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Autonomous Mobility-on-Demand Systems for Future Urban Mobility404 (i.e., the probability that at least one vehicle is available to provide immediate service) or average wait times to receive service. The model is portrayed in Figure 19.2. Controlling a spatial queueing system involves a joint task allocation and scheduling problem, whereby vehicle routes should be dynamically designed to allocate vehicles to transportation requests so as to minimize, for example, wait times. In such a dynamic and stochastic setup, one needs to design a closed-loop control policy, as opposed to open-loop preplanned routes. The problem combines aspects of networked control, queueing theory, combinatorial optimization, and geometric probability (i.e., probabilistic analysis in a geo- metrical setting). This precludes the direct application of “traditional” queueing theory due to the complexity added by the spatial component (these complexities include, for example, congestion effects on network edges, energy constraints, and statistical couplings induced by the vehicles’ motion [17, 19, 20]). It also precludes the direct application of combinato- rial static optimization, as the dynamic aspect of the problem implies that the problem instance is incrementally revealed over time and static methods can no longer be applied. As a consequence, researchers have devised a number of alternative approaches, as detailed in the next section. 19.2.2 Approaches for controlling AMoD systems This section presents two recent, yet promising approaches for the control of spatial queue- ing systems as models for AMoD systems, namely the lumped approach and the distribut- ed approach. Both approaches employ a number of relaxations and approximations to Fig. 19.2 A spatial queueing model of an AMoD system entails an exogenous dynamical process that generates “transportation requests” (yellow dots) at spatially-localized queues. Self-driving vehicles (represented by small car icons) travel among such locations according to a given network topology to transport passengers.