In this research report we study both the throughput and the energy optimization problem for a distributed system subject to failures that executes a workflow at different speed levels. The application is modeled as a directed acyclic graph composed of typed tasks linked by dependency constraints. A continuous flow, or a great number of application instances has to be processed optimizing the collaborative system performance which implies to increase the throughput -- the number of application instances processed by time unit --~or to decrease the period~-- the time needed to output one instance of the system. The system is designed as a collaborative platform of distributed machines. Each machine collaborates with others by performing all the instances of at least one task of the DAG. The problem we tackle is to optimize the configuration of the platform. In this report we propose two polynomial algorithms that optimize the two objectives of period (i.e., throughput) and energy minimization and we prove that the proposed algorithms give optimal results. Our optimization approach is hierarchic in the sens that we either minimize the energy consumption for an optimal period or minimize the period for the optimal energy consumption.
