This study considers a production cell consisting of two machines and a material handling robot. The robot has a buffer space that moves with it. Identical parts are to be produced repetitively in this flowshop environment. The problem is to determine the cyclic schedule of the robot moves that maximizes the throughput rate. After developing the necessary framework to analyze such cells, we separately consider the single-, double-, and infinite-capacity buffer cases. For single- and double-capacity cases, consistent with the literature, we consider one-unit cycles that produce a single part in one repetition. We compare these cycles with each other and determine the set of undominated cycles. For the single-capacity case, we determine the parameter regions where each cycle is optimal, whereas for the double-capacity case, we determine efficient cycles and their worst-case performance bounds. For the infinite-capacity buffer case, we define a new class of cycles that better utilizes the benefits of the buffer space. We derive all such cycles and determine the set of undominated ones.We perform a computational study where we investigate the benefits of robots with a buffer space and the effects of the size of the buffer space on the performance. We compare the performances of self-buffered robots, dual-gripper robots, and robots with swap ability.