This paper models and analyzes tier-captive autonomous vehicle storage and retrieval systems. While previous models assume sequential commissioning of the lift and vehicles, we propose a parallel processing policy for the system, under which an arrival transaction can request the lift and the vehicle simultaneously. To investigate the performance of this policy, we formulate a fork-join queueing network in which an arrival transaction will be split into a horizontal movement task served by the vehicle and a vertical movement task served by the lift. We develop an approximation method based on decomposition of the fork-join queueing network to estimate the system performance. We build simulation models to validate the effectiveness of analytical models. The results show that the fork-join queueing network is accurate in estimating the system performance under the parallel processing policy. Numerical experiments and a real case are carried out to compare the system response time of retrieval transactions under parallel and sequential processing policies. The results show that, in systems with less than 10 tiers, the parallel processing policy outperforms the sequential processing policy by at least 5.51 percent. The advantage of parallel processing policy is decreasing with the rack height and the aisle length. In systems with more than 10 tiers and a length to height ratio larger than 7, we can find a critical retrieval transaction arrival rate, below which the parallel processing policy outperforms the sequential processing policy. 
Modeling parallel movement of lifts and vehicles in tier-captive vehicle-based warehousing systems