RTA for EDF with Bounded Non-Preemptive Segments

In this section we instantiate the Abstract RTA for EDF-schedulers with Bounded Priority Inversion to EDF-schedulers for ideal uni-processor model of real-time tasks with arbitrary arrival models and bounded non-preemptive segments. Recall that Abstract RTA for EDF-schedulers with Bounded Priority Inversion does not specify the cause of priority inversion. In this section, we prove that the priority inversion caused by execution of non-preemptive segments is bounded. Thus the Abstract RTA for EDF-schedulers is applicable to this instantiation.

Consider any type of tasks ...

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

We assume the classic (i.e., Liu & Layland) model of readiness without jitter or self-suspensions, wherein pending jobs are always ready.

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 valid ideal uni-processor schedule of this arrival sequence ...

In addition, we assume the existence of a function mapping jobs to their preemption points ...

... and assume that it defines a valid preemption model with bounded non-preemptive segments.

Next, we assume that the schedule is a work-conserving schedule...

... and the schedule respects the scheduling policy at every preemption point.

Consider an arbitrary task set ts, ...

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

... and the cost of a job cannot be larger than the task cost.

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_rtct tsk is (1) no bigger than tsk's cost, (2) for any job of task tsk job_rtct is bounded by task_rtct.

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

Next, we introduce task_rbf as an abbreviation for 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).

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

Let's define some local names for clarity.

For a job with the relative arrival offset A within its busy window, we define the following blocking bound. Only other tasks that potentially release non-zero-cost jobs are relevant, so we define a predicate to exclude pathological cases.

Search Space

If priority inversion is caused exclusively by non-preemptive sections, then we do not need to consider the priority-inversion bound in the search space. Hence we define the following search space, which refines the more general bounded_pi.is_in_search_space for our specific setting.

For the following proof, we exploit the fact that the blocking bound is monotonically decreasing in A, which we note here.

To use the refined search space with the abstract theorem, we must show that it still includes all relevant points. To this end, we first observe that a step in the blocking bound implies the existence of a task that could release a job with an absolute deadline equal to the absolute deadline of the job under analysis.

With the above setup in place, we can show that the search space defined above by is_in_search_space covers the the more abstract search space defined by bounded_pi.is_in_search_space.

Priority inversion is bounded

In this section, we prove that a priority inversion for task tsk is bounded by the maximum length of non-preemptive segments among the tasks with lower priority.

First, we observe that the maximum non-preemptive segment length of any task that releases a job with an earlier absolute deadline (w.r.t. a given job j) and non-zero execution cost upper-bounds the maximum possible length of priority inversion (of said job j).

Second, we prove that the maximum length of a priority inversion of a given job j is indeed bounded by defined the blocking bound.

Using the lemma above, we prove that the priority inversion of the task is bounded by the maximum length of a nonpreemptive section of lower-priority tasks.

Response-Time Bound

In this section, we prove that the maximum among the solutions of the response-time bound recurrence is a response-time bound for tsk.

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

Consider any value R, and assume that for any given arrival offset A in the search space, there is a solution of the response-time bound recurrence which is bounded by R.

Then, using the results for the general RTA for EDF-schedulers, we establish a response-time bound for the more concrete model of bounded nonpreemptive segments. Note that in case of the general RTA for EDF-schedulers, we just assume that the priority inversion is bounded. In this module we provide the preemption model with bounded nonpreemptive segments and prove that the priority inversion is bounded.