Optimizing over FP/EDF Execution Times: Known Results and Open Problems

TL;DR

This paper reviews existing results on the formulation of both the Fixed Priority and Earliest Deadline First exact schedulability constraints and expresses the formulations by a combination of linear constraints, which enables then optimization routines.

Abstract

In many use cases the execution time of tasks is unknown and can be chosen by the designer to increase or decrease the application features depending on the availability of processing capacity. If the application has real-time constraints, such as deadlines, then the necessary and sufficient schedulability test must allow the execution times to be left unspecified. By doing so, the designer can then perform optimization of the execution times by picking the schedulable values that minimize any given cost. In this paper, we review existing results on the formulation of both the Fixed Priority and Earliest Deadline First exact schedulability constraints. The reviewed formulations are expressed by a combination of linear constraints, which enables then optimization routines.

Figures (7)

• Figure 1: An example of the schedulability points $\mathcal{S}_3$ and $\mathcal{P}_2(D_3)$ for a set of $3$ tasks with $T_1=3$, $T_2=8$, and $D_3=19$. Notice that the set of points does not depend on the execution times.
• Figure 2: Region of RM schedulable execution times. We draw $C_2$ along the horizontal axis and $C_1$ along the vertical axis. In this example, we assume $n=2$ tasks and parameters: $T_1=4$, $D_1=3$, and $D_2=5$ and any $T_2\geq D_2$. From (\ref{['eq:iff_RM_schedP']}), when $i=1$ we have $\mathcal{P}_0(D_1)=\{D_1\}=\{3\}$ which gives $C_1\leq 3$ as the only (and trivial) necessary and sufficient constraint to guarantee the schedulability of $\tau_1$, the highest priority task. When $i=2$ the schedulability points are $\mathcal{P}_2(D_2)=\{5,4\}$, which yield the constraints $2C_1+C_2\leq 5$ and $C_1+C_2\leq 4$, respectively, both represented by thin lines. We need to make union among these constraints, as required by the logical OR of (\ref{['eq:iff_RM_schedP']}), thus getting the two oblique thick segments. Since the overall schedulability region is given by the intersection of the single-task regions, we find that the RM-schedulable execution times are the ones represented in the cyan area.
• Figure 3: The tight set of necessary and sufficient constraints for EDF. In this example, the parameters are $T_1=2, D_2=3$, $T_2=5, D_2=5$, and $T_3=7, D_3=6$. Job releases are represented by upward black arrows. Job deadlines are represented by downward red arrows. We represent the deadlines until $H+\max\{D_i\}=70+6=76$, as required by (\ref{['eq:def_dlSet']}). The number of total constraints to be checked is $49$ corresponding to $48$ deadlines plus the utilization constraint of (\ref{['eq:tot_uleq1']}). The reduced number of constraints, however, is only $5$: $4$ deadlines (circled in green) plus the utilization constraint.
• Figure 4: Region of EDF schedulable execution times. As in Figure \ref{['fig:workRM']}, we draw $C_2$ along the $x$ axis and $C_1$ along the $y$ axis. Using the same example of Figure \ref{['fig:workRM']}, we assume $2$ tasks with parameters: $T_1=4$, $D_1=3$, and $T_2=D_2=5$. In this case, the set $\mathcal{D}$ from (\ref{['eq:def_dlSet']}) of all deadlines to be considered is $\mathcal{D}=\{3, 5, 7, 10, 11, 15, 19, 0\}$. We remind that the "deadline $0$" represents the utilization constraint of (\ref{['eq:tot_uleq1']}). All constraints are represented by a thin line, whereas the boundary of their intersection is represented by a thicker line. It can be observed that a large majority of the constraints does not contribute to determine the boundary of EDF-schedulable execution times. In this example, only the $2$ deadlines at $3$ and at $15$ are needed to characterize the exact region.
• Figure 5: In this experiment, randomly generated integer task period and deadlines for $n=2$ tasks. For each experiment, we are plotting a dot at the corresponding hyperperiod $H$ along the horizontal axis in log scale, and the number of necessary and sufficient constraints along the vertical axis. At the right, also the density of the minimal number of constraints over the sample space. We also plot the upper envelope of the points as such an envelope represents the hardest instances to be tested.

Theorems & Definitions (3)

• Theorem 1: from Leh89
• Theorem 2: Corollary 1 in Bar90a
• Corollary 1