1.8 Organization
2.1.1 Skippable Task Model
Among the early deterministic models is the skippable periodic/sporadic task model of Koren and Shasha [73]. In this model, a job of a task may be skipped, i.e., may not execute at all or have its execution aborted mid-way, as long as there is a minimum separation, referred to as the task’s skip parameter, between consecutive skips. Task τi’s skip parameter is denoted
si, where si ≥2, and has the interpretation that for every job of τi that is skipped, at least
si−1 following jobs are not to be skipped. Using Koren and Shasha’s terminology, a job that may be skipped is said to be blue, and one that has to complete by its deadline, red. This model was developed to support some signal-processing systems, where some samples can be skipped, and streaming applications that are tolerant to packet losses.
Let UR def= PNi=1
ei·(si−1)
pi·si denote the total utilization considering only jobs that may not
be skipped. One might reasonably expect that UR ≤ 1 is a sufficient feasibility condition. However, consider the task system τ = {τ1(3.5,4), τ2(1,4)}. Clearly, this task system is not
feasible on a uniprocessor if no job of any task may be skipped. It is also not feasible if only jobs of τ1 may be skipped regardless of the value of τ1’s skip parameter, s1. (Note that s1
should be at least two.) This is because if bothτ1 and τ2 release a red job each at the same
time, then at most one can complete execution by its deadline. For this example, withs1 = 2,
i.e., when every other job of τ1 may be skipped, UR = 38.5 + 14 = 58.5, which is less than 1.0. Hence, UR ≤ 1 is not a sufficient condition for ensuring the feasibility of a skippable task system.
1It should be noted that algorithms with provable properties have been developed for some very special
τ
2(3,4)τ
1(3,4) Red Red 0 2 4 6 8 0 2 4 6 8 Blue Blue Aborted Blue Blue Red Red Red time time 10 12(b)
(a)
Figure 2.1: Schedules for two concrete instances of a skippable sporadic task system.
As another example, task system τ ={τ1(3,4), τ2(3,4)} with s1 =s2 = 2 can be success-
fully scheduled. If τ is periodic, then a schedule in which odd jobs of τ1 and even jobs ofτ2,
or vice versa, are skipped, would not violate any constraints. Note that for this task system,
UR= 38+38 = 34, which is higher that the corresponding value for the previous example. Hence, skippable task systems cannot be compared for feasibility based on theirUR values.
Recall that the model allows jobs to be aborted mid-way and that an aborted job is also considered to be a skipped job. Allowing such behavior may be necessary for sporadic task systems because in such systems, whether a job is red or blue may have to be determined at run-time, based on how jobs of other tasks are released. For illustration, let the task system in the above example be sporadic and consider two partial sets of job releases as shown in insets (a) and (b) of Figure 2.1. In each inset, at most one of τ1,1 and τ1,2 can meet its deadline
and τ1,1 is chosen for completion. Therefore, in each inset, τ2,2 is a red instance, and as far
asτ1 is concerned, τ1,2 is skippable, and hence, can be blue. Also, if τ1,2 is skipped, then τ1,3
would be a red instance, whose deadline must be met. τ1,3’s deadline can be met if τ2,2 is
released as in inset (a). On the other hand, if the release of τ2,2 is postponed as in inset (b)
and τ1,3 is released in time, then at most one of these two jobs can meet its deadline, leading
to a timing-constraint violation. However, this scenario can be avoided becauseτ2,2’s release is
sufficiently delayed thatτ1,2’s deadline can be met andτ1,3can be turned into a blue instance.
Therefore, when tasks are sporadic, schedulability can be improved by dynamically choosing to not skip certain jobs.
It turns out that optimally determining which jobs to skip is NP-hard even if all job releases are knowna priori. Hence, Koren and Shasha proposed the use of algorithms based on simple heuristics. In their first algorithm, called red tasks only (RTO), every (k·si)th job, for every
k ≥ 1, is considered blue, and is skipped. The remaining jobs are scheduled by either EDF or RM. Note that the task system in the second example above is not schedulable by RTO, even if it is periodic. Their second algorithm is calledblue when possible (BWP), and tries to provide improved service in comparison to RTO in a best-effort manner by scheduling some blue job (chosen according to one of several heuristics) when no red job is pending. They also provided associated schedulability conditions for bothEDF andRM.
2.1.2 (m, k)-Firm Model
The (m, k)-firm model is similar to, and, in fact, subsumes the skippable task model, and was proposed by Hamdaoui and Ramanathan [62].2 In this model, each task is associated with two parameters m and k with the interpretation that at least m jobs in every window of k
consecutive jobs must meet their deadlines. Note that a skippable task with a skip parameter
sis also an (m, k)-task with parameters m=s−1 and k=s, and that while every skippable task can be represented as an (m, k) task the converse does not hold. Also, the (m, k) model can allow skips to be consecutive, and in that sense, can provide weaker guarantees than the skippable model.
Hamdaoui and Ramanathan proposed an algorithm called distance based priority (DBP) for scheduling (m, k)-tasks. Unlike Koren and Shasha’s algorithms, scheduling under DBP is dependent on run-time conditions. Hence,DBPmaintains a state, which captures the current history of met and missed deadlines, with each task. The task states are used to determine how “unsafe” each task is, i.e., how close it is to having its (m, k) constraint violated, and accord higher priority to tasks that are more unsafe. Tasks within the same priority class are scheduled on anEDF basis.
The goal of DBP is to schedule tasks so that the average probability of violating the (m, k) constraints is minimized and the realization of this goal is evaluated in [62] through simulations. The model and the algorithm were initially developed for scheduling streams of network packets, and hence, in the evaluations, the arrival pattern for jobs is neither periodic nor sporadic, but is either Poisson or bursty. In [63], an analytic model of DBP that can be
used to provide probabilistic guarantees on meeting the (m, k) constraint is developed, and its accuracy is evaluated through simulations.
The application of the (m, k) model to overloaded periodic task systems is considered in [98]. Here, as with RTO, static rules are provided for skipping jobs, and unskipped jobs are scheduled usingRM. A sufficient condition for deterministically meeting (m, k) constraints is also derived.