RTSOPS Steering Committee

Certain control computations may be modeled as periodic tasks with the correctness requirement that for each task, the fraction of jobs of the task that complete execution by their respective deadlines be no smaller than a specified value. This appears to be a correctness requirement that has not previously been studied in the real-time scheduling theory community. In feedback (or closed loop) control, the state of a plant is monitored, the error — deviation of the monitored value from a desired value – is determined, and a control signal that should decrease the error is computed and applied to the plant; this entire process is repeated periodically. The computation of the control signal in such control loops is often modeled as a periodic task, with each iteration of the loop represented by a job of the task. In many application systems there are multiple control loops, each responsible for controlling a different aspect of the system’s behavior, that run simultaneously upon a shared platform; determining an appropriate strategy for scheduling these control computations is thus a periodic scheduling problem. The property of bounded input bounded output (BIBO) stability is desired of certain control systems. (Loosely speaking, a control system is BIBO stable if every bounded input to the system results in a bounded output.) Branicky, Phillips, and Zhang [2] explored the effect upon stability of skipping the computation of the control signal during some iterations and instead reusing the value that was used during the previous iteration. They derived techniques for determining a minimal fraction of control-signal computations that must be completed in order to ensure stability; in follow-up work, Majumdar, Saha, and Zamani [5] derived techniques for determining the minimal fraction of such computations that must be completed in order to additionally achieve optimal disturbance rejection performance. The work in [2], [5] suggests the following task model for the scheduling of control computations. Task Model: Each control loop is modeled as a control task τi that is characterized by three parameters: τi = (Ci, Ti, ri), where Ci is the WCET, Ti the period, and ri a positive real number ≤ 1 denoting the desired asymptotic completion rate for task τi. Such a task generates jobs; the k’th job generated by τi has a release time1 (k − 1)× Ti, an execution requirement no greater than Ci, and a deadline k × Ti, for all integer k ≥ 1. It is not required that all the jobs generated by τi execute. Let k1, k2 denote positive integers with k1 ≤ k2, and let COMPi(k1, k2) denote the number of jobs out of the k1’th, (k1 + 1)’th, (k1 + 2)’th, · · · , (k2 − 1)’th, k2’th jobs generated by τi that do complete execution by their deadlines in some schedule. For the schedule to be considered correct for τi, it is required that lim k→∞ (COMPi(1, k) k ) ≥ ri (1) Note that this is an asymptotic notion of correctness, which does not restrict what may happen with any sequence of successive jobs of any particular length. Consider, for example, a task τi with ri = 12 ; as per the definition above, a schedule that skips every alternate job is equally acceptable as one that skips the first million jobs and then schedules the next million.2 THE OPEN PROBLEM Given a collection of n control tasks of the kind described above, determine whether it can be scheduled correctly upon a preemptive uniprocessor. Some obvious extensions of this open problem are also of interest, both individually and in combinations: 1) Determine appropriate algorithms for scheduling collections of control tasks, and design efficient schedulability conditions for these scheduling algorithms. 2) Consider the multiprocessor version of this problem. 1. I.e., the tasks are periodic rather than sporadic, and all tasks have an initial offset equal to zero. 2. Clearly, this fact indicates that something was lost in translation in [2], [5] from control loops to periodic tasks – some unstated assumption that bounds, for example, the maximum deviation of ( COMPi(k1, k2)/(k2 − k1 + 1) ) from ri for all k1, k2. 3 3) Consider more general task models. This could include periodic tasks with offsets; sporadic (rather than periodic) arrivals; models in which tasks are characterized by a relative deadline parameter in addition to WCET and period; and further generalizations. As stated in footnote 2, the control tasks model appears inadequate for truly expressing desired properties of schedules for control tasks. We believe that it is important and potentially interesting to construct a more realistic task model for control loops that does indeed specify some such additional constraint. We consider the construction of such a model as an important open problem, which is probably best tackled by scheduling-theory and controltheory researchers working collaboratively. RELATIONSHIP TO PRIOR WORK Several models have previously been proposed in the real-time systems literature that allow for the specification of the fact that some jobs of a task may be skipped during execution. Such models include the (m, k)-firm model [3], models that allow for the specification of a skip factor [4], and the weakly-hard task model [1]. However, it should be evident that the control tasks model described above differs widely from these prior models, all of which specify allowed job-drops within an interval of a certain number of successive jobs. In contrast, the control tasks model only requires that the fraction of correctly-scheduled jobs be no smaller than a specified ratio asymptotically. This characteristic of a real-time requirement that is only required to hold over arbitrarily large time intervals (as time approaches infinity), appears to be novel and previously completely unexplored. A necessary (but not sufficient) schedulability condition. Observe that the quantity ( ri× Ci Ti ) denotes a tight upper bound upon the fraction of the computing capacity of a shared unit-speed processor that may need to be devoted to executing the i’th control task τi; hence, N ∑ i+1 ri Ci Ti ≤ 1 (2) is clearly a necessary condition for the system {τ1, τ2, . . . , τN} to be schedulable upon a unit-speed processor. However, this condition is not sufficient: consider the system {τ1, τ2, τ3} with τ1 = τ2 = τ3 = (6, 10, 1/2). The summation evaluates to 0.9, but this instance is not schedulable since only one job can “fit” into each period while each of the three tasks desires to be scheduled during half the periods. (This example serves as a counter-example to [5, (Proposition 1)], which had claimed that Condition 2 is an exact test.) A sufficient (but not necessary) schedulability condition. A trivial sufficient schedulability condition is obtained by simply not exploiting the ability to skip jobs. In that case, it is obvious that