among jobs is reflexive at every point in time.
    Definition reflexive_priorities := t, reflexive (hep_job_at t).

A JLDP policy is transitive if the relation among jobs is transitive at every point in time.
    Definition transitive_priorities := t, transitive (hep_job_at t).

A JLDP policy is total if the relation among jobs is total at every point in time.
    Definition total_priorities := t, total (hep_job_at t).

  End JLDP.

Next, we define a property of JLFP policies.
  Section JLFP.

Consider any JLFP policy.
    Context `{JLFP_policy Job}.

Recall that jobs of a sequential task are necessarily executed in the order that they arrive.
An arbitrary JLFP policy, however, can violate the sequential tasks hypothesis. For example, consider two jobs j1, j2 of the same task such that job_arrival j1 < job_arrival j2. It is possible that a JLFP priority policy π will assign a higher priority to the second job π j2 j1 = true. But such a situation would contradict the natural execution order of sequential tasks. It is therefore sometimes necessary to restrict the space of JLFP policies to those that assign priorities in a way that is consistent with sequential tasks.
To this end, we say that a policy respects sequential tasks if, for any two jobs j1, j2 of the same task, job_arrival j1 job_arrival j2 implies π j1 j2 = true.
    Definition policy_respects_sequential_tasks :=
       j1 j2,
        job_task j1 == job_task j2
        job_arrival j1 job_arrival j2
        hep_job j1 j2.

  End JLFP.

Finally, we we define and observe two properties of FP policies.
  Section FP.

Consider any FP policy.
    Context `{FP_policy Task}.

To express the common assumption that task priorities are unique, we define whether the given FP policy is antisymmetric over a task set ts.
    Definition antisymmetric_over_taskset (ts : seq Task) :=
      antisymmetric_over_list hep_task ts.

Further, we observe that any FP_policy respects the sequential tasks hypothesis, meaning that later-arrived jobs of a task don't have higher priority than earlier-arrived jobs of the same task (assuming that task priorities are reflexive).
    Remark respects_sequential_tasks :
      reflexive_priorities policy_respects_sequential_tasks.
      moveREFL j1 j2 /eqP EQ LT.
      rewrite /hep_job /FP_to_JLFP EQ.
        by eapply (REFL 0).

  End FP.

End Priorities.

We add the above observation into the "Hint Database" basic_facts, so Coq will be able to apply it automatically.
Hint Resolve respects_sequential_tasks : basic_facts.