Evolución del concepto de soberanía alimentaria

Some tasks with widely varying inter-arrival times or execution requirements, such as those in video-conferencing applications or animation games, require soft real-time guarantees such as bounds on tardiness or limits on the percentage of deadlines missed. Hence, it may not be appropriate to schedule such systems using the server-based approaches described earlier in which a single server serves more than one task or in which needed guarantees cannot be provided. For such systems, one alternative to reserving resources based on worst-case param- eters, which can be extremely wasteful, is to model task parameters probabilistically (using probability distributions for inter-arrival times or execution times) and to provide probabilis- tic guarantees on meeting deadlines. In this section, we describe some efforts taken in this direction.

2.2.1 Semi-Periodic Task Model

In [115], Tia et al. considered scheduling tasks whose jobs have highly varying execution requirements but are released periodically. Tia et al. referred to such tasks as semi-periodic tasksand characterized their execution requirements using generic probability density functions (pdfs). They then extended the time-demand and generalized time-demand analyses [77, 75]

5_{It should be noted that if the relative deadlines of all tasks are not increased by the same amount, then job}

used with static-priority algorithms for semi-periodic task systems under the assumption that the worst-case response time for each task can be found in a busy interval that begins when every task releases a job. Whereas standard time-demand analysis simply sums the demand due to individual jobs that execute in a busy interval, in the probabilistic analysis proposed by them, a convolution of the pdf’s of the execution times of the jobs is taken. The outcome of the convolution is a pdf for the total demand at any time within the busy interval. This pdf can then be used to compute the probability that a given job’s response time exceeds its deadline. Tiaet al. noted that their approach is valid only if job execution times are independent and discussed how to correctly compute failure probabilities in the absence of this independence assumption.

In the same paper, an alternative approach called task transformation is proposed for scheduling semi-periodic tasks. The intention is to guarantee 100% schedulability for short jobs and some fraction of each long job. Hence, it is proposed that each job of a semi-periodic task be logically decomposed into two components: a fixed-size periodic component and a variable-size sporadic component. The execution requirement is constant for the periodic components of all the jobs and may vary for the sporadic components. Also, the periodic components are guaranteed deterministically, whereas the sporadic components are served by special server tasks of appropriate utilizations and are provided probabilistic guarantees. (A single server task may serve multiple sporadic component tasks.) Note that, in some sense, the periodic and sporadic components are reminiscent of the mandatory and optional parts of the imprecise computation model.

The semi-periodic task model was also considered by Diaz et al. in [55]. However, the analysis they provided is different from that of Tiaet al.. Diazet al.modeled the varying exe- cution costs using discrete random variables characterized by probability mass functions. They showed how the state of a priority-driven system can be modeled as a discrete-time Markov chain (DTMC) and how a response-time distribution may be determined (both numerically and analytically) using it.

2.2.2 Statistical Rate-Monotonic Scheduling

The goal of statistical rate-monotonic scheduling (SRMS) proposed by Atlas and Bestavros in [18] is to schedule periodic task systems such that the statistical quality-of-service (QoS) guarantee needed by each task is met. The task model considered in this work is similar to and is based on the semi-periodic model described above, and associates with each task τi

a constant inter-arrival time, a pdf characterizing the utilization of its jobs (using which the execution requirement may be computed), and a permissible QoS. The QoS of a task is defined as a lower bound on the probability that an arbitrary job will meet its deadline.

As explained in [18], the main tenet of SRMS is that variability in task execution require- ments can be smoothened by aggregating the requirements of successive jobs. SRMSis based on RM, and for each taskτi, treats jobs that fit within asuperperiod as a unit. The length of

τi’s superperiod is given by that of the period of the next lower-priority task,τi+1. UnderRM,

at most _dpi+1

pi e jobs of τi can interfere with any job ofτi+1. Also,τi+1 is not impacted byhow

the aggregate cost is distributed amongτi’s jobs as long as the distribution is among the jobs within a superperiod. SRMSmakes use of these properties to ensure that statistical guarantees are met. Hence, at run-time, for each task, allocations made to jobs in its current superperiod are maintained; this information, along with the resource needs of all higher-priority tasks, is used to determine whether the deadline of a newly arriving job can be met. (It is assumed that the execution requirement of a job is known when it is released.) A job whose deadline cannot be met is simply rejected, and thus, every job that is admitted is guaranteed to meet its deadline. Offline probabilistic analysis ensures that the percentage of discarded jobs does not violate the QoS requirement of the task.

The main differences between SRMS and the approaches considered in Section 2.2.1 are that in SRMS, only jobs guaranteed to meet their deadlines are admitted to the system, and higher-priority tasks cannot overrun and infringe on lower-priority tasks. One critique of this model, however, is that the assumption that job execution requirements are known at their release times is somewhat questionable.

2.2.3 Constant-Bandwidth Server

In the work considered so far in this section, only execution costs of the jobs of a task are allowed to vary. Tasks whose job inter-arrival times may also vary were considered by Abeni and Buttazzo in [5], [6], and [7]. Unlike the previous approaches, Abeni and Buttazzo considered scheduling each variable-parameter task using a separate, dedicated server, referred to as a

constant bandwidth server (CBS). Recall that a server is a periodic or a sporadic task with a budget (i.e., execution cost) and a period. Under CBS, a server task’s budget and period are set to the mean execution cost and the mean inter-arrival time, respectively, of the task it serves. Server rules are defined such that, in the long run, each client task is allocated a fraction of the total processor time approximately equal to its mean utilization, and the

different clients are temporally isolated6 _{from one another. Thus, in effect, an appropriate}

fraction of the processor isreserved for each variable-parameter task.

Abeni and Buttazzo also presented analysis for tasks scheduled using CBS servers. Their analysis can, in general, be applied to any reservation-based system. However, their analysis is restricted to allowing only one of the two parameters (either execution cost or inter-arrival time) to vary. They showed how to model the state of a CBS-served task with one varying parameter as a discrete-time Markov chain, and how to compute a steady-state probability for any given tardiness for the task. Because each task is reserved a fraction of a processor, they were able to model each task as an independent stochastic process and analyze it independently, without consideration of interference from the other tasks. In our opinion, this considerably simplifies their analysis in comparison to those discussed for semi-periodic tasks in Section 2.2.1.

2.2.4 Real-Time Queueing Theory

The probabilistic analyses that we have considered so far allow at most one parameter of each real-time task to be stochastic. In contrast, the grand aim ofreal-time queueing theory seems to be to allow every aspect of the traditional real-time system model to be stochastic, if needed, and develop tools for analyzing such systems scheduled under priority-driven algorithms. Real- time queueing theory strives to achieve this aim by combining the timing elements of real-time scheduling theory with the stochastic elements of queueing theory. This theory was first proposed by Lehoczky in [76], where a semi-formal analysis is presented for systems with a single queue under heavy-traffic conditions.7 (Each queue corresponds to a task, so a single- queue system is essentially a system with a single task serving aperiodic jobs.) Fully-developed theory for single-server, single-queue systems scheduled under EDF and FIFO is presented in [56], and that for acyclic networks of servers with multiple independent queues in [74]. It should be noted that apart from inter-arrival times and service times, job deadlines are also modeled as independent and identically distributed random variables. However, it is not clear whether the assumption that all times are independent is realistic, and what removing this assumption entails.

6_{A set of tasks is said to betemporally isolated if execution overruns of any task cannot impact any of the}

remaining tasks.

7_{Under heavy traffic conditions, the traffic intensity or the average processor utilization converges to one.}

According to Lehoczky, the heavy-traffic case is the worst case for real-time systems, and hence, can be used as an upper bound for lighter conditions.

One critique of the probabilistic models by proponents of other models is that percentage of deadline misses or probability of meeting a deadline by itself is not sufficient to assess quality of service unless the distribution of misses is also known. With that note, we conclude our discussion on probabilistic models.