# A Backward Algorithm for the Multiprocessor Online Feasibility of Sporadic Tasks

## TL;DR

This paper introduces an optimized backward algorithm using antichain techniques to efficiently solve the online feasibility problem for multiprocessor sporadic tasks, significantly improving performance over previous methods.

## Contribution

It adapts and enhances the classical attractor algorithm with antichain techniques to address the intractability of the game graph in multiprocessor feasibility testing.

## Key findings

- Dramatic performance improvements in solving the feasibility game.
- Effective handling of large game graphs with antichain optimization.
- Potential for practical application in real-time multiprocessor scheduling.

## Abstract

The online feasibility problem (for a set of sporadic tasks) asks whether there is a scheduler that always prevents deadline misses (if any), whatever the sequence of job releases, which is a priori} unknown to the scheduler. In the multiprocessor setting, this problem is notoriously difficult. The only exact test for this problem has been proposed by Bonifaci and Marchetti-Spaccamela: it consists in modelling all the possible behaviours of the scheduler and of the tasks as a graph; and to interpret this graph as a game between the tasks and the scheduler, which are seen as antagonistic players. Then, computing a correct scheduler is equivalent to finding a winning strategy for the `scheduler player', whose objective in the game is to avoid deadline misses. In practice, however this approach is limited by the intractable size of the graph. In this work, we consider the classical attractor algorithm to solve such games, and introduce antichain techniques to optimise its performance in practice and overcome the huge size of the game graph. These techniques are inspired from results from the formal methods community, and exploit the specific structure of the feasibility problem. We demonstrate empirically that our approach allows to dramatically improve the performance of the game solving algorithm.