A Robotic Mobile Fulfillment System is a recently developed automated, parts-to-picker material handling system. Robots can move storage shelves, also known as inventory pods, between the storage area and the workstations and can continually reposition them during operations. This article shows how to optimize three key decision variables: (i) the number of pods per SKU; (ii) the ratio of the number of pick stations to replenishment stations; and (iii) the replenishment level per pod. Our results show that throughput performance improves substantially when inventory is spread across multiple pods, when an optimum ratio between the number of pick stations to replenishment stations is achieved and when a pod is replenished before it is completely empty. This article contributes methodologically by introducing a new type of Semi-Open Queueing Network (SOQN): cross-class matching multi-class SOQN, by deriving necessary stability conditions, and by introducing a novel interpretation of the classes.