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Sophie Quinton
rtproofs
Commits
0be6dd67
Commit
0be6dd67
authored
Sep 20, 2018
by
Sergey Bozhko
Committed by
Sergey Bozhko
Apr 05, 2019
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Add notion of arrival curves
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Require
Import
rt
.
util
.
all
.
Require
Import
rt
.
model
.
arrival
.
basic
.
arrival_sequence
rt
.
model
.
arrival
.
basic
.
task_arrival
.
From
mathcomp
Require
Import
ssreflect
ssrbool
eqtype
ssrnat
seq
div
.
(* In this section, we define the notion of arrival curves, which
can be used to reason about the frequency of job arrivals. *)
Module
ArrivalCurves
.
Import
ArrivalSequence
TaskArrival
.
Section
DefiningArrivalCurves
.
Context
{
Task
:
eqType
}.
Context
{
Job
:
eqType
}.
Variable
job_task
:
Job
>
Task
.
(* Consider any job arrival sequence. *)
Variable
arr_seq
:
arrival_sequence
Job
.
(* Recall the job arrivals of tsk in a given interval [t1, t2). *)
Let
arrivals_of_tsk
tsk
:
=
arrivals_of_task_between
job_task
arr_seq
tsk
.
Let
num_arrivals_of_tsk
tsk
:
=
num_arrivals_of_task
job_task
arr_seq
tsk
.
(* First, we define what constitutes an arrival bound for task tsk. *)
Section
ArrivalBound
.
(* Let max_arrivals denote any function that takes an interval length and
returns the associated number of job arrivals of task.
(This corresponds to the eta+ function in the literature.) *)
Variable
max_arrivals
:
Task
>
time
>
nat
.
(* Then, we say that max_arrivals is an arrival bound iff, for any interval [t1, t2),
[num_arrivals (t2  t1)] bounds the number of jobs of tsk that arrive in that interval. *)
Definition
is_arrival_bound
(
tsk
:
Task
)
:
=
forall
(
t1
t2
:
time
),
t1
<=
t2
>
num_arrivals_of_tsk
tsk
t1
t2
<=
max_arrivals
tsk
(
t2

t1
).
(* We say that max_arrivals is an arrival bound for taskset ts
iff max_arrival is an arrival bound for any task from ts. *)
Definition
is_arrival_bound_for_taskset
(
ts
:
seq
Task
)
:
=
forall
(
tsk
:
Task
),
tsk
\
in
ts
>
is_arrival_bound
tsk
.
(* We provide the notion of an arrival curve that equals 0 for the empty interval. *)
Definition
zero_arrival_curve
(
tsk
:
Task
)
:
=
max_arrivals
tsk
0
=
0
.
(* Next, we provide the notion of a monotonic arrival curve. *)
Definition
monotonic_arrival_curve
(
tsk
:
Task
)
:
=
monotone
(
max_arrivals
tsk
)
leq
.
(* We say that max_arrivals is a proper arrival curve for task tsk iff
[max_arrivals tsk] is an arrival bound for task tsk and [max_arrivals tsk]
is a monotonic function that equals 0 for the empty interval delta = 0. *)
Definition
proper_arrival_curve
(
tsk
:
Task
)
:
=
is_arrival_bound
tsk
/\
zero_arrival_curve
tsk
/\
monotonic_arrival_curve
tsk
.
(* We say that max_arrivals is a family of proper arrival curves iff
for all tsk in ts [max_arrival tsk] is a proper arrival curve. *)
Definition
family_of_proper_arrival_curves
(
ts
:
seq
Task
)
:
=
forall
(
tsk
:
Task
),
tsk
\
in
ts
>
proper_arrival_curve
tsk
.
End
ArrivalBound
.
(* Next, we define the notion of a separation bound for task tsk, i.e., the smallest
interval length in which a certain number of jobs of tsk can be spawned. *)
Section
SeparationBound
.
(* Let min_length denote any function that takes a number of jobs and
returns an associated interval length.
(This corresponds to the delta function in the literature.) *)
Variable
min_length
:
Task
>
nat
>
time
.
(* Then, we say that min_length is a separation bound iff, for any number of jobs
of tsk, min_separation lowerbounds the minimum interval length in which that number
of jobs can be spawned. *)
Definition
is_separation_bound
tsk
:
=
forall
t1
t2
,
t1
<=
t2
>
min_length
tsk
(
num_arrivals_of_tsk
tsk
t1
t2
)
<=
t2

t1
.
End
SeparationBound
.
End
DefiningArrivalCurves
.
End
ArrivalCurves
.
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