# Parameter Assignment And Schedulability Analysis For Real-Time Multiframe Task Systems

#### Abstract

Schedulability analysis has been considered as one of the most important subjects in real-time systems. Schedulability analysis decides whether all tasks work correctly and safely in a system. For example, the schedulability analysis of an Air Traffic Control (ATC) system should ensure that all airplanes do not have conflicts on departure lanes and are scheduled on time. In a modern car system, it has been shown that there are more than one hundred engine control units (ECUs), and more than twenty million lines of code in a typical modern car [19]. The scheduling of such complex systems is required to be well developed. As more sensors and functions (e.g., self-driving) will be added in a car, the scheduling of a car system faces more challenges that are caused by the dependent behaviors of functions, the suspending behaviors of functions, and randomness of parameters [19]. We tackle these challenges in this thesis, and we believe that the scheduling of similar large systems can also benefit from the techniques in this thesis.

The input of schedulability analysis is the information of task parameters such as execution times, deadlines, periods, etc. Parameters in a real-time system are typically immutable and assigned before the launch of schedulability analysis. Such immutable parameters lack flexibility and may lead to the failure of schedulability analysis. In order to tackle this problem, we let a subset of parameters be flexible to be chosen, specifically in tasks each of which contains dependable subtasks/frames. In this thesis, we introduce new flexible models GMF-PA (the generalized multiframe model with parameter adaptation) and dGMF-PA (the distributed GMF-PA) which let frame periods and deadlines be flexible to be chosen under certain constraints.

The GMF-PA and dGMF-PA models generalize the GMF model which extends the sporadic task model and multiframe task model. Each frame in the GMF model contains an execution time, a relative deadline, and a period (minimum inter-arrival time). These parameters are fixed after task specification time in the GMF model. However, systems such as multimedia and adaptive control systems may be overloaded and no longer stabilized when the task parameters in such systems are not flexible. In order to address this problem, task deadlines and periods may change to alleviate temporal overload, for example in the parameter adaptation and elastic scheduling model.

Our GMF-PA (dGMF-PA) model allows frame periods and deadlines to be flexible in arbitrary (constrained) -deadline systems. A necessary schedulability test based on mixed-integer linear programming (MILP) is given to check the schedulability under EDF scheduling and optimally assign frame deadlines and periods at the same time. We also prove that the test is a sufficient and necessary schedulability test when task parameters must be integers. A MILP-based approximation algorithm is also deployed to reduce computational running time and is a sufficient schedulability test in general. The speed-up factor of our approximation algorithm is 1 + ε where ε can be arbitrarily small, with respect to the exact schedulability test of GMF-PA (dGMF-PA) tasks under EDF scheduling.

We also present a pseudo-polynomial linear programming (LP)-based heuristic algorithm guided

by a concave approximation algorithm to achieve a feasible parameter assignment at a fraction of the time overhead of the MILP-based approach. The concave programming approximation algorithm closely approximates the MILP algorithm, and we prove its speed-up factor is (1 + δ)2 where δ > 0 can be arbitrarily small, with respect to the exact schedulability test of GMF-PA tasks under EDF. The LP-based heuristic algorithm takes shorter running time than the MILP-based heuristic algorithm, but the MILP-based heuristic algorithm has a lower speed-up factor in general.

In uniprocessor systems, we apply the GMF-PA model to self-suspending tasks. By extending the work on scheduling self-suspending tasks, we remove the assumption that deadlines are fixed after system specification time in self-suspending tasks, and the system is extended from constrained-deadline systems to arbitrary-deadline systems. We have done extensive experiments to show that the schedulability ratio is improved using our techniques in our GMF-PA model. We also analyze a case study on a robot car to show the effectiveness of the algorithms.

In distributed systems, we apply the dGMF-PA model to transactions (end-to-end tasks). By applying our algorithms on scheduling transactions in distributed systems, the schedulability ratio is improved compared to state-of-the-art algorithms. Since our parameter assignment is jointly considered with schedulability analysis, this combined technique dominates the previous parameter assignment algorithms based on trial and error.

*This paper has been withdrawn.*