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40719.3 Evaluating AMoD systems ters. For example, one can show that a necessary and VXI¿FLHQW FRQGLWLRQ IRU VWDELOLW\ LV that the load factor ȡ := Ȝ [EijO ijD [Y í X ] + EMD(ijO , ijD )] / (vm) (19.1) is strictly less than one, where m is the number of servicing vehicles, v is the average speed of the vehicles, EijO ijD [Y í X ] is the expected distance between origin and destination locations, and EMD(ijO ijD) is the earth mover’s distance between densities ijO and ijD [32], representing the minimum distance, on average, a vehicle must travel to realign itself with an asymmetrical travel demand [16]. Intuitively, if distributions ijO and ijD are imagined as describing two piles each consisting of a unit of “dirt” (i.e., earth), then EMD(ijO ijD) can be thought of as the minimum work (dirt × distance) required to reshape ijO into ijD VHH > @ IRU D IRUPDO GH¿QLWLRQ 2QH FDQ XVH WKH DERYH IRUPXOD WR HVWLPDWH WKH UHTXLUHG ÀHHW VL]H WR HQVXUH VWDELOLW\ ± DQ H[DPSOH DSSOLFDWLRQ WR D FDVH VWXG\ RI 6LQJDSRUH is presented in Section 19.3. With this approach, it is also possible to obtain formal perfor- mance bounds (i.e., factors of sub-optimality) for receding horizon control policies, in the asymptotic regimes ȡ ĺ í (heavy-load, system saturated) and ȡ ĺ + (light-load, system empty of customers) [33, 17]. 19.2.3 Comparison The lumped approach and the distributed approach are complementary in a number of ways. Both models provide formal guarantees for stability and performance. The former is more realistic (a road topology can be readily mapped into this model) and provides a natural pathway to synthesize control policies. The latter provides significant mathematical sim- plifications (as one only needs to study a spatially-averaged queue) and enables the deter- mination of analytical scaling laws that can be used to select system parameters (e.g., fleet sizing). In the next section we exploit the interplay between these two approaches to char- acterize AMoD systems for case studies of New York City and Singapore. Both approaches appear to be promising tools to systematically tackle the problem of system-wide control of AMoD systems. Several research questions, however, still need to be addressed to fulfill this objective, particularly with respect to inclusion of congestion effects (in Section 19.2.2.1, roads are modeled as infinite server queues, so the travel time for each vehicle is independent of all other vehicles), predictive accuracy, and control syn- thesis for complex scenarios, as detailed in Section 19.4. 19.3 Evaluating AMoD systems Leveraging models and methods from Section 19.2, this section studies hypothetical deployments of AMoD systems in two major cities, namely New York City and Singapore.