Clusters and computational grids are opened environments on which a great number of different users can submit computational requests. Some privileged users may have strong Quality of Service requirements whereas others may be less demanding. Common mapping algorithms are not well suited to guarantee a defined quality of service, they propose at best priority systems in order to favour some applications without any guaranty. We propose a new mapping algorithm, dealing with the notion of quality of service for scheduling applications over clusters and grids over different classes of service. This algorithm uses information on the application to map, all the unfinished applications previously mapped, the state of the execution support, and the processor access model (round robin model) to suggest a mapping which guarantees all the expressed constraints. The mapping decision is taken on-line based on the release date of all applications and the memory space used. To finish, the validation of the algorithm is performed with real log files entries simulated with Simgrid.