From rt.restructuring.analysis.transform Require Export prefix swap.
From rt.util Require Export search_arg.
(** In this file we define the "EDF-ification" of a schedule, the
operation at the core of the EDF optimality proof. *)
Section EDFTransformation.
(* Consider any given type of jobs... *)
Context {Job : JobType} `{JobCost Job} `{JobDeadline Job} `{JobArrival Job}.
(* ... and ideal uniprocessor schedules. *)
Let PState := ideal.processor_state Job.
Let SchedType := schedule (PState).
(* We say that a state s1 "has an earlier or equal deadline" than a
state s2 if the job scheduled in state s1 has has an earlier or
equal deadline than the job scheduled in state s2. This function
is never used on idle states, so the default values are
irrelevant. *)
Definition earlier_deadline (s1 s2: PState) :=
(oapp job_deadline 0 s1) <= (oapp job_deadline 0 s2).
(* We say that a state is relevant (for the purpose of the
transformation) if it is not idle and if the job scheduled in it
has arrived prior to some given reference time. *)
Definition relevant_pstate (reference_time: instant) (s: PState) :=
match s with
| None => false
| Some j' => job_arrival j' <= reference_time
end.
(* Next, we define a central element of the "EDF-ification"
procedure: given a job scheduled at a time t1, find a later time
t2 before the job's deadlineat which a job with an
earlier-or-equal deadline is scheduled. In other words, find a
job that causes the [EDF_at] property to be violated at time
t1. *)
Definition find_swap_candidate (sched: SchedType) (t1: instant) (j: Job) :=
if search_arg sched (relevant_pstate t1) earlier_deadline t1 (job_deadline j) is Some t
then t
else 0.
(* The point-wise EDF-ification procedure: given a schedule and a
time t1, ensure that the schedule satisfies [EDF_at] at time
t1. *)
Definition make_edf_at (sched: SchedType) (t1: instant): SchedType :=
match sched t1 with
| None => sched (* leave idle instants alone *)
| Some j =>
let
t2 := find_swap_candidate sched t1 j
in swapped sched t1 t2
end.
(* To transform a finite prefix of a given reference schedule, apply
[make_edf_at] to every point up to the given finite horizon. *)
Definition edf_transform_prefix (sched: SchedType) (horizon: instant): SchedType :=
prefix_map sched make_edf_at horizon.
(* Finally, a full EDF schedule (i.e., one that satisfies [EDF_at]
at any time) is obtained by first computing an EDF prefix up to
and including the requested time t, and by then looking at the
last point of the prefix. *)
Definition edf_transform (sched: SchedType) (t: instant): ideal.processor_state Job :=
let
edf_prefix := edf_transform_prefix sched t.+1
in edf_prefix t.
End EDFTransformation.