# Library prosa.analysis.abstract.IBF.task

Require Export prosa.analysis.definitions.task_schedule.
Require Export prosa.analysis.facts.model.rbf.
Require Export prosa.analysis.facts.model.task_schedule.
Require Export prosa.analysis.facts.model.sequential.
Require Export prosa.analysis.abstract.abstract_rta.
In this section, we define a notion of task interference-bound function task_IBF. Function task_IBF bounds interference that excludes interference due to self-interference.
Section TaskInterferenceBound.
Consider any type of job associated with any type of tasks...
Context {Job : JobType}.
Context {Task : TaskType}.
Context `{JobTask Job Task}.
... with arrival times and costs.
Context `{JobArrival Job}.
Context `{JobCost Job}.
Consider any kind of processor state model.
Context {PState : ProcessorState Job}.
Consider any arrival sequence ...
Variable arr_seq : arrival_sequence Job.
... and any schedule of this arrival sequence.
Variable sched : schedule PState.
Let tsk be any task that is to be analyzed.
Variable tsk : Task.
Assume we are provided with abstract functions for interference and interfering workload.
Context `{Interference Job}.
Context `{InterferingWorkload Job}.
Next we introduce the notion of task interference. Intuitively, task tsk incurs interference when some of the jobs of task tsk incur interference. As a result, tsk cannot make any progress.
More formally, consider a job j of task tsk. The task experiences interference at time t if job j experiences interference (interference j t = true) and task tsk is not scheduled at time t.
Let us define a predicate stating that the task of a job j is not scheduled at a time instant t.
Definition nonself (j : Job) (t : instant) :=
~~ task_served_at arr_seq sched (job_task j) t.
We define task interference as conditional interference where nonself is used as the predicate. This way, task_interference j t is false if the interference experienced by a job j is caused by a job of the same task.
Definition task_interference (j : Job) (t : instant) :=
cond_interference nonself j t.
Next, we define the cumulative task interference.
Definition cumul_task_interference j t1 t2 :=
cumul_cond_interference nonself j t1 t2.
Consider an interference bound function task_IBF.
Variable task_IBF : duration → duration → work.
We say that task interference is bounded by task_IBF iff for any job j of task tsk cumulative task interference within the interval `[t1, t1 + R)` is bounded by function task_IBF(tsk, A, R).
Definition task_interference_is_bounded_by :=
cond_interference_is_bounded_by
arr_seq sched tsk task_IBF (relative_arrival_time_of_job_is_A sched) nonself.
End TaskInterferenceBound.
In the following section, we prove that, under certain assumptions defined next, the fact that a function task_IBF tsk A R satisfies hypothesis task_interference_is_bounded_by implies that a function (task_rbf (A + ε) - task_cost tsk) + task_IBF tsk A R satisfies job_interference_is_bounded_by. In other words, the self-interference can be bounded by the term (task_rbf (A + ε) - task_cost tsk), where A is the relative arrival time of a job under analysis.
Section TaskIBFtoJobIBF.
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 Job Task}.
Context `{JobArrival Job}.
Context {jc : JobCost Job}.
Consider any kind of ideal uni-processor state model.
Context `{PState : ProcessorState Job}.
Hypothesis H_uniprocessor_proc_model : uniprocessor_model PState.
Hypothesis H_unit_service_proc_model : unit_service_proc_model PState.
Consider any valid arrival sequence with consistent, non-duplicate arrivals...
Variable arr_seq : arrival_sequence Job.
Hypothesis H_valid_arrival_sequence : valid_arrival_sequence arr_seq.
... and any ideal schedule of this arrival sequence.
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 nor 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 that the job costs are no larger than the task costs.
Hypothesis H_valid_job_cost : arrivals_have_valid_job_costs arr_seq.
Consider an arbitrary task set.
Variable ts : list Task.
Let tsk be any task in ts that is to be analyzed.
Variable tsk : Task.
Hypothesis H_tsk_in_ts : tsk \in ts.
Let max_arrivals be a family of valid 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.
Context `{MaxArrivals Task}.
Hypothesis H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals.
Hypothesis H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts.
Assume we are provided with abstract functions for interference and interfering workload.
Context `{Interference Job}.
Context `{InterferingWorkload Job}.
We assume that the schedule is work-conserving.
Hypothesis H_work_conserving : work_conserving arr_seq sched.
Let us abbreviate task tsk's RBF for clarity.
Let task_rbf := task_request_bound_function tsk.
When assuming sequential tasks, we need to introduce an additional hypothesis to ensure that the values of interference and workload remain consistent with the priority policy.
To understand why, let us consider a case where the sequential task assumption does not hold, and then work backwards. Consider a fully preemptive policy that assigns the highest priority to the job that was released last. Then, if there is a pending job j of a task tsk, it is not a part of a busy interval of the next job j' of the same task. In this case, both the interference and the interfering workload of job j' will simply ignore job j. Thus, it is possible to have a quiet time (for j') even though j is pending.
Now, assuming that our priority policy ensures that tasks are sequential, the situation described above is impossible (job j will always have a higher-or-equal priority than job j'). Hence, we need to rule our interference and interfering workload instantiations that do not conform to the sequential tasks assumption.
We ensure the consistency by assuming that, when a busy interval of a job j of task tsk starts, both the cumulative task workload and the task service must be equal within the interval `[0, t1)`. This implies that a busy interval for job j cannot start if there is another pending job of the same task tsk.
Definition interference_and_workload_consistent_with_sequential_tasks :=
∀ (j : Job) (t1 t2 : instant),
arrives_in arr_seq j →
job_of_task tsk j →
job_cost j > 0 →
busy_interval sched j t1 t2 →
task_workload_between arr_seq tsk 0 t1
= task_service_of_jobs_in sched tsk (arrivals_between arr_seq 0 t1) 0 t1.
To prove the reduction from task_IBF to job_IBF, we need to ensure that the scheduling policy, interference, and interfering workload all respect the sequential tasks hypothesis. For this, we assume that (1) tasks are sequential and (2) functions interference and interfering_workload are consistent with the hypothesis of sequential tasks.
Hypothesis H_sequential_tasks : sequential_tasks arr_seq sched.
Hypothesis H_interference_and_workload_consistent_with_sequential_tasks :
interference_and_workload_consistent_with_sequential_tasks.
Next, we assume that task_IBF is a bound on interference incurred by the task.
Variable task_IBF : duration → duration → duration.
Hypothesis H_task_interference_is_bounded :
task_interference_is_bounded_by arr_seq sched tsk task_IBF.
Before proceed to the main proof, we first show a few simple lemmas about the completion of jobs from the task considering the busy interval of the job under consideration.
Section CompletionOfJobsFromSameTask.
Consider any two jobs j1 j2 of tsk.
Variable j1 j2 : Job.
Hypothesis H_j1_arrives : arrives_in arr_seq j1.
Hypothesis H_j2_arrives : arrives_in arr_seq j2.
Hypothesis H_j1_from_tsk : job_of_task tsk j1.
Hypothesis H_j2_from_tsk : job_of_task tsk j2.
Hypothesis H_j1_cost_positive : job_cost_positive j1.
Consider the busy interval `[t1, t2)` of job j1.
Variable t1 t2 : instant.
Hypothesis H_busy_interval : busy_interval sched j1 t1 t2.
We prove that if a job from task tsk arrived before the beginning of the busy interval, then it must be completed before the beginning of the busy interval
Lemma completed_before_beginning_of_busy_interval :
job_arrival j2 < t1 →
completed_by sched j2 t1.
Next we prove that if a job is pending after the beginning of the busy interval `[t1, t2)` then it arrives after t1.
Lemma arrives_after_beginning_of_busy_interval :
∀ t,
t1 ≤ t →
pending sched j2 t →
arrived_between j2 t1 t.+1.
End CompletionOfJobsFromSameTask.
In this section we show that there exists a bound for cumulative interference for any job of task tsk.
Section BoundOfCumulativeJobInterference.
Consider any job j of tsk.
Variable j : Job.
Hypothesis H_j_arrives : arrives_in arr_seq j.
Hypothesis H_job_of_tsk : job_of_task tsk j.
Hypothesis H_job_cost_positive : job_cost_positive j.
Consider the busy interval `[t1, t2)` of job j.
Variable t1 t2 : instant.
Hypothesis H_busy_interval : busy_interval sched j t1 t2.
Let's define A as a relative arrival time of job j (with respect to time t1).
Let A : duration := job_arrival j - t1.
Consider an arbitrary time x ...
Variable x : duration.
... such that t1 + x is inside the busy interval ...
Hypothesis H_inside_busy_interval : t1 + x < t2.
... and job j is not completed by time t1 + x.
Hypothesis H_job_j_is_not_completed : ~~ completed_by sched j (t1 + x).
In the following, we show that the cumulative interference of job j in the interval `[t1, t1 + x)` is bounded by the sum of the task workload in the interval `[t1, t1 + A + ε)` and the cumulative interference of j's task in the interval `[t1, t1 + x)`. Note that the task workload is computed only on the interval `[t1, t1 + A + ε)`. Thanks to the hypothesis about sequential tasks, jobs of task tsk that arrive after t1 + A + ε cannot interfere with j.
We start by proving a simpler analog of the lemma that states that at any time instant `t ∈ [t1, t1 + x)` the sum of interference j t and scheduled_at j t is no larger than the sum of the service received by jobs of task tsk at time t and task_iterference tsk t.
Next we consider 4 cases.
Section CaseAnalysis.
Consider an arbitrary time instant t ∈ `[t1, t1 + x)`.
Variable t : instant.
Hypothesis H_t_in_interval : t1 ≤ t < t1 + x.
Section Case1.
Assume the processor is idle at time t.
Hypothesis H_idle : is_idle arr_seq sched t.
In case when the processor is idle, one can show that interference j t = 1, service_at j t = 0. But since interference doesn't come from a job of task tsk task_interference tsk = 1. Which reduces to 1 ≤ 1.
Lemma interference_plus_sched_le_serv_of_task_plus_task_interference_idle :
interference j t + service_at sched j t
≤ service_of_jobs_at sched (job_of_task tsk) (arrivals_between arr_seq t1 (t1 + A + ε)) t
+ task_interference arr_seq sched j t.
End Case1.
Section Case2.
Assume a job j' from another task is scheduled at time t.
Variable j' : Job.
Hypothesis H_sched : scheduled_at sched j' t.
Hypothesis H_not_job_of_tsk : ~~ job_of_task tsk j'.
If a job j' from another task is scheduled at time t, then interference j t = 1, served_at j t = 0. But since interference doesn't come from a job of task tsk task_interference tsk = 1. Which reduces to 1 ≤ 1.
Lemma interference_plus_sched_le_serv_of_task_plus_task_interference_task :
interference j t + service_at sched j t
≤ service_of_jobs_at sched (job_of_task tsk) (arrivals_between arr_seq t1 (t1 + A + ε)) t
+ task_interference arr_seq sched j t.
End Case2.
Section Case3.
Assume a job j' of task tsk is scheduled at time t but receives no service.
Variable j' : Job.
Hypothesis H_sched : scheduled_at sched j' t.
Hypothesis H_not_job_of_tsk : job_of_task tsk j'.
Hypothesis H_serv : service_at sched j' t = 0.
If a job j' of task tsk is scheduled at time t, then interference j t = 1, service_at j t = 0. Moreover, since interference comes from a job of the same task task_interference tsk = 0. However, in this case service_of_jobs of tsk = 1. Which reduces to 1 ≤ 1.
Lemma interference_plus_sched_le_serv_of_task_plus_task_interference_job :
interference j t + service_at sched j t
≤ service_of_jobs_at sched (job_of_task tsk) (arrivals_between arr_seq t1 (t1 + A + ε)) t
+ task_interference arr_seq sched j t.
End Case3.
Section Case4.
Before proceeding to the last case, let us note that the sum of interference and the service of j at t always equals to 1.
Fact interference_and_service_eq_1 :
interference j t + service_at sched j t = 1.
Assume that a job j' is scheduled at time t and receives service.
Variable j' : Job.
Hypothesis H_sched : scheduled_at sched j' t.
Hypothesis H_not_job_of_tsk : job_of_task tsk j'.
Hypothesis H_serv : service_at sched j' t = 1.
If job j' is served at time t, then service_of_jobs of tsk = 1. With the fact that interference + service_at = 1, we get the inequality to 1 ≤ 1.
Lemma interference_plus_sched_le_serv_of_task_plus_task_interference_j :
interference j t + service_at sched j t
≤ service_of_jobs_at sched (job_of_task tsk) (arrivals_between arr_seq t1 (t1 + A + ε)) t
+ task_interference arr_seq sched j t.
End Case4.
We use the above case analysis to prove that any time instant `t ∈ [t1, t1 + x)` the sum of interference j t and scheduled_at j t is no larger than the sum of the service received by jobs of task tsk at time t and task_iterference tsk t.
Lemma interference_plus_sched_le_serv_of_task_plus_task_interference :
interference j t + service_at sched j t
≤ service_of_jobs_at sched (job_of_task tsk) (arrivals_between arr_seq t1 (t1 + A + ε)) t
+ task_interference arr_seq sched j t.
End CaseAnalysis.
Next we prove cumulative version of the lemma above.
Lemma cumul_interference_plus_sched_le_serv_of_task_plus_cumul_task_interference :
cumulative_interference j t1 (t1 + x)
≤ (task_service_of_jobs_in sched tsk (arrivals_between arr_seq t1 (t1 + A + ε)) t1 (t1 + x)
- service_during sched j t1 (t1 + x)) + cumul_task_interference arr_seq sched j t1 (t1 + x).
As the next step, the service terms in the inequality above can be upper-bound by the workload terms.
Lemma serv_of_task_le_workload_of_task_plus :
task_service_of_jobs_in sched tsk (arrivals_between arr_seq t1 (t1 + A + ε)) t1 (t1 + x)
- service_during sched j t1 (t1 + x) + cumul_task_interference arr_seq sched j t1 (t1 + x)
≤ (task_workload_between arr_seq tsk t1 (t1 + A + ε) - job_cost j)
+ cumul_task_interference arr_seq sched j t1 (t1 + x).
Finally, we show that the cumulative interference of job j in the interval `[t1, t1 + x)` is bounded by the sum of the task workload in the interval `[t1, t1 + A + ε)` and the cumulative interference of j's task in the interval `[t1, t1 + x)`.
Lemma cumulative_job_interference_le_task_interference_bound :
cumulative_interference j t1 (t1 + x)
≤ (task_workload_between arr_seq tsk t1 (t1 + A + ε) - job_cost j)
+ cumul_task_interference arr_seq sched j t1 (t1 + x).
We use the lemmas above to obtain the bound on interference in terms of task_rbf and task_interference.
Lemma cumulative_job_interference_bound :
cumulative_interference j t1 (t1 + x)
≤ (task_rbf (A + ε) - task_cost tsk) + cumul_task_interference arr_seq sched j t1 (t1 + x).
End BoundOfCumulativeJobInterference.
Finally, we show that one can construct a job IBF from the given task IBF.
Lemma task_IBF_implies_job_IBF :
job_interference_is_bounded_by
arr_seq sched tsk
(fun A R ⇒ (task_rbf (A + ε) - task_cost tsk) + task_IBF A R)
(relative_arrival_time_of_job_is_A sched).
End TaskIBFtoJobIBF.