Probabilistic Analysis

Abstract

The classical model of a real-time system consists of a number of tasks, each of which has an execution time which is upper bounded by a constant, referred to as the worst-case execution time (WCET). Further, jobs of each task execute periodically or sporadically, subject to some minimum inter-arrival time. Task execution is controlled by a real-time scheduler that determines, at any given time, which of the ready jobs the processor will execute. For such a model, schedulability analysis provides an a priori mathematical verification indicating whether or not all of the jobs of each task can be guaranteed to meet their deadlines under the particular scheduling policy used. This analysis is typically achieved by determining the worst-case scenario that leads to the worst-case response time (from the release to the completion of any job of the task), calculating the worst-case response time, and comparing it with the task’s deadline. Probabilistic real-time systems differ from this classical model in two main ways. Firstly, at least one parameter of the tasks (e.g., execution time) is modeled as a random variable, i.e., described by a probability distribution. Secondly, rather than requiring an absolute guarantee that all deadlines must be met, timing constraints are specified in terms of a threshold on the acceptable probability of a deadline miss for each task. This chapter focuses on research into scheduling and specifically schedulability analysis for probabilistic real-time systems.

Appendix: Task Set Generation

This appendix details a simple approach to generating task sets with probabilistic parameters that are suitable for empirical assessment of the performance of different scheduling algorithms and probabilistic schedulability analyses.

STEP 1:

Generate the worst-case utilizations ($U_i = C_i^{max}/T_i$) for each of the n tasks using the UUnifast algorithm (Bini and Buttazzo 2005) to give an unbiased distribution of maximum utilization values.

STEP 2:

Generate the task periods according to a log-uniform distribution (Emberson et al. 2010). For example, the range of task periods may span two orders of magnitude, e.g., from 10 to 1000 ms.

STEP 3:

Obtain the worst-case execution time of each task from its utilization and period as follows: $C_i^{max}=U_i T_i$.

STEP 4:

The best case execution time of each task may be obtained by using a fixed multiplier on the worst-case execution time, $C_i^{min}={SF} \cdot C_i^{max}$, where SF is the scaling factor.

STEP 5:

Task deadlines can be implicit, i.e., equal to the task period or constrained, i.e., no larger than the period. Constrained deadlines may be chosen from a uniform distribution in the range $[C_i^{max}, T_i]$.

STEP 6:

The size of the pWCET distribution is given as an input parameter to the probabilistic real-time task generator. If the size is 1, then the distribution has a single value, i.e., $C_i^{max} = C_i^{min}$, with probability equal to 1.

STEP 7:

The probability associated with $C_i^{max}$ can also be given as input to the task generator. It is expected that this value is small, for example, in the range [10⁻⁶, 10⁻¹²], since it is expected that the probability of extreme execution times is very small (Cucu-Grosjean et al. 2012).

The pWCET distribution for each task can then be generated via extrapolation from the $C_i^{min}$ and $C_i^{max}$ parameter values, using the probability for the maximum value, and assuming that the distribution has an exponential tail. Thus the 1-CDF of the pWCET, plotted on an exceedance graph with probabilities given on a log scale, is as depicted in Fig. 2. Each line ends with the right most point at $C_i^{max}$ and connects the intermediate points via a straight line (exponential tail). The left most point, at $C_i^{min}$, collects the remaining part of the distribution so that the probability mass sums to 1. (Note the longer lines are for a scaling factor of SF = 0.33 and thus show more execution time variation than the shorter lines which are for SF = 0.73.)

Fig. 2

Example of possible pWCET distributions

STEP 8:

Task priorities may be set using the algorithm presented in Sect. 3.