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Sophie Quinton
rtproofs
Commits
3fd984a4
Commit
3fd984a4
authored
Apr 04, 2019
by
Sergey Bozhko
Committed by
Sergey Bozhko
Apr 05, 2019
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Finish FP instantiation of abstract RTA
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model/schedule/uni/limited/fixed_priority/nonpr_reg/fixed/response_time_bound.v
...ited/fixed_priority/nonpr_reg/fixed/response_time_bound.v
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model/schedule/uni/limited/fixed_priority/nonpr_reg/floating/response_time_bound.v
...d/fixed_priority/nonpr_reg/floating/response_time_bound.v
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model/schedule/uni/limited/fixed_priority/nonpr_reg/nonpreemptive/response_time_bound.v
...ed_priority/nonpr_reg/nonpreemptive/response_time_bound.v
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model/schedule/uni/limited/fixed_priority/nonpr_reg/preemptive/response_time_bound.v
...fixed_priority/nonpr_reg/preemptive/response_time_bound.v
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model/schedule/uni/limited/fixed_priority/nonpr_reg/response_time_bound.v
...ni/limited/fixed_priority/nonpr_reg/response_time_bound.v
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model/schedule/uni/limited/fixed_priority/response_time_bound.v
...schedule/uni/limited/fixed_priority/response_time_bound.v
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model/schedule/uni/limited/fixed_priority/nonpr_reg/nonpreemptive/response_time_bound.v
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Require
Import
rt
.
util
.
all
.
Require
Import
rt
.
model
.
arrival
.
basic
.
job
rt
.
model
.
arrival
.
basic
.
task_arrival
rt
.
model
.
priority
.
Require
Import
rt
.
model
.
schedule
.
uni
.
service
rt
.
model
.
schedule
.
uni
.
workload
rt
.
model
.
schedule
.
uni
.
schedule
rt
.
model
.
schedule
.
uni
.
response_time
.
Require
Import
rt
.
model
.
schedule
.
uni
.
nonpreemptive
.
schedule
.
Require
Import
rt
.
model
.
schedule
.
uni
.
limited
.
schedule
rt
.
model
.
schedule
.
uni
.
limited
.
fixed_priority
.
nonpr_reg
.
response_time_bound
.
Require
Import
rt
.
model
.
arrival
.
curves
.
bounds
.
Require
Import
rt
.
analysis
.
uni
.
arrival_curves
.
workload_bound
.
Require
Import
rt
.
model
.
schedule
.
uni
.
limited
.
platform
.
nonpreemptive
.
From
mathcomp
Require
Import
ssreflect
ssrbool
eqtype
ssrnat
seq
path
fintype
bigop
.
(** * RTA for fully nonpreemptive FP model *)
(** In this module we prove the RTA theorem for the fully nonpreemptive FP model. *)
Module
RTAforFullyNonPreemptiveFPModelwithArrivalCurves
.
Import
Epsilon
Job
ArrivalCurves
TaskArrival
Priority
UniprocessorSchedule
NonpreemptiveSchedule
Workload
Service
FullyNonPreemptivePlatform
ResponseTime
MaxArrivalsWorkloadBound
LimitedPreemptionPlatform
RTAforFPwithBoundedNonpreemptiveSegmentsWithArrivalCurves
.
Section
Analysis
.
Context
{
Task
:
eqType
}.
Variable
task_cost
:
Task
>
time
.
Context
{
Job
:
eqType
}.
Variable
job_arrival
:
Job
>
time
.
Variable
job_cost
:
Job
>
time
.
Variable
job_task
:
Job
>
Task
.
(* Consider any arrival sequence with consistent, nonduplicate arrivals. *)
Variable
arr_seq
:
arrival_sequence
Job
.
Hypothesis
H_arrival_times_are_consistent
:
arrival_times_are_consistent
job_arrival
arr_seq
.
Hypothesis
H_arr_seq_is_a_set
:
arrival_sequence_is_a_set
arr_seq
.
(* Consider an arbitrary task set ts. *)
Variable
ts
:
list
Task
.
(* Assume that all jobs come from the task set... *)
Hypothesis
H_all_jobs_from_taskset
:
forall
j
,
arrives_in
arr_seq
j
>
job_task
j
\
in
ts
.
(* ...and the cost of a job cannot be larger than the task cost. *)
Hypothesis
H_job_cost_le_task_cost
:
cost_of_jobs_from_arrival_sequence_le_task_cost
task_cost
job_cost
job_task
arr_seq
.
(* Let max_arrivals be a family of proper arrival curves, i.e., for any task tsk in ts
[max_arrival tsk] is (1) an arrival bound of tsk, and (2) it is a monotonic function
that equals 0 for the empty interval delta = 0. *)
Variable
max_arrivals
:
Task
>
time
>
nat
.
Hypothesis
H_family_of_proper_arrival_curves
:
family_of_proper_arrival_curves
job_task
arr_seq
max_arrivals
ts
.
(* Let tsk be any task in ts. *)
Variable
tsk
:
Task
.
Hypothesis
H_tsk_in_ts
:
tsk
\
in
ts
.
(* Next, consider any uniprocessor nonpreemptive schedule of this arrival sequence...*)
Variable
sched
:
schedule
Job
.
Hypothesis
H_jobs_come_from_arrival_sequence
:
jobs_come_from_arrival_sequence
sched
arr_seq
.
Hypothesis
H_nonpreemptive_sched
:
is_nonpreemptive_schedule
job_cost
sched
.
(* ... where jobs do not execute before their arrival nor after completion. *)
Hypothesis
H_jobs_must_arrive_to_execute
:
jobs_must_arrive_to_execute
job_arrival
sched
.
Hypothesis
H_completed_jobs_dont_execute
:
completed_jobs_dont_execute
job_cost
sched
.
(* Assume we have sequential jobs, i.e, jobs from the same
task execute in the order of their arrival. *)
Hypothesis
H_sequential_jobs
:
sequential_jobs
job_arrival
job_cost
sched
job_task
.
(* Consider an FP policy that indicates a higherorequal priority relation,
and assume that the relation is reflexive and transitive. *)
Variable
higher_eq_priority
:
FP_policy
Task
.
Hypothesis
H_priority_is_reflexive
:
FP_is_reflexive
higher_eq_priority
.
Hypothesis
H_priority_is_transitive
:
FP_is_transitive
higher_eq_priority
.
(* Next, we assume that the schedule is a workconserving schedule which
respects the FP policy under a fully nonpreemptive model. *)
Hypothesis
H_work_conserving
:
work_conserving
job_arrival
job_cost
arr_seq
sched
.
Hypothesis
H_respects_policy
:
respects_FP_policy_at_preemption_point
job_arrival
job_cost
job_task
arr_seq
sched
(
can_be_preempted_for_fully_nonpreemptive_model
job_cost
)
higher_eq_priority
.
(* Let's define some local names for clarity. *)
Let
response_time_bounded_by
:
=
is_response_time_bound_of_task
job_arrival
job_cost
job_task
arr_seq
sched
.
Let
task_rbf
:
=
task_request_bound_function
task_cost
max_arrivals
tsk
.
Let
total_hep_rbf
:
=
total_hep_request_bound_function_FP
task_cost
higher_eq_priority
max_arrivals
ts
tsk
.
Let
total_ohep_rbf
:
=
total_ohep_request_bound_function_FP
task_cost
higher_eq_priority
max_arrivals
ts
tsk
.
(* Next, we define a bound for the priority inversion caused by tasks of lower priority. *)
Definition
blocking
:
=
\
max_
(
tsk_other
<
ts

~~
higher_eq_priority
tsk_other
tsk
)
(
task_cost
tsk_other

ε
).
(* Let L be any positive fixed point of the busy interval recurrence, determined by
the sum of blocking and higherorequalpriority workload. *)
Variable
L
:
time
.
Hypothesis
H_L_positive
:
L
>
0
.
Hypothesis
H_fixed_point
:
L
=
blocking
+
total_hep_rbf
L
.
(* To reduce the time complexity of the analysis, recall the notion of search space. *)
Let
is_in_search_space
A
:
=
(
A
<
L
)
&&
(
task_rbf
A
!=
task_rbf
(
A
+
ε
)).
(* Next, consider any value R, and assume that for any given arrival A from search space
there is a solution of the responsetime bound recurrence which is bounded by R. *)
Variable
R
:
nat
.
Hypothesis
H_R_is_maximum
:
forall
A
,
is_in_search_space
A
>
exists
F
,
A
+
F
=
blocking
+
(
task_rbf
(
A
+
ε
)

(
task_cost
tsk

ε
))
+
total_ohep_rbf
(
A
+
F
)
/\
F
+
(
task_cost
tsk

ε
)
<=
R
.
(* Now, we can reuse the results for the abstract model with fixed preemption points to
establish a responsetime bound for the more concrete model of fully nonpreemptive scheduling. *)
Theorem
uniprocessor_response_time_bound_fully_nonpreemptive_fp
:
response_time_bounded_by
tsk
R
.
Proof
.
move
:
(
posnP
(
task_cost
tsk
))
=>
[
ZERO

POS
].
{
intros
j
ARR
TSK
.
have
ZEROj
:
job_cost
j
=
0
.
{
move
:
(
H_job_cost_le_task_cost
j
ARR
)
=>
NEQ
.
rewrite
/
job_cost_le_task_cost
TSK
ZERO
in
NEQ
.
by
apply
/
eqP
;
rewrite

leqn0
.
}
rewrite
/
is_response_time_bound_of_job
/
completed_by
eqn_leq
;
apply
/
andP
;
split
.

by
apply
H_completed_jobs_dont_execute
.

by
rewrite
ZEROj
.
}
eapply
uniprocessor_response_time_bound_fp_with_bounded_nonpreemptive_segments
with
(
job_max_nps
:
=
fun
j
=>
job_cost
j
)
(
task_max_nps
:
=
fun
tsk
=>
task_cost
tsk
)
(
job_lock_in_service
:
=
fun
j
=>
ε
)
(
task_lock_in_service
:
=
fun
tsk
=>
ε
)
(
L0
:
=
L
)
;
eauto
2
.

by
eapply
fully_nonpreemptive_model_is_correct
;
eauto
2
.

eapply
fully_nonpreemptive_model_is_model_with_bounded_nonpreemptive_regions
;
eauto
2
.

repeat
split
;
try
done
.

intros
j
t
t'
ARR
LE
SERV
NCOMPL
.
rewrite
/
service
in
SERV
;
apply
incremental_service_during
in
SERV
.
move
:
SERV
=>
[
t_first
[/
andP
[
_
H1
]
[
H2
H3
]]].
apply
H_nonpreemptive_sched
with
t_first
;
try
done
.
by
apply
leq_trans
with
t
;
first
apply
ltnW
.

repeat
split
;
try
done
.
Qed
.
End
Analysis
.
End
RTAforFullyNonPreemptiveFPModelwithArrivalCurves
.
\ No newline at end of file
model/schedule/uni/limited/fixed_priority/nonpr_reg/preemptive/response_time_bound.v
0 → 100644
View file @
3fd984a4
Require
Import
rt
.
util
.
all
.
Require
Import
rt
.
model
.
arrival
.
basic
.
job
rt
.
model
.
arrival
.
basic
.
task_arrival
rt
.
model
.
priority
.
Require
Import
rt
.
model
.
schedule
.
uni
.
service
rt
.
model
.
schedule
.
uni
.
workload
rt
.
model
.
schedule
.
uni
.
schedule
rt
.
model
.
schedule
.
uni
.
response_time
.
Require
Import
rt
.
model
.
schedule
.
uni
.
limited
.
platform
.
preemptive
rt
.
model
.
schedule
.
uni
.
limited
.
schedule
rt
.
model
.
schedule
.
uni
.
limited
.
fixed_priority
.
nonpr_reg
.
response_time_bound
.
Require
Import
rt
.
model
.
arrival
.
curves
.
bounds
.
Require
Import
rt
.
analysis
.
uni
.
arrival_curves
.
workload_bound
.
From
mathcomp
Require
Import
ssreflect
ssrbool
eqtype
ssrnat
seq
path
fintype
bigop
.
(** * RTA for fully preemptive FP model *)
(* In this module we prove the RTA theorem for fully preemptive FP model *)
Module
RTAforFullyPreemptiveFPModelwithArrivalCurves
.
Import
Epsilon
Job
ArrivalCurves
TaskArrival
Priority
UniprocessorSchedule
Workload
Service
ResponseTime
MaxArrivalsWorkloadBound
FullyPreemptivePlatform
LimitedPreemptionPlatform
RTAforFPwithBoundedNonpreemptiveSegmentsWithArrivalCurves
.
(* In this section we prove that the maximum among the solutions of the responsetime bound
recurrence for some set of parameters is a response time bound for tsk. *)
Section
Analysis
.
Context
{
Task
:
eqType
}.
Variable
task_cost
:
Task
>
time
.
Context
{
Job
:
eqType
}.
Variable
job_arrival
:
Job
>
time
.
Variable
job_cost
:
Job
>
time
.
Variable
job_deadline
:
Job
>
time
.
Variable
job_task
:
Job
>
Task
.
(* Consider any arrival sequence with consistent, nonduplicate arrivals. *)
Variable
arr_seq
:
arrival_sequence
Job
.
Hypothesis
H_arrival_times_are_consistent
:
arrival_times_are_consistent
job_arrival
arr_seq
.
Hypothesis
H_arr_seq_is_a_set
:
arrival_sequence_is_a_set
arr_seq
.
(* Consider an arbitrary task set ts. *)
Variable
ts
:
list
Task
.
(* Assume that all jobs come from the task set... *)
Hypothesis
H_all_jobs_from_taskset
:
forall
j
,
arrives_in
arr_seq
j
>
job_task
j
\
in
ts
.
(* ...and the cost of a job cannot be larger than the task cost. *)
Hypothesis
H_job_cost_le_task_cost
:
cost_of_jobs_from_arrival_sequence_le_task_cost
task_cost
job_cost
job_task
arr_seq
.
(* Let max_arrivals be a family of proper arrival curves, i.e., for any task tsk in ts
[max_arrival tsk] is (1) an arrival bound of tsk, and (2) it is a monotonic function
that equals 0 for the empty interval delta = 0. *)
Variable
max_arrivals
:
Task
>
time
>
nat
.
Hypothesis
H_family_of_proper_arrival_curves
:
family_of_proper_arrival_curves
job_task
arr_seq
max_arrivals
ts
.
(* Let tsk be any task in ts. *)
Variable
tsk
:
Task
.
Hypothesis
H_tsk_in_ts
:
tsk
\
in
ts
.
(* Next, consider any uniprocessor schedule of the arrival sequence...*)
Variable
sched
:
schedule
Job
.
Hypothesis
H_jobs_come_from_arrival_sequence
:
jobs_come_from_arrival_sequence
sched
arr_seq
.
(* ... where jobs do not execute before their arrival nor after completion. *)
Hypothesis
H_jobs_must_arrive_to_execute
:
jobs_must_arrive_to_execute
job_arrival
sched
.
Hypothesis
H_completed_jobs_dont_execute
:
completed_jobs_dont_execute
job_cost
sched
.
(* Assume we have sequential jobs. *)
Hypothesis
H_sequential_jobs
:
sequential_jobs
job_arrival
job_cost
sched
job_task
.
(* Consider an FP policy that indicates a higherorequal priority relation,
and assume that the relation is reflexive and transitive. *)
Variable
higher_eq_priority
:
FP_policy
Task
.
Hypothesis
H_priority_is_reflexive
:
FP_is_reflexive
higher_eq_priority
.
Hypothesis
H_priority_is_transitive
:
FP_is_transitive
higher_eq_priority
.
(* Next, we assume that the schedule is a workconserving schedule which
respects the FP policy under a fully preemptive model. *)
Hypothesis
H_work_conserving
:
work_conserving
job_arrival
job_cost
arr_seq
sched
.
Hypothesis
H_respects_policy
:
respects_FP_policy_at_preemption_point
job_arrival
job_cost
job_task
arr_seq
sched
(
can_be_preempted_for_fully_preemptive_model
)
higher_eq_priority
.
(* Let's define some local names for clarity. *)
Let
response_time_bounded_by
:
=
is_response_time_bound_of_task
job_arrival
job_cost
job_task
arr_seq
sched
.
Let
task_rbf
:
=
task_request_bound_function
task_cost
max_arrivals
tsk
.
Let
total_hep_rbf
:
=
total_hep_request_bound_function_FP
task_cost
higher_eq_priority
max_arrivals
ts
tsk
.
Let
total_ohep_rbf
:
=
total_ohep_request_bound_function_FP
task_cost
higher_eq_priority
max_arrivals
ts
tsk
.
(* Let L be any positive fixed point of the busy interval recurrence, determined by
the sum of blocking and higherorequalpriority workload. *)
Variable
L
:
time
.
Hypothesis
H_L_positive
:
L
>
0
.
Hypothesis
H_fixed_point
:
L
=
total_hep_rbf
L
.
(* To reduce the time complexity of the analysis, recall the notion of search space. *)
Let
is_in_search_space
A
:
=
(
A
<
L
)
&&
(
task_rbf
A
!=
task_rbf
(
A
+
ε
)).
(* Next, consider any value R, and assume that for any given arrival A from search space
there is a solution of the responsetime bound recurrence which is bounded by R. *)
Variable
R
:
nat
.
Hypothesis
H_R_is_maximum
:
forall
A
,
is_in_search_space
A
>
exists
F
,
A
+
F
=
task_rbf
(
A
+
ε
)
+
total_ohep_rbf
(
A
+
F
)
/\
F
<=
R
.
(* Now, we can reuse the results for the abstract model with fixed preemption
points to establish a responsetime bound for the more concrete model
of fullypreemptive scheduling. *)
Theorem
uniprocessor_response_time_bound_fully_preemptive_fp
:
response_time_bounded_by
tsk
R
.
Proof
.
have
BLOCK
:
RTAforFPwithBoundedNonpreemptiveSegmentsWithArrivalCurves
.
blocking_bound
(
fun
_
=>
ε
)
higher_eq_priority
ts
tsk
=
0
.
{
by
rewrite
/
RTAforFPwithBoundedNonpreemptiveSegmentsWithArrivalCurves
.
blocking_bound
subnn
big1_eq
.
}
eapply
uniprocessor_response_time_bound_fp_with_bounded_nonpreemptive_segments
with
(
task_max_nps
:
=
fun
_
=>
ε
)
(
can_be_preempted
:
=
fun
j
prog
=>
true
)
(
task_lock_in_service
:
=
fun
tsk
=>
task_cost
tsk
)
(
job_lock_in_service
:
=
fun
j
=>
job_cost
j
)
(
job_max_nps
:
=
fun
j
=>
ε
)
;
eauto
2
;
try
done
.

by
eapply
fully_preemptive_model_is_model_with_bounded_nonpreemptive_regions
.

repeat
split
;
try
done
.
intros
?
?
?
ARR
;
move
=>
LE
COMPL
/
negP
NCOMPL
.
exfalso
;
apply
:
NCOMPL
.
apply
completion_monotonic
with
t
;
try
done
.
rewrite
/
completed_by
eqn_leq
;
apply
/
andP
;
split
;
try
done
.

repeat
split
;
try
done
.
rewrite
/
task_lock_in_service_le_task_cost
.
by
done
.
unfold
task_lock_in_service_bounds_job_lock_in_service
.
by
intros
?
ARR
TSK
;
rewrite

TSK
;
apply
H_job_cost_le_task_cost
.

by
rewrite
BLOCK
add0n
.

move
=>
A
/
andP
[
LT
NEQ
].
specialize
(
H_R_is_maximum
A
)
;
feed
H_R_is_maximum
.
{
by
apply
/
andP
;
split
.
}
move
:
H_R_is_maximum
=>
[
F
[
FIX
BOUND
]].
exists
F
;
split
.
+
by
rewrite
BLOCK
add0n
subnn
subn0
.
+
by
rewrite
subnn
addn0
.
Qed
.
End
Analysis
.
End
RTAforFullyPreemptiveFPModelwithArrivalCurves
.
\ No newline at end of file
model/schedule/uni/limited/fixed_priority/nonpr_reg/response_time_bound.v
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model/schedule/uni/limited/fixed_priority/response_time_bound.v
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