RTA for EDF with Floating Non-Preemptive Regions

In this module we prove the RTA theorem for floating non-preemptive regions EDF model. Throughout this file, we assume the EDF priority policy, ideal uni-processor schedules, and the basic (i.e., Liu & Layland) readiness model.

Furthermore, we assume the task model with floating non-preemptive regions.

Setup and Assumptions

Consider any type of tasks ...

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

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

Assume we have the model with floating non-preemptive regions. I.e., for each task only the length of the maximal non-preemptive segment is known _and_ each job level is divided into a number of non-preemptive segments by inserting preemption points.

Consider an arbitrary task set ts, ...

... assume that all jobs come from this 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.

Next, consider any ideal uni-processor schedule with limited preemptions of this arrival sequence ...

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

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

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

... and the schedule respects the policy defined by the job_preemptable function (i.e., jobs have bounded non-preemptive segments).

Total Workload and Length of Busy Interval

We introduce the 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).

We define a bound for the priority inversion caused by jobs with lower priority.

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

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

Response-Time Bound

To reduce the time complexity of the analysis, recall the notion of search space.

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.

Now, we can leverage the results for the abstract model with bounded nonpreemptive segments to establish a response-time bound for the more concrete model with floating nonpreemptive regions.