To improve operational flexibility, throughput capacity, and responsiveness in order fulfillment operations, several distribution centers are implementing autonomous vehicle-based storage and retrieval system (AVS/RS) in their high-density storage areas. In such systems, vehicles are self-powered to travel in horizontal directions (x- and y- axes), and use lifts or conveyors for vertical motion (z-axis). In this research, we propose a multi-tier queuing modeling framework for the performance analysis of such vehicle-based warehouse systems. We develop an embedded Markov chain based analysis approach to estimate the first and second moment of inter-departure times from the load-dependent station within a semi-open queuing network. The linking solution approach uses traffic process approximations to analyze the performance of sub-models corresponding to individual tiers (semi-open queues) and the vertical transfer units (open queues). These sub-models are linked to form an integrated queuing network model, which is solved using an iterative algorithm. Performance estimates such as expected transaction cycle times and resource (vehicle and vertical transfer unit) utilization are determined using this algorithm, and can be used to evaluate a variety of design configurations during the conceptualization phase.