Performance analysis of parallel programs based on directed acyclic graphs

Abstract.

Task precedence graphs are widely used for modeling and evaluation of parallel applications. Their nodes represent the subtasks of the parallel program and the edges represent the precedence relations between the subtasks. The execution times of the subtasks are described by random variables and their distributions. In our paper we introduce a new class of distributions, particularly suited for the modeling and evaluation of parallel programs. Exponential polynomials introduced by Sahner and Trivedi have the disadvantage that a large number of parameters is needed for the representation of realistic task execution times, which usually have a small value of variation. We extend this class to derive the class of truncated $\theta$-exponential polynomials which allow the representation of realistic task execution times with fewer parameters. Additionally this class of distributions has the advantage that minimum as well as maximum execution times can be guaranteed. Models with a large number of subtasks $n$ can not be evaluated on a computer using exact analytical methods because of memory requirements and numerical inaccuracies, which accumulate, when the operations of analysis are applied. Using extreme value theory we derive approximate formulas for the parallel independent execution of $n$ subtasks, a structure, which can be found in every parallel program. The obtained results for truncated and not truncated distributions show, that distributions with an infinite domain are not suitable, particularly for massively parallel structures.