Autonomous vehicle-based storage and retrieval systems are commonly used in many fulfillment centers (e.g., e-commerce warehouses), because they allow a high- and flexible-throughput capacity. In these systems, roaming robots transport loads between a storage location and a workstation. Two main variants exist: horizontal, where the robots only move horizontally and use lifts for vertical transport, and a new variant vertical, where the robots can also travel vertically in the rack. This paper builds a framework to analyze the performance of the vertical system and compare its throughput capacity with the horizontal system. We build closed queueing network models for this that, in turn, are used to optimize the design. The results show that the optimal height-to-width ratio of a vertical system is around one. Because a large number of system robots may lead to blocking and delays, we compare the effects of different robot blocking protocols on the system throughput: robot Recirculation and Wait-on-Spot. The Wait-on-Spot policy produces a higher system throughput when the number of robots in the system is small. However, for a large number of robots in the system, the Recirculation policy dominates the Wait-on-Spot policy. Finally, we compare the operational costs of the vertical and horizontal transport systems. For systems with one load/unload (L/U) point, the vertical system always produces a similar or higher system throughput with a lower operating cost compared with the horizontal system with a discrete lift. It also outperforms the horizontal system with a continuous lift in systems with two L/U points.