In this paper, we consider the execution of a complex application on a heterogeneous "grid" computing platform. The complex application consists of a suite of identical, independent problems to be solved. In turn, each problem consists of a set of tasks. There are dependences (precedence constraints) between these tasks. A typical example is the repeated execution of the same algorithm on several distinct data samples. We use a non-oriented graph to model the grid platform, where resources have different speeds of computation and communication. We show how to determine the optimal steady-state scheduling strategy for each processor (the fraction of time spent computing and the fraction of time spent communicating with each neighbor) and how to build such as schedule. This result holds for a quite general framework, allowing for cycles and multiple paths in the platform graph.