RTA for FIFO Scheduling on Uniprocessors under the Periodic Resource Model

In the following, we derive a response-time analysis for FIFO schedulers, assuming a workload of sporadic real-time tasks, characterized by arbitrary arrival curves, executing upon a uniprocessor under the periodic resource model (see "Periodic Resource Model for Compositional Real-Time Guarantees" by Shin & Lee, RTSS 2003). To this end, we instantiate the sequential variant of abstract Restricted-Supply Analysis (aRSA) as provided in the prosa.analysis.abstract.restricted_supply module.

Defining the System Model

Before any formal claims can be stated, an initial setup is needed to define the system model under consideration. To this end, we next introduce and define the following notions using Prosa's standard definitions and behavioral semantics:

• tasks, jobs, and their parameters,
• the task set and the task under analysis,
• the processor model,
• the sequence of job arrivals,
• the absence of self-suspensions,
• an arbitrary schedule of the task set, and finally,
• the resource-supply model.

Tasks and Jobs

Consider tasks characterized by a WCET task_cost, an arrival curve max_arrivals, and a run-to-completion threshold task_rtct, ...

... and their associated jobs, where each job has a corresponding task job_task, an execution time job_cost, an arrival time job_arrival, and a list of preemption points job_preemptable.

The Task Set and The Task Under Analysis

Consider an arbitrary task set ts, and ...

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

Processor Model

Consider any kind of fully-supply-consuming, unit-supply processor state model.

The Job Arrival Sequence

Allow for any possible arrival sequence arr_seq consistent with the parameters of the task set ts. That is, arr_seq is a valid arrival sequence in which each job's cost is upper-bounded by its task's WCET, every job stems from a task in ts, and the number of arrivals respects each task's max_arrivals bound.

We assume that tsk is described by a valid task run-to-completion threshold. That is, there exists a task parameter task_rtct such that task_rtct tsk is

• (1) no larger than tsk's WCET, and
• (2) for any job of task tsk, the job's run-to-completion threshold job_rtct is bounded by task_rtct tsk.

Absence of Self-Suspensions

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

The Schedule

Consider a work-conserving, valid uniprocessor schedule that corresponds to the given arrival sequence arr_seq (and hence the given task set ts).

We assume that the schedule complies with some valid preemption model. Under FIFO scheduling, the specific choice of preemption model does not matter, since the resulting schedule behaves as if it were non-preemptive: jobs are executed in arrival order without preemption.

We assume that the schedule respects the FIFO scheduling policy.

Periodic Resource Model

Assume that the processor supply follows the *periodic resource model* as defined in "Periodic Resource Model for Compositional Real-Time Guarantees" by Shin & Lee (RTSS 2003). Under this model, the supply is distributed periodically with a resource period Π and a resource allocation time γ: in every interval [Π⋅k, Π⋅(k+1)), the processor provides at least γ units of supply. Furthermore, let prm_sbf Π γ denote the corresponding SBF defined in the paper, which, as proven in the same paper and verified in prosa.analysis.facts.model.sbf.periodic, is a valid SBF.

Maximum Length of a Busy Interval

In order to apply aRSA, we require a bound on the maximum busy-window length. To this end, let L be any positive solution of the busy-interval "recurrence" (i.e., inequality) prm_sbf Π γ L ≥ total_request_bound_function ts L, as defined below. As the lemma busy_intervals_are_bounded_rs_jlfp shows, under FIFO scheduling, this condition is sufficient to guarantee that the maximum busy-window length is at most L, i.e., the length of any busy interval is bounded by L.

Response-Time Bound

Having established all necessary preliminaries, it is finally time to state the claimed response-time bound R. A value R is a response-time bound for task tsk if, for any given offset A in the search space (w.r.t. the busy-window bound L), the response-time bound "recurrence" (i.e., inequality) has a solution F not exceeding A + R.

Finally, using the abstract restricted-supply analysis, we establish that any R that satisfies the stated equation is a sound response-time bound for FIFO scheduling on a unit-speed uniprocessor under the periodic resource model.