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Shared autonomous vehicles (SAVs) have been widely studied in the recent literature. Agent-based simulations and theoretical models have extensively explored the effects on travel service, fleet size, and congestion using heuristic dispatching strategies to match SAVs with on-demand passengers. A major question that simulations have sought to address is the service rate or replacement rate: the number of passengers each SAV can serve. Thus far, the service rate has mostly been estimated through simulation. This paper investigates an analytical max-pressure dispatch policy, which aims to maximize passenger throughput under any stochastic demand pattern, which takes the form of a model predictive control algorithm. An analytical proof using Lyapunov drift techniques is presented to show that the dispatch policy achieves maximum stability. The service rate and minimum fleet sizes are derived analytically …