Robotic tape libraries are popular for applications with very high storage requirements, such as video servers. Here, we study the throughput of a tape library system, we design a new scheduling algorithm, the so-called Relief, and compare it against some older/straightforward ones, like FCFS, Maximum Queue Length (MQL) and an unfair one (Bypass), roughly equivalent to Shortest Job First. The proposed algorithm incorporates an aging mechanism in order to attain fairness and we prove that, under certain assumptions, it minimizes the average start-up latency. Extensive simulation experiments show that Relief outperforms its competitors (fair and unfair alike), with up to 203% improvement in throughput, for the same rejection ratio.