In this paper, we consider steady-state scheduling techniques for mapping a collection of task graphs onto heterogeneous systems, such as clusters and grids. We advocate the use of steady-state scheduling to solve this difficult problem. Due to space limitations, we concentrate on complexity results. We show that the problem of optimizing the steady-state throughput is NP-complete in the general case. We formulate a compact version of the problem that belongs to the NP complexity class but which does not restrict the optimality of the solution. We provide many positive results in the extended version (Beaumont et al., 2004). Indeed, we show how to determine in polynomial time the best steady-state scheduling strategy for a large class of application graphs and for an arbitrary platform graphs, using a linear programming approach.