Library prosa.model.schedule.tdma

Require Export prosa.model.task.concept.
Require Export prosa.util.seqset.
Require Export prosa.util.rel.
From mathcomp Require Import div.

Throughout this file, we assume ideal uniprocessor schedules.

TDMA Scheduling

The TDMA scheduling policy is based on two properties. (1) Each task has a fixed, reserved time slot for execution; (2) These time slots are ordered in sequence to form a TDMA cycle, which repeats along the timeline. An example of TDMA schedule is illustrated in the following.
    | s1 |  s2  |s3| s1 |  s2  |s3|...
    0                                            t

Task Parameter for TDMA Scheduling

We define the TDMA policy as follows.
Section TDMAPolicy.

  Variable Task: eqType.
With each task, we associate the duration of the corresponding TDMA slot.
  Definition TDMA_slot := Task duration.

Moreover, within each TDMA cycle, task slots are ordered according to some relation (i.e, slot_order slot1 slot2 means that slot1 comes before slot2 in a TDMA cycle).
  Definition TDMA_slot_order := rel Task.

End TDMAPolicy.

We introduce slots and the slot order as task parameters.
Class TDMAPolicy (T : TaskType) :=
    task_time_slot : TDMA_slot T;
    slot_order : TDMA_slot_order T

TDMA Policy Validity

First, we define the properties of a valid TDMA policy.
Section ValidTDMAPolicy.

  Context {Task : eqType}.

Consider any task set ts...
  Variable ts : {set Task}.

...and a TDMA policy.
  Context `{TDMAPolicy Task}.

Time slot order must be transitive...
..., totally ordered over the task set...
  Definition total_slot_order :=
    total_over_list slot_order ts.

... and antisymmetric over task set.
A valid time slot must be positive

TDMA Cycles, Sots, and Offsets

In this section, we define key TDMA concepts.
Section TDMADefinitions.

  Context {Task : eqType}.

Consider any task set ts...
  Variable ts : {set Task}.

...and a TDMA policy.
  Context `{TDMAPolicy Task}.

We define the TDMA cycle as the sum of all the tasks' time slots
  Definition TDMA_cycle:=
    \sum_(tsk <- ts) task_time_slot tsk.

We define the function returning the slot offset for each task: i.e., the distance between the start of the TDMA cycle and the start of the task time slot.
The following function tests whether a task is in its time slot at instant t.

TDMA Schedule Validity

Section TDMASchedule.

  Context {Task : TaskType} {Job : JobType}.

  Context `{JobArrival Job} `{JobCost Job} `{JobReady Job (option Job)} `{JobTask Job Task}.

Consider any job arrival sequence...
  Variable arr_seq : arrival_sequence Job.

..., any uniprocessor ideal schedule ...
... and any sporadic task set.
  Variable ts: {set Task}.

  Context `{TDMAPolicy Task}.

In order to characterize a TDMA policy, we first define whether a job is executing its TDMA slot at time t.
  Definition job_in_time_slot (job:Job) (t:instant):=
    task_in_time_slot ts (job_task job) t.

We say that a TDMA policy is respected by the schedule iff 1. when a job is scheduled at time t, then the corresponding task is also in its own time slot...
2. when a job is backlogged at time t, the corresponding task isn't in its own time slot or another previous job of the same task is scheduled
  Definition backlogged_implies_not_in_slot_or_other_job_sched j t:=
    backlogged sched j t
    ¬ job_in_time_slot j t
     j_other, arrives_in arr_seq j_other
                    job_arrival j_other < job_arrival j
                    job_task j = job_task j_other
                    scheduled_at sched j_other t.

  Definition respects_TDMA_policy:=
     (j:Job) (t:instant),
      arrives_in arr_seq j
      sched_implies_in_slot j t
      backlogged_implies_not_in_slot_or_other_job_sched j t.
End TDMASchedule.