# Library prosa.analysis.facts.busy_interval.existence

Require Export prosa.model.job.properties.
Require Export prosa.model.schedule.work_conserving.
Require Export prosa.analysis.facts.model.service_of_jobs.
Require Export prosa.analysis.definitions.readiness.
Require Export prosa.analysis.facts.priority.inversion.

# Existence of Busy Interval for JLFP-models

In this module we derive a sufficient condition for existence of busy intervals for uni-processor for JLFP schedulers.
Section ExistsBusyIntervalJLFP.
Consider any type of tasks ...
Context {Task : TaskType}.
Context `{TaskCost Task}.
... and any type of jobs associated with these tasks.
Context {Job : JobType}.
Context {JobTask : JobTask Job Task}.
Context {Arrival : JobArrival Job}.
Context {Cost : JobCost Job}.
Consider any valid arrival sequence.
Variable arr_seq : arrival_sequence Job.
Hypothesis H_valid_arrival_time : valid_arrival_sequence arr_seq.
Next, consider any schedule of this arrival sequence ...
Context {PState : ProcessorState Job}.
Variable sched : schedule PState.
Hypothesis H_jobs_come_from_arrival_sequence :
jobs_come_from_arrival_sequence sched arr_seq.
... where jobs do not execute before their arrival or after completion.
Hypothesis H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched.
Hypothesis H_completed_jobs_dont_execute : completed_jobs_dont_execute sched.
Assume a given JLFP policy.
Context {JLFP : JLFP_policy Job}.
Further, allow for any work-bearing notion of job readiness.
Context {RM : JobReady Job PState}.
Hypothesis H_job_ready : work_bearing_readiness RM arr_seq sched.
For simplicity, let's define some local names.
Let job_pending_at := pending sched.
Let job_completed_by := completed_by sched.
Consider an arbitrary task tsk.
Variable tsk : Task.
Consider an arbitrary job j.
Variable j : Job.
Hypothesis H_from_arrival_sequence : arrives_in arr_seq j.
Hypothesis H_job_task : job_of_task tsk j.
Hypothesis H_job_cost_positive : job_cost_positive j.
Recall the list of jobs that arrive in any interval.
Let quiet_time t1 := quiet_time arr_seq sched j t1.
Let quiet_time_dec t1 := quiet_time_dec arr_seq sched j t1.
Let busy_interval_prefix t1 t2 := busy_interval_prefix arr_seq sched j t1 t2.
Let busy_interval t1 t2 := busy_interval arr_seq sched j t1 t2.
Let is_priority_inversion_bounded_by K := priority_inversion_of_job_is_bounded_by arr_seq sched j K.
We begin by proving a few basic lemmas about busy intervals.
Section BasicLemmas.
Assume that the priority relation is reflexive.
Hypothesis H_priority_is_reflexive : reflexive_job_priorities JLFP.
Consider any busy interval `[t1, t2)` of job j.
Variable t1 t2 : instant.
Hypothesis H_busy_interval : busy_interval t1 t2.
We prove that job j completes by the end of the busy interval.
Lemma job_completes_within_busy_interval :
job_completed_by j t2.
End BasicLemmas.
In this section, we prove that during a busy interval there always exists a pending job.
Section ExistsPendingJob.
Let `[t1, t2]` be any interval where time t1 is quiet and time t2 is not quiet.
Variable t1 t2 : instant.
Hypothesis H_interval : t1 ≤ t2.
Hypothesis H_quiet : quiet_time t1.
Hypothesis H_not_quiet : ¬ quiet_time t2.
Then, we prove that there is a job pending at time t2 that has higher or equal priority (with respect to tsk).
Lemma not_quiet_implies_exists_pending_job :
∃ j_hp,
arrives_in arr_seq j_hp
∧ arrived_between j_hp t1 t2
∧ hep_job j_hp j
∧ ¬ job_completed_by j_hp t2.
End ExistsPendingJob.
In this section, we prove that during a busy interval the processor is never idle.
Section ProcessorAlwaysBusy.
Assume that the schedule is work-conserving ...
Hypothesis H_work_conserving : work_conserving arr_seq sched.
... and the priority relation is reflexive and transitive.
Hypothesis H_priority_is_reflexive : reflexive_job_priorities JLFP.
Hypothesis H_priority_is_transitive : transitive_job_priorities JLFP.
Consider any busy interval prefix `[t1, t2)`.
Variable t1 t2 : instant.
Hypothesis H_busy_interval_prefix : busy_interval_prefix t1 t2.
We prove that if the processor is idle at a time instant t, then the next time instant t+1 will be a quiet time.
Lemma idle_time_implies_quiet_time_at_the_next_time_instant :
∀ (t : instant),
is_idle arr_seq sched t →
quiet_time t.+1.
Next, we prove that at any time instant t within the busy interval there exists a job jhp such that (1) job jhp is pending at time t and (2) job jhp has higher-or-equal priority than task tsk.
Lemma pending_hep_job_exists :
∀ t,
t1 ≤ t < t2 →
∃ jhp,
arrives_in arr_seq jhp
∧ job_pending_at jhp t
∧ hep_job jhp j.
We prove that at any time instant t within `[t1, t2)` the processor is not idle.
Lemma not_quiet_implies_not_idle :
∀ t,
t1 ≤ t < t2 →
¬ is_idle arr_seq sched t.
End ProcessorAlwaysBusy.
In section we prove a few auxiliary lemmas about quiet time and service.
Section QuietTimeAndServiceOfJobs.
Assume that the schedule is work-conserving ...
Hypothesis H_work_conserving : work_conserving arr_seq sched.
... and there are no duplicate job arrivals.
Hypothesis H_arrival_sequence_is_a_set :
arrival_sequence_uniq arr_seq.
Let t1 be a quiet time.
Variable t1 : instant.
Hypothesis H_quiet_time : quiet_time t1.
Assume that there is no quiet time in the interval `(t1, t1 + Δ]`.
Variable Δ : duration.
Hypothesis H_no_quiet_time : ∀ t, t1 < t ≤ t1 + Δ → ¬ quiet_time t.
For clarity, we introduce a notion of the total service of jobs released in time interval `[t_beg, t_end)` during the time interval `[t1, t1 + Δ)`.
Let service_received_by_hep_jobs_released_during t_beg t_end :=
service_of_higher_or_equal_priority_jobs
sched (arrivals_between arr_seq t_beg t_end) j t1 (t1 + Δ).
We prove that jobs with higher-than-or-equal priority that released before time instant t1 receive no service after time instant t1.
Lemma hep_jobs_receive_no_service_before_quiet_time :
service_received_by_hep_jobs_released_during t1 (t1 + Δ)
= service_received_by_hep_jobs_released_during 0 (t1 + Δ).
Next, assume that the processor is an ideal-progress, unit-speed uniprocessor.
Hypothesis H_uni : uniprocessor_model PState.
Hypothesis H_unit : unit_service_proc_model PState.
Hypothesis H_progress : ideal_progress_proc_model PState.
Under this assumption, we prove that the total service within a "non-quiet" time interval `[t1, t1 + Δ)` is exactly Δ.
Lemma no_idle_time_within_non_quiet_time_interval :
total_service_of_jobs_in sched (arrivals_between arr_seq 0 (t1 + Δ)) t1 (t1 + Δ) = Δ.
End QuietTimeAndServiceOfJobs.
In this section, we show that the length of any busy interval is bounded, as long as there is enough supply to accommodate the workload of tasks with higher or equal priority.
Section BoundingBusyInterval.
Assume that the schedule is work-conserving, ...
Hypothesis H_work_conserving : work_conserving arr_seq sched.
... and there are no duplicate job arrivals, ...
Hypothesis H_arrival_sequence_is_a_set :
arrival_sequence_uniq arr_seq.
... and the priority relation is reflexive and transitive.
Hypothesis H_priority_is_reflexive : reflexive_job_priorities JLFP.
Hypothesis H_priority_is_transitive : transitive_job_priorities JLFP.
Next, we recall the notion of workload of all jobs released in a given interval `[t1, t2)` that have higher-or-equal priority w.r.t. the job j being analyzed.
Let hp_workload t1 t2 :=
workload_of_hep_jobs arr_seq j t1 t2.
With regard to the jobs with higher-or-equal priority that are released in a given interval `[t1, t2)`, we also recall the service received by these jobs in the same interval `[t1, t2)`.
Let hp_service t1 t2 :=
service_of_higher_or_equal_priority_jobs
sched (arrivals_between arr_seq t1 t2) j t1 t2.
Now we begin the proof. First, we show that the busy interval is bounded.
Section BoundingBusyInterval.
Suppose that job j is pending at time t_busy.
Variable t_busy : instant.
Hypothesis H_j_is_pending : job_pending_at j t_busy.
First, we show that there must exist a busy interval prefix.
Since job j is pending, there is a (potentially unbounded) busy interval that starts no later than with the arrival of j.
Lemma exists_busy_interval_prefix :
∃ t1,
busy_interval_prefix t1 t_busy.+1
∧ t1 ≤ job_arrival j ≤ t_busy.
Next we prove that, if there is a point where the requested workload is upper-bounded by the supply, then the busy interval eventually ends.
Section UpperBound.
The following proofs assume that the processor is an ideal-progress, unit-speed uniprocessor.
Hypothesis H_uni : uniprocessor_model PState.
Hypothesis H_unit : unit_service_proc_model PState.
Hypothesis H_progress : ideal_progress_proc_model PState.
Consider any busy interval prefix of job j.
Variable t1 : instant.
Hypothesis H_is_busy_prefix : busy_interval_prefix t1 t_busy.+1.
Let's define A as the relative arrival time of job j (with respect to time t1).
Let A := job_arrival j - t1.
Let priority_inversion_bound be a constant that bounds the length of any priority inversion.
Variable priority_inversion_bound : instant → instant.
Hypothesis H_priority_inversion_is_bounded :
is_priority_inversion_bounded_by priority_inversion_bound.
Next, assume that for some positive delta, the sum of requested workload at time t1 + delta and constant priority_inversion_bound is bounded by delta (i.e., the supply).
Variable delta : duration.
Hypothesis H_delta_positive : delta > 0.
Hypothesis H_workload_is_bounded :
priority_inversion_bound A + hp_workload t1 (t1 + delta) ≤ delta.
If there is a quiet time by time t1 + delta, it trivially follows that the busy interval is bounded. Thus, let's consider first the harder case where there is no quiet time, which turns out to be impossible.
Section CannotBeBusyForSoLong.
Assume that there is no quiet time in the interval `(t1, t1 + delta]`.
Hypothesis H_no_quiet_time :
∀ t, t1 < t ≤ t1 + delta → ¬ quiet_time t.
Since the interval is always non-quiet, the processor is always busy with tasks of higher-or-equal priority or some lower priority job which was scheduled, i.e., the sum of service done by jobs with actual arrival time in `[t1, t1 + delta)` and priority inversion equals delta.
Lemma busy_interval_has_uninterrupted_service :
delta ≤ priority_inversion_bound A + hp_service t1 (t1 + delta).
Moreover, the fact that the interval is not quiet also implies that there's more workload requested than service received.
Lemma busy_interval_too_much_workload :
hp_workload t1 (t1 + delta) > hp_service t1 (t1 + delta).
Using the two lemmas above, we infer that the workload is larger than the interval length. However, this contradicts the assumption H_workload_is_bounded.
Corollary busy_interval_workload_larger_than_interval :
priority_inversion_bound A + hp_workload t1 (t1 + delta) > delta.
End CannotBeBusyForSoLong.
Since the interval cannot remain busy for so long, we prove that the busy interval finishes at some point t2 ≤ t1 + delta.
Lemma busy_interval_is_bounded :
∃ t2,
t2 ≤ t1 + delta
∧ busy_interval t1 t2.
End UpperBound.
End BoundingBusyInterval.
The following proofs assume that the processor is an ideal-progress, unit-speed uniprocessor.
Hypothesis H_uni : uniprocessor_model PState.
Hypothesis H_unit : unit_service_proc_model PState.
Hypothesis H_progress : ideal_progress_proc_model PState.
In this section, we show that from a workload bound we can infer the existence of a busy interval.
Section BusyIntervalFromWorkloadBound.
Let priority_inversion_bound be a constant that bounds the length of a priority inversion.
Variable priority_inversion_bound : duration → duration.
Hypothesis H_priority_inversion_is_bounded :
is_priority_inversion_bounded_by priority_inversion_bound .
Assume that for some positive delta, the sum of requested workload at time t1 + delta and priority inversion is bounded by delta (i.e., the supply).
Variable delta : duration.
Hypothesis H_delta_positive : delta > 0.
Hypothesis H_workload_is_bounded :
∀ t, priority_inversion_bound (job_arrival j - t) + hp_workload t (t + delta) ≤ delta.
Next, we assume that job j has positive cost, from which we can infer that there is a time in which j is pending.
Hypothesis H_positive_cost : job_cost j > 0.
Therefore there must exists a busy interval `[t1, t2)` that contains the arrival time of j.
Corollary exists_busy_interval :
∃ t1 t2,
t1 ≤ job_arrival j < t2
∧ t2 ≤ t1 + delta
∧ busy_interval t1 t2.
End BusyIntervalFromWorkloadBound.
If we know that the workload is bounded, we can also use the busy interval to infer a response-time bound.
Section ResponseTimeBoundFromBusyInterval.
Let priority_inversion_bound be a constant that bounds the length of a priority inversion.
Variable priority_inversion_bound : duration → duration.
Hypothesis H_priority_inversion_is_bounded :
is_priority_inversion_bounded_by priority_inversion_bound.
Assume that for some positive delta, the sum of requested workload at time t1 + delta and priority inversion is bounded by delta (i.e., the supply).
Variable delta : duration.
Hypothesis H_delta_positive : delta > 0.
Hypothesis H_workload_is_bounded :
∀ t, priority_inversion_bound (job_arrival j - t) + hp_workload t (t + delta) ≤ delta.
Then, job j must complete by job_arrival j + delta.
Lemma busy_interval_bounds_response_time :
job_completed_by j (job_arrival j + delta).
End ResponseTimeBoundFromBusyInterval.
End BoundingBusyInterval.
End ExistsBusyIntervalJLFP.