Average-case performance analysis of scheduling random parallel tasks with precedence constraints on mesh connected multicomputer systems

Abstract

We investigate the problem of scheduling parallel tasks with precedence constraints on mesh connected multicomputer systems. It is still an open problem on whether there exists an approximation algorithm with finite asymptotic worst-case and/or average-case performance bound for this scheduling problem. As an early attempt to solve our problem, we propose and analyze the performance of a level-by-level scheduling algorithm LL. In fact, we solve a special case of the problem when all tasks request for square submeshes and run on a square mesh system whose size is a power of 2. There are three basic techniques in algorithm LL, i.e., the level-by-level scheduling strategy for handling precedence constraints, the largest-task-first algorithm for scheduling tasks in the same level, and the two-dimensional buddy system for system partitioning and processor allocation. Algorithm LL does not have a finite worst-case performance bound; however, it has quite acceptable average-case performance. The main contribution of the paper is to show that under the assumptions that task sizes are independent and identically distributed (i.i.d.) random variables with a common probability distribution, and that task execution times are i.i.d. random variables with finite mean and variance, and that the probability distributions of task sizes and execution times are independent of each other, for wide task graphs and typical task size distributions, algorithm LL has an asymptotic average-case performance bound about two for all probability distributions of task execution times.