Throughout this file, we assume ideal uni-processor schedules.

Throughout this file, we assume the basic (i.e., Liu & Layland) readiness model.

Abstract RTA for EDF-schedulers with Bounded Priority Inversion

In this module we instantiate the Abstract Response-Time analysis (aRTA) to EDF-schedulers for ideal uni-processor model of real-time tasks with arbitrary arrival models. Given EDF priority policy and an ideal uni-processor scheduler model, we can explicitly specify interference, interfering_workload, and interference_bound_function. In this settings, we can define natural notions of service, workload, busy interval, etc. The important feature of this instantiation is that we can induce the meaningful notion of priority inversion. However, we do not specify the exact cause of priority inversion (as there may be different reasons for this, like execution of a non-preemptive segment or blocking due to resource locking). We only assume that that a priority inversion is bounded.

Consider any type of tasks ...

... and any type of jobs associated with these tasks.

For clarity, let's denote the relative deadline of a task as D.

Consider the EDF policy that indicates a higher-or-equal priority relation. Note that we do not relate the EDF policy with the scheduler. However, we define functions for Interference and Interfering Workload that actively use the concept of priorities.

Consider any arrival sequence with consistent, non-duplicate arrivals.

Next, consider any ideal uniprocessor schedule of this arrival sequence ...

... where jobs do not execute before their arrival or after completion.

Note that we differentiate between abstract and classical notions of work conserving schedule.

We assume that the schedule is a work-conserving schedule in the classical sense, and later prove that the hypothesis about abstract work-conservation also holds.

Assume we have sequential tasks, i.e, jobs from the same task execute in the order of their arrival.

Assume that a job cost cannot be larger than a task cost.

Consider an arbitrary task set ts.

Next, we assume that all jobs come from the task set.

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.

Let tsk be any task in ts that is to be analyzed.

Consider a valid preemption model...

...and a valid task run-to-completion threshold function. That is, task_run_to_completion_threshold tsk is (1) no bigger than tsk's cost, (2) for any job of task tsk job_run_to_completion_threshold is bounded by task_run_to_completion_threshold.

We introduce rbf as an abbreviation of the task request bound function, which is defined as task_cost(T) × max_arrivals(T,Δ) for some task T.

Next, we introduce task_rbf as an abbreviation of the task request bound function of task tsk.

Using the sum of individual request bound functions, we define the request bound function of all tasks (total request bound function).

For simplicity, let's define some local names.

Assume that there exists a constant priority_inversion_bound that bounds the length of any priority inversion experienced by any job of tsk. Since we analyze only task tsk, we ignore the lengths of priority inversions incurred by any other tasks.

Let L be any positive fixed point of the busy interval recurrence.

Next, we define an upper bound on interfering workload received from jobs of other tasks with higher-than-or-equal priority.

To reduce the time complexity of the analysis, we introduce the notion of search space for EDF. Intuitively, this corresponds to all "interesting" arrival offsets that the job under analysis might have with regard to the beginning of its busy-window. In case of search space for EDF we ask whether task_rbf A ≠ task_rbf (A + ε)...

...or there exists a task tsko from ts such that tsko ≠ tsk and rbf(tsko, A + D tsk - D tsko) ≠ rbf(tsko, A + ε + D tsk - D tsko). Note that we use a slightly uncommon notation has (λ tsko ⇒ P tskₒ) ts which can be interpreted as follows: task-set ts contains a task tsko such that a predicate P holds for tsko.

The final search space for EDF is a set of offsets that are less than L and where task_rbf or bound_on_total_hep_workload changes.

Let R be a value that upper-bounds the solution of each response-time recurrence, i.e., for any relative arrival time A in the search space, there exists a corresponding solution F such that F + (task cost - task lock-in service) ≤ R.

To use the theorem uniprocessor_response_time_bound_seq from the Abstract RTA module, we need to specify functions of interference, interfering workload and IBF. Instantiation of Interference We say that job j incurs interference at time t iff it cannot execute due to a higher-or-equal-priority job being scheduled, or if it incurs a priority inversion.

Instantiation of Interfering Workload The interfering workload, in turn, is defined as the sum of the priority inversion function and interfering workload of jobs with higher or equal priority.

Finally, we define the interference bound function as the sum of the priority interference bound and the higher-or-equal-priority workload.

Filling Out Hypothesis Of Abstract RTA Theorem

In this section we prove that all hypotheses necessary to use the abstract theorem are satisfied.

First, we prove that in the instantiation of interference and interfering workload, we really take into account everything that can interfere with tsk's jobs, and thus, the scheduler satisfies the abstract notion of work conserving schedule.

Next, we prove that the interference and interfering workload functions are consistent with sequential tasks.

Recall that L is assumed to be a fixed point of the busy interval recurrence. Thanks to this fact, we can prove that every busy interval (according to the concrete definition) is bounded. In addition, we know that the conventional concept of busy interval and the one obtained from the abstract definition (with the interference and interfering workload) coincide. Thus, it follows that any busy interval (in the abstract sense) is bounded.

Next, we prove that IBF is indeed an interference bound.

We show that task_interference_is_bounded_by is bounded by IBF by constructing a sequence of inequalities.

Consider any busy interval [t1, t2) of job j.

Let's define A as a relative arrival time of job j (with respect to time t1).

Consider an arbitrary shift Δ inside the busy interval ...

... and the set of all arrivals between t1 and t1 + Δ.

Next, we define two predicates on jobs by extending EDF-priority relation. Predicate EDF_from tsk holds true for any job jo of task tsk such that job_deadline jo ≤ job_deadline j.

Predicate EDF_not_from tsk holds true for any job jo such that job_deadline jo ≤ job_deadline j and job_task jo ≠ tsk.

Recall that IBF(A, R) := priority_inversion_bound + bound_on_total_hep_workload(A, R). The fact that priority_inversion_bound bounds cumulative priority inversion follows from assumption H_priority_inversion_is_bounded.

Next, we show that bound_on_total_hep_workload(A, R) bounds interference from jobs with higher-or-equal priority. From lemma instantiated_cumulative_interference_of_hep_tasks_equal_total_interference_of_hep_tasks it follows that cumulative interference from jobs with higher-or-equal priority from other tasks is equal to the total service of jobs with higher-or-equal priority from other tasks. Which in turn means that cumulative interference is bounded by service.

By lemma service_of_jobs_le_workload, the total service of jobs with higher-or-equal priority from other tasks is at most the total workload of jobs with higher-or-equal priority from other tasks.

Next, we reorder summation. So the total workload of jobs with higher-or-equal priority from other tasks is equal to the sum over all tasks tsk_o that are to equal to task tsk of workload of jobs with higher-or-equal priority task tsk_o.

Next we focus on one task tsk_o ≠ tsk and consider two cases. Case 1: Δ ≤ A + ε + D tsk - D tsk_o.

Consider an arbitrary task tsk_o ≠ tsk from ts.

And assume that Δ ≤ A + ε + D tsk - D tsk_o.

Then by definition of rbf, the total workload of jobs with higher-or-equal priority from task tsk_o is bounded rbf(tsk_o, Δ).

Case 2: A + ε + D tsk - D tsk_o ≤ Δ.

Consider an arbitrary task tsk_o ≠ tsk from ts.

And assume that A + ε + D tsk - D tsk_o ≤ Δ.

Important step. Next we prove that the total workload of jobs with higher-or-equal priority from task tsk_o over time interval t1, t1 + Δ is bounded by workload over time interval t1, t1 + A + ε + D tsk - D tsk_o. The intuition behind this inequality is that jobs which arrive after time instant t1 + A + ε + D tsk - D tsk_o has smaller priority than job j due to the term D tsk - D tsk_o.

And similarly to the previous case, by definition of rbf, the total workload of jobs with higher-or-equal priority from task tsk_o is bounded rbf(tsk_o, Δ).

By combining case 1 and case 2 we prove that total workload of tasks is at most bound_on_total_hep_workload(A, Δ).

Recall that in module abstract_seq_RTA hypothesis task_interference_is_bounded_by expects to receive a function that maps some task t, the relative arrival time of a job j of task t, and the length of the interval to the maximum amount of interference. However, in this module we analyze only one task -- tsk, therefore it is “hard-coded” inside the interference bound function IBF. Therefore, in order for the IBF signature to match the required signature in module abstract_seq_RTA, we wrap the IBF function in a function that accepts, but simply ignores the task.

Finally, we show that there exists a solution for the response-time recurrence.

Consider any job j of tsk.

Given any job j of task tsk that arrives exactly A units after the beginning of the busy interval, the bound of the total interference incurred by j within an interval of length Δ is equal to task_rbf (A + ε) - task_cost tsk + IBF(A, Δ).

Next, consider any A from the search space (in abstract sense).

We prove that A is also in the concrete search space.

Then, there exists solution for response-time recurrence (in the abstract sense).

Final Theorem

Based on the properties established above, we apply the abstract analysis framework to infer that R is a response-time bound for tsk.