Magnetic tape provides a cost-effective way to retain the exponentially increasing volumes of data being produced. The low cost per gigabyte and the low energy consumption render tape a preferred option over hard disk drives and flash for infrequently accessed data. Assessing the performance of tape library systems is central to achieving appropriate storage provisioning and dimensioning. Performance is affected by the number and the operational characteristics of the tape drives and the robotic arms, and the mount and unmount policies deployed. In this paper, we develop a novel analytical model that accurately captures the principal aspects of tape library operation. Several relevant performance measures including the mean waiting time and the mount/unmount rates are derived. The model provides useful insights into the behavior of the tape libraries and yields results that enable a better understanding of the design tradeoffs. The validity of the model developed is confirmed by demonstrating a good agreement of the predicted performance with that obtained by simulation across various configurations. To mitigate the burden on the robotic mechanism, a scheme of accumulating multiple requests before sending them to the tape library is proposed and studied.