Library prosa.analysis.abstract.definitions

Require Export prosa.model.task.concept.

Definitions for Abstract Response-Time Analysis

In the following, we propose a set of definitions for the general framework for response-time analysis (RTA) of uni-processor scheduling of real-time tasks with arbitrary arrival models.
We are going to introduce two main variables of the analysis: (a) interference, and (b) interfering workload.

a) Interference

Execution of a job may be postponed by the environment and/or the system due to different factors (preemption by higher-priority jobs, jitter, black-out periods in hierarchical scheduling, lack of budget, etc.), which we call interference.
Besides, note that even the subsequent activation of a task can suffer from interference at the beginning of its busy interval (despite the fact that this job hasn’t even arrived at that moment!). Thus, it makes more sense (at least for the current busy-interval analysis) to think about interference of a job as any interference within the corresponding busy interval, and not just after the release of the job.
Based on that rationale, assume a predicate that expresses whether a job j under consideration incurs interference at a given time t (in the context of the schedule under consideration). This will be used later to upper-bound job j's response time. Note that a concrete realization of the function may depend on the schedule, but here we do not require this for the sake of simplicity and generality.

b) Interfering Workload

In addition to interference, the analysis assumes that at any time t, we know an upper bound on the potential cumulative interference that can be incurred in the future by any job (i.e., the total remaining potential delays). Based on that, assume a function interfering_workload that indicates for any job j, at any time t, the amount of potential interference for job j that is introduced into the system at time t. This function will be later used to upper-bound the length of the busy window of a job. One example of workload function is the "total cost of jobs that arrive at time t and have higher-or-equal priority than job j". In some task models, this function expresses the amount of the potential interference on job j that "arrives" in the system at time t.
Next we introduce all the abstract notions required by the analysis.
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}.

In order to bound the response time of a job, we must consider the cumulative interference and cumulative interfering workload.
Definition of Busy Interval Further analysis will be based on the notion of a busy interval. The overall idea of the busy interval is to take into account the workload that cause a job under consideration to incur interference. In this section, we provide a definition of an abstract busy interval.
  Section BusyInterval.

We say that time instant t is a quiet time for job j iff two conditions hold. First, the cumulative interference at time t must be equal to the cumulative interfering workload. Intuitively, this condition indicates that the potential interference seen so far has been fully "consumed" (i.e., there is no more higher-priority work or other kinds of delay pending). Second, job j cannot be pending at any time earlier than t and at time instant t (i.e., either it was pending earlier but is no longer pending now, or it was previously not pending and may or may not be released now). The second condition ensures that the busy window captures the execution of job j.
    Definition quiet_time (j : Job) (t : instant) :=
      cumulative_interference j 0 t = cumulative_interfering_workload j 0 t
      ~~ pending_earlier_and_at sched j t.

Based on the definition of quiet time, we say that an interval [t1, t2) is a (potentially unbounded) busy-interval prefix w.r.t. job j iff the interval (a) contains the arrival of job j, (b) starts with a quiet time and (c) remains non-quiet.
    Definition busy_interval_prefix (j : Job) (t1 t2 : instant) :=
      t1 job_arrival j < t2
      quiet_time j t1
      ( t, t1 < t < t2 ¬ quiet_time j t).

Next, we say that an interval [t1, t2) is a busy interval iff [t1, t2) is a busy-interval prefix and t2 is a quiet time.
    Definition busy_interval (j : Job) (t1 t2 : instant) :=
      busy_interval_prefix j t1 t2
      quiet_time j t2.

Note that the busy interval, if it exists, is unique.
    Fact busy_interval_is_unique :
       j t1 t2 t1' t2',
        busy_interval j t1 t2
        busy_interval j t1' t2'
        t1 = t1' t2 = t2'.
      movej t1 t2 t1' t2' BUSY BUSY'.
      have EQ: t1 = t1'.
      { apply/eqP.
        apply/negPn/negP; intros CONTR.
        move: BUSY ⇒ [[IN [QT1 NQ]] _].
        move: BUSY' ⇒ [[IN' [QT1' NQ']] _].
        move: CONTR; rewrite neq_ltn; move ⇒ /orP [LT|GT].
        { apply NQ with t1' ⇒ //; clear NQ.
          apply/andP; split⇒ [//|].
          move: IN IN' ⇒ /andP [_ T1] /andP [T2 _].
            by apply leq_ltn_trans with (job_arrival j).
        { apply NQ' with t1 ⇒ [|//]; clear NQ'.
          apply/andP; split⇒ [//|].
          move: IN IN' ⇒ /andP [T1 _] /andP [_ T2].
          by apply leq_ltn_trans with (job_arrival j).
      subst t1'.
      have EQ: t2 = t2'.
      { apply/eqP.
        apply/negPn/negP; intros CONTR.
        move: BUSY ⇒ [[IN [_ NQ]] QT2].
        move: BUSY' ⇒ [[IN' [_ NQ']] QT2'].
        move: CONTR; rewrite neq_ltn; move ⇒ /orP [LT|GT].
        { apply NQ' with t2 ⇒ //; clear NQ'.
          apply/andP; split⇒ [|//].
          move: IN IN' ⇒ /andP [_ T1] /andP [T2 _].
          by apply leq_ltn_trans with (job_arrival j).
        { apply NQ with t2' ⇒ //; clear NQ.
          apply/andP; split⇒ [|//].
          move: IN IN' ⇒ /andP [T1 _] /andP [_ T2].
          by apply leq_ltn_trans with (job_arrival j).
      by subst t2'.

  End BusyInterval.

In this section, we introduce some assumptions about the busy interval that are fundamental to the analysis.
  Section BusyIntervalProperties.

We say that a schedule is "work-conserving" (in the abstract sense) iff, for any job j from task tsk and at any time t within a busy interval, there are only two options: either (a) interference(j, t) holds or (b) job j is scheduled at time t.
    Definition work_conserving :=
       j t1 t2 t,
        arrives_in arr_seq j
        job_cost j > 0
        busy_interval_prefix j t1 t2
        t1 t < t2
        ¬ interference j t receives_service_at sched j t.

Next, we say that busy intervals of task tsk are bounded by L iff, for any job j of task tsk, there exists a busy interval with length at most L. Note that the existence of such a bounded busy interval is not guaranteed if the schedule is overloaded with work. Therefore, in the later concrete analyses, we will have to introduce an additional condition that prevents overload.
    Definition busy_intervals_are_bounded_by L :=
        arrives_in arr_seq j
        job_of_task tsk j
        job_cost j > 0
         t1 t2,
          t1 job_arrival j < t2
          t2 t1 + L
          busy_interval j t1 t2.

Although we have defined the notion of cumulative interference of a job, it cannot be used in a (static) response-time analysis because of the dynamic variability of job parameters. To address this issue, we define the notion of an interference bound.
As a first step, we introduce a notion of an "interference bound function" IBF. An interference bound function is any function with a type Task duration duration work that bounds cumulative interference of a job of a task under analysis (a precise definition will be presented below).
Note that the function has three parameters. The first and the last parameters are a task under analysis and the length of an interval in which the interference is supposed to be bounded, respectively. These are quite intuitive; so, we will not explain them in more detail. However, the second parameter deserves more thoughtful explanation.
The second parameter of IBF allows one to organize a case analysis over a set of values that are known only during the computation. For example, the most common parameter is the relative arrival time A of a job (of a task under analysis). Strictly speaking, A is now known at a time of computing a fixpoint; however, one can consider a set of A that covers all the relevant cases There can be other valid properties such as "a time instant when a job under analysis has received enough service to become non-preemptive."
    Variable IBF : Task duration duration work.

To make the second parameter customizable, we introduce a predicate P : Job instant Prop that connects the second parameter to its semantics. More precisely, consider an expression IBF(tsk, X, delta), and assume that we instantiated P as some predicate P0. Then, it is assumed that IBF(tsk, X, delta) bounds interference of a job under analysis j tsk if P0 j X holds.
    Variable P : Job nat Prop.

Next, let us define this reasoning formally. We say that the job interference is bounded by an "interference bound function" IBF iff for any job j of task tsk the cumulative interference incurred by j in the sub-interval [t1, t1 + delta) of busy interval [t1, t2)>> does not exceed IBF(tsk, X, delta), where X is a constant that satisfies a predefined predicate P. Note that according to the definition of an abstract work conservation, interference does not include execution of a job under analysis itself. Therefore, an interference bound is not obliged to take into account the execution of this job.
    Definition job_interference_is_bounded_by :=
Consider a job j of task tsk, a busy interval [t1,t2) of j, and an arbitrary interval [t1, t1 + Δ) t1, t2)>>.
       t1 t2 Δ j,
        arrives_in arr_seq j
        job_of_task tsk j
        busy_interval j t1 t2
We require the IBF to bound the interference only within the interval t1, t1 + Δ).
        t1 + Δ < t2
Next, we require the IBF to bound the interference only until the job is completed, after which the function can behave arbitrarily.
        ~~ completed_by sched j (t1 + Δ)
And finally, the IBF function might depend not only on the length of the interval, but also on a constant X satisfying predicate P.