# Library prosa.model.task.preemption.limited_preemptive

Require Export prosa.model.task.preemption.parameters.

Require Export prosa.model.preemption.limited_preemptive.

Require Export prosa.model.preemption.limited_preemptive.

# Limited-Preemptive Task Model

## Model Validity

Consider any type of tasks with WCET bounds and given preemption points ...

... and any type of jobs associated with these tasks, ...

... where each job has an arrival time, an execution cost, and some
preemption points.

Consider any arrival sequence.

Consider an arbitrary task set ts.

First, we describe structural properties that a sequence of preemption
points of a task should satisfy.
(1) We require the sequence of preemption points to contain the beginning ...

Definition task_beginning_of_execution_in_preemption_points :=

∀ tsk, tsk \in ts → first0 (task_preemption_points tsk) = 0.

∀ tsk, tsk \in ts → first0 (task_preemption_points tsk) = 0.

... and (2) the end of execution.

Definition task_end_of_execution_in_preemption_points :=

∀ tsk, tsk \in ts → last0 (task_preemption_points tsk) = task_cost tsk.

∀ tsk, tsk \in ts → last0 (task_preemption_points tsk) = task_cost tsk.

(3) Furthermore, we require the sequence of preemption points to be a
non-decreasing sequence.

Definition nondecreasing_task_preemption_points :=

∀ tsk, tsk \in ts → nondecreasing_sequence (task_preemption_points tsk).

∀ tsk, tsk \in ts → nondecreasing_sequence (task_preemption_points tsk).

(4) We also require the number of nonpreemptive segments of a job to be
equal to the number of nonpreemptive segments of its task. Note that
some of nonpreemptive segments of a job can have zero length;
nonetheless the number of segments should match.

Definition consistent_job_segment_count :=

∀ j,

arrives_in arr_seq j →

size (job_preemptive_points j) = size (task_preemption_points (job_task j)).

∀ j,

arrives_in arr_seq j →

size (job_preemptive_points j) = size (task_preemption_points (job_task j)).

(5) We require the lengths of the nonpreemptive segments of a job to be
bounded by the lengths of the corresponding segments of its task.

Definition job_respects_segment_lengths :=

∀ j n,

arrives_in arr_seq j →

nth 0 (distances (job_preemptive_points j)) n

≤ nth 0 (distances (task_preemption_points (job_task j))) n.

∀ j n,

arrives_in arr_seq j →

nth 0 (distances (job_preemptive_points j)) n

≤ nth 0 (distances (task_preemption_points (job_task j))) n.

(6) Lastly, we ban empty nonpreemptive segments at the task level.

Definition task_segments_are_nonempty :=

∀ tsk n,

(tsk \in ts) →

n < size (distances (task_preemption_points tsk)) →

ε ≤ nth 0 (distances (task_preemption_points tsk)) n.

∀ tsk n,

(tsk \in ts) →

n < size (distances (task_preemption_points tsk)) →

ε ≤ nth 0 (distances (task_preemption_points tsk)) n.

We define a valid task-level model with fixed preemption points as the
conjunction of the hypotheses above.

Definition valid_fixed_preemption_points_task_model :=

task_beginning_of_execution_in_preemption_points ∧

task_end_of_execution_in_preemption_points ∧

nondecreasing_task_preemption_points ∧

consistent_job_segment_count ∧

job_respects_segment_lengths ∧

task_segments_are_nonempty.

task_beginning_of_execution_in_preemption_points ∧

task_end_of_execution_in_preemption_points ∧

nondecreasing_task_preemption_points ∧

consistent_job_segment_count ∧

job_respects_segment_lengths ∧

task_segments_are_nonempty.

Finally, a model with fixed preemption points is valid if it is both valid
a the job and task levels.

Definition valid_fixed_preemption_points_model :=

valid_limited_preemptions_job_model arr_seq ∧

valid_fixed_preemption_points_task_model.

End ValidModelWithFixedPreemptionPoints.

valid_limited_preemptions_job_model arr_seq ∧

valid_fixed_preemption_points_task_model.

End ValidModelWithFixedPreemptionPoints.

## Run-to-Completion Threshold

Consider any type of tasks with WCET bounds and fixed preemption points.

Given fixed preemption points, no job can be preempted after a job
reaches its last non-preemptive segment. Thus, we can set the task-level
run-to-completion threshold to task_cost tsk - (task_last_nonpr_seg tsk - ε),
which safely bounds job_cost j - (job_last_nonpr_seg j - ε).

Definition limited_preemptions_rtc_threshold : TaskRunToCompletionThreshold Task :=

fun tsk : Task ⇒ task_cost tsk - (task_last_nonpr_segment tsk - ε).

End TaskRTCThresholdLimitedPreemptions.

fun tsk : Task ⇒ task_cost tsk - (task_last_nonpr_segment tsk - ε).

End TaskRTCThresholdLimitedPreemptions.