Enrico Bini

Measuring the Performance of Schedulability Tests

The high computational complexity required for performing an exact schedulability analysis of fixed priority systems has led the research community to investigate new feasibility tests which are less complex than exact tests, but still provide a reasonable performance in terms of acceptance ratio. The performance of a test is typically evaluated by generating a huge number of synthetic task sets and then computing the fraction of those that pass the test with respect to the total number of feasible ones. The resulting ratio, however, depends on the metrics used for evaluating the performance and on the method for generating random task parameters. In particular, an important factor that affects the overall result of the simulation is the probability density function of the random variables used to generate the task set parameters. In this paper we discuss and compare three different metrics that can be used for evaluating the performance of schedulability tests. Then, we investigate how the random generation procedure can bias the simulation results of some specific scheduling algorithm. Finally, we present an efficient method for generating task sets with uniform distribution in a given space, and show how some intuitive solutions typically used for task set generation can bias the simulation results.

Resource partitioning among real-time applications

When executing different real-time applications on a single processor system, one problem is how to compose these applications and guarantee at the same time that their timing requirements are not violated. A possible way of composing applications is through the resource reservation approach. Each application is handled by a dedicated server that is assigned a fraction of the processor. Using this approach, the system can be seen as a two-level hierarchical scheduler. A considerable amount of work has been recently addressed to the analysis of this kind of hierarchical systems. However, a question is still unanswered: given a set of real-time tasks to be handled by a server, how to assign the server parameters so that the task set is feasible? In this paper, we answer to the previous question for the case of fixed priority local scheduler by presenting a methodology for computing the class of server parameters that make the task set feasible.

Schedulability analysis of periodic fixed priority systems

Feasibility analysis of fixed priority systems has been widely studied in the real-time literature and several acceptance tests have been proposed to guarantee a set of periodic tasks. They can be divided in two main classes: polynomial time tests and exact tests. Polynomial time tests can efficiently be used for online guarantee of real-time applications, where tasks are activated at runtime. These tests introduce a negligible overhead, when executed upon a new task arrival, however provide only a sufficient schedulability condition, which may cause a poor processor utilization. On the other hand, exact tests, which are based on response time analysis, provide a necessary and sufficient schedulability condition, but are too complex to be executed on line for large task sets. As a consequence, for large task sets, they are often executed off line. This paper proposes a novel approach for analyzing the schedulability of periodic task sets on a single processor under an arbitrary fixed priority assignment: Using this approach, we derive a new schedulability test which can be tuned through a parameter to balance complexity versus acceptance ratio, so that it can be used on line to better exploit the processor, based on the available computational power. Extensive simulations show that our test, when used in its exact form, is significantly faster than the current response time analysis methods. Moreover the proposed approach, for its elegance and compactness, offers an explanation of some known phenomena of fixed priority scheduling and could be helpful for further work on schedulability analysis.

Rate monotonic analysis: the hyperbolic bound

We propose a novel schedulability analysis for verifying the feasibility of large periodic task sets under the rate monotonic algorithm when the exact test cannot be applied on line due to prohibitively long execution times. The proposed test has the same complexity as the original Liu and Layland (1973) bound, but it is less pessimistic, thus allowing it to accept task sets that would be rejected using the original approach. The performance of the proposed approach is evaluated with respect to the classical Liu and Layland method and theoretical bounds are derived as a function of n (the number of tasks) and for the limit case of n tending to infinity. The analysis is also extended to include aperiodic servers and blocking times due to concurrency control protocols. Extensive simulations on synthetic tasks sets are presented to compare the effectiveness of the proposed test with respect to the Liu and Layland method and the exact response time analysis.

Biasing effects in schedulability measures

The performance of a schedulabilty test is typically evaluated by generating a huge number of synthetic task sets and then computing the fraction of those that pass the test with respect to the total number of feasible ones. The resulting ratio, however, depends on the metrics used for evaluating the performance and on the method for generating random task parameters. In particular, an important factor that affects the overall result of the simulation is the probability density function of the random variables used to generate the task set parameters. In this paper we discuss and compare three different metrics that can be used for evaluating the performance of schedulability tests. Then, we investigate how the random generation procedure can bias the simulation results of some specific scheduling algorithm. Finally, we present an efficient method for generating task sets with uniform distribution in a given space, and show how some intuitive solutions typically used for task set generation can bias the simulation results.

Sensitivity analysis for fixed-priority real-time systems

At early stages in the design of real-time embedded applications, the timing attributes of the computational activities are often incompletely specified or subject to changes. Later in the development cycle, schedulability analysis can be used to check the feasibility of the task set. However, the knowledge of the worst-case response times of tasks is often not sufficient to precisely determine the actions that would correct a non-schedulable design. In these situations, sensitivity analysis provides useful information for changing the implementation, by giving a measure of those computation times that must be reduced to achieve feasibility, or those that can be increased in case of a product extension, or providing the range of feasible periods for selecting the proper task activation rates. In this work, we exploit the concept of feasibility region to propose a faster and more concise solution to the sensitivity analysis problem with respect to existing techniques based on binary search. Furthermore, we show how the formalization of other problems in the feasibility domain, such as managing overloads through elastic scheduling, can be extended to the exact analysis.

A hyperbolic bound for the rate monotonic algorithm

In this paper we propose a novel schedulability analysis for verifying the feasibility of large periodic task sets under the rate monotonic algorithm, when the exact test cannot be applied on line due to prohibitively long execution times. The proposed test has the same complexity as the original Liu and Layland bound but it is less pessimistic, so allowing to accept task sets that would be rejected using the original approach. The performance of the proposed approach is evaluated with respect to the classical Liu and Layland method, and theoretical bounds are derived as a function of n (the number of tasks) and for the limit case of n tending to infinity. The analysis is also extended to include aperiodic servers and blocking times due to concurrency control protocols. Extensive simulations on synthetic tasks sets are presented to compare the effectiveness of the proposed test with respect to the Liu and Layland method and the exact response time analysis.

Delay-Aware Period Assignment in Control Systems

We consider the problem of optimal static period assignment for multiple independent control tasks executing on the same CPU. Previous works have assumed that the control performance can be expressed as a function of the sampling rate only. Arguing that the control delay has a large impact on the control performance, in this work we include the control delay in the cost function. The delay is estimated using an approximate response-time analysis. Assuming linear cost functions for the controllers then allows us to solve the optimal period assignment problem analytically. The performance improvements over previous methods are verified in evaluations on synthetic task sets as well as detailed co- simulations of the controllers, the plants, and the scheduler.

Speed modulation in energy-aware real-time systems

This paper presents a general framework for analyzing and designing embedded systems with energy and timing requirements. A set of realistic assumptions is considered in the model in order to apply the results in practical realtime applications. For example, the processor is assumed to have as a set of discrete operating modes, each characterized by speed, power consumption. The transition delay between modes is considered. To take I/O operations into account, task computation times are modeled with a part that scales with the speed and a part having a fixed duration. Given a set of real-time tasks, the proposed method allows to compute the optimal sequence of voltage/speed changes that approximates the minimum continuous speed which guarantees the feasibility of the system. The analysis is performed both under fixed and dynamic priority assignments.

On Lyapunov sampling for event-driven controllers

This paper investigates an event condition for event-driven controllers based on Lyapunov functions. Considering that constant values of a Lyapunov function define contour curves that form closed regions around the equilibrium point, in this paper we present a sampling mechanism that enforces job executions (sampling, control algorithm computation and actuation) each time the system trajectory reaches a given contour curve. By construction, the sequence of generated samples is stable in the discrete Lyapunov sense. However, in order to ensure that the system trajectory will tend to zero as time tends to infinity, it must be ensured that the sequence of samples is infinite. We provide conditions to ensure this property. The approach is illustrated by simulated examples.