The Parametric Polytope and its applications to a Scheduling Problem

An important feature in Real-time systems is parameter impreciseness, i.e. the inability to accurately determine certain parameter values. The most common such parameter is task execution time. A second feature is the presence of complex relationships between tasks that constrain their execution. Traditional models do not accomodate either feature completely: (a) Variable execution times are modeled through a xed value (worst-case), and (b) Relationships are limited to those that can be represented by precedence graphs. We present a task model that eeectively captures variable task execution time, while simultaneously permitting arbitrary linear relationships between tasks. Our model nds applications in diverse areas such as real-time task scheduling, compiler scheduling, real-time database scheduling and machine control. This paper focuses primarily on the computational complexity of answering queries posed in our model; in particular we demonstrate the existence of constraint classes that make the scheduling problem hard.