RTA for Fully Preemptive FP Scheduling on Uniprocessors with Overheads

In the following, we derive a response-time analysis for FP schedulers, assuming a workload of sporadic real-time tasks, characterized by arbitrary arrival curves, executing upon a uniprocessor subject to scheduling overheads. 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:

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

Processor Model

Consider a unit-speed uniprocessor subject to scheduling overheads.

Tasks and Jobs

Consider any type of tasks, each characterized by a WCET task_cost and an arrival curve max_arrivals, ...

... and any type of jobs associated with these tasks, where each job has a corresponding task job_task, an execution time job_cost, and an arrival time job_arrival.

Furthermore, assume that jobs and tasks are fully preemptive.

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.

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.

Additionally, we assume that all jobs in arr_seq have positive execution costs. This requirement is not fundamental to the analysis approach itself but reflects an artifact of the current proof structure specific to upper bounds on the total duration of overheads.

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 an arbitrary fixed-priority policy FP that indicates a higher-or-equal priority relation among the tasks in ts, and assume that the relation is reflexive and transitive.

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

We assume that the schedule respects the given FP scheduling policy.

Furthermore, we require that the schedule ensures two additional properties: jobs of the same task are executed in the order of their arrival, ...

... and preemptions occur only when strictly required by the scheduling policy (specifically, a job is never preempted by another job of equal priority).

Overhead-Aware Supply-Bound Function

We assume that the scheduling overheads encountered in the schedule sched are bounded by the following upper bounds:

• the maximum dispatch overhead is bounded by DB,
• the maximum context-switch overhead is bounded by CSB, and
• the maximum cache-related preemption delay is bounded by CRPDB.

In order to account for the maximum cumulative delay that task tsk may experience due to the presence of scheduling overheads, we introduce a supply-bound function SBF that conservatively models the restriction of processor supply due to overheads (i.e., it lower-bounds the amount of "useful" processor cycles not lost to overheads). The overhead-aware SBF works by determining a blackout bound, which upper-bounds the maximum cumulative duration during which processor cycles are "lost" to scheduling overheads in a given interval. Under FP scheduling, the blackout bound in an interval of length Δ is determined by the arrivals of tasks with higher-or-equal priority relative to tsk. In this case, up to n such arrivals can lead to at most 1 + 2n segments without a schedule change, each potentially incurring dispatch, context-switch, and preemption-related overhead. We let SBF denote the supply-bound function expressing this bound for the task under analysis tsk.

Workload Abbreviations

For brevity in the following definitions, we introduce a number of local abbreviations. We let rbf denote the task request-bound function, which is defined as task_cost(T) × max_arrivals(T,Δ) for a task T.

Additionally, we let total_hep_rbf denote the cumulative request-bound function w.r.t. all tasks with higher-or-equal priority ...

... and use total_ohep_rbf as an abbreviation for the cumulative request-bound function w.r.t. all tasks with higher-or-equal priority other than task tsk itself.

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) total_hep_rbf L ≤ SBF tsk L, as defined below. As the lemma busy_intervals_are_bounded_rs_fp shows, under FP 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 R.

Finally, using the sequential variant of abstract restricted-supply analysis, we establish that, given a bound on the maximum busy-window length L, any such R is indeed a sound response-time bound for task tsk under fully-preemptive fixed-priority scheduling on a unit-speed uniprocessor subject to scheduling overheads.