In an approach analogous to that used to treat electronic currents in semiconductor quantum dots, we investigate the exciton current in a pigment network that is sandwiched between two exciton reservoirs, also known as the emitter and the acceptor. Employing the master equation for the reduced density matrix, the exciton current is obtained analytically for a two-site model, and numerically for an eight-site Fenna-Matthews-Olson (FMO) subunit model. It is found that, to maximize the exciton current with a specific network configuration, there exist optimal emitter temperatures and exciton transfer rates between the network and the reservoirs. The steady state current in the FMO model is consistent with the trapping time calculated by network optimization in the one-exciton picture. The current optimization with respect to various control parameters is discussed for the FMO model. At and below the biologically relevant transfer rate 1 ps−1, the FMO network is more efficient for excitation energy transfer than the two-site model. Beyond this scale, the FMO network shows robustness with respect to the interplay with the reservoirs.