We present a quantum statistical model for nanoscale graphene field-effect transistors (FETs) on a SiC substrate. In the model, the scattering events as well as ballistic transport are taken into account. The channel charge and current are calculated with the Keldysh non-equilibrium Green's function technique. The model, which is meant for the design of the graphene FETs, is semianalytical, but it is easy to reduce the model to a simple fully analytical one, which then can be applied to the design of the integrated circuits made of graphene. It is shown that the neglect of the scattering would lead to too large values for the saturation current. The calculated transconductance and the on-off ratio for the saturation current are in agreement with the experimental results.