In the design of real-time systems, it is often the case that certain process parameters ( such as {\em execution time} ) are not known precisely. The challenge in real-time system design is to develop techniques that efficiently meet the requirements of impreciseness. Traditional models tend to simplify the issue of impreciseness by assuming {\em worst-case} times. This assumption is unrealistic and at the same time, may cause certain constraints to be violated at run-time. In this paper, we shall study the problem of scheduling a set of ordered, non-preemptive processes under non-constant execution times. Typical applications for variable execution time scheduling include process scheduling in Real-time Operating Systems such as Maruti, compiler scheduling, database transaction scheduling and automated machine control. An important feature of application areas such as robotics is the interaction between execution times of various processes. We explicitly model this interaction through the representation of execution time vectors as points in convex sets. We present both sequential and parallel algorithms for determining the existence of a static schedule. (Also cross-referenced as UMIACS-TR-2000-14)