Data with vastly different access characteristics is efficiently stored in multi-tiered storage systems. A cost-effective way to retain large volumes of infrequently accessed data is to store it on tape. Steady developments in tape technology deliver ever increasing storage capacities at low cost. This has established tape as a viable solution to cope with the extreme data growth in the context of Big Data. Assessing the performance of the various tiers is central to achieving appropriate tier dimensioning and storage provisioning. To that end, we develop an analytical model to evaluate the performance of a tape library system that considers various relevant aspects, such as the number of cartridges and tape drives as well as different mount/unmount policies. Closed-form expressions for the corresponding mean waiting times are derived. The validity of the model developed is confirmed by demonstrating that the predicted performance matches well with that obtained by simulation across a wide range of system parameter values.