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# Library prosa.analysis.definitions.blocking_bound.elf

Require Export prosa.model.task.preemption.parameters.
Require Export prosa.model.task.arrival.curves.
Require Import prosa.model.priority.elf.

# Blocking Bound for ELF

In this section, we define a bound on the length of a non-preemptive segment of a lower-priority job (w.r.t. a task tsk) under ELF scheduling.
Section MaxNPSegmentBlockingBound.
Consider any type of tasks ...
Context {Job : JobType}{Task : TaskType}.
Context `{TaskCost Task}.
Context `{TaskMaxNonpreemptiveSegment Task}.
... with relative priority points.
Context `{PriorityPoint Task}.
Consider an arbitrary task set ts.
Variable ts : list Task.
Let max_arrivals be a family of arrival curves.
Context `{MaxArrivals Task}.
Consider any FP policy.
Context {FP : FP_policy Task}.
Only other tasks that potentially release non-zero-cost jobs are relevant, so we define a predicate to exclude pathological cases.
Let blocking_relevant (tsk_o : Task) :=
(max_arrivals tsk_o ε > 0) && (task_cost tsk_o > 0).
Now we define a predicate to only include blocking_relevant tasks that have a lower priority than tsk.
Let lp_tsk_blocking_relevant (tsk : Task) (tsk_o : Task) :=
(hp_task tsk tsk_o) && (blocking_relevant tsk_o).
Next we define a predicate for filtering blocking_relevant tasks that have the same priority as tsk.
Let ep_tsk_blocking_relevant (tsk : Task) (tsk_o : Task) (A : duration) :=
let ep_tsk_j_blocking_relevant := (task_priority_point tsk_o > A%:R + task_priority_point tsk)%R in
(ep_task tsk tsk_o) && ep_tsk_j_blocking_relevant && (blocking_relevant tsk_o).
For a job of a given task tsk, given the relative arrival offset A within its busy window, we define the following blocking bound. This bound assumes that task are independent (i.e., it does not account for possible blocking due to locking protocols).
Definition blocking_bound (tsk : Task) (A : duration) :=
\max_(tsk_o <- ts | lp_tsk_blocking_relevant tsk tsk_o || ep_tsk_blocking_relevant tsk tsk_o A)
(task_max_nonpreemptive_segment tsk_o - ε).
End MaxNPSegmentBlockingBound.