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 and these dependences are organized as a tree. 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). This result holds for a quite general framework, allowing for cycles and multiple paths in the platform graph.