Traffic congestion: models, costs and optimal transport

We develop two models of highway traffic: (i) a deterministic fluid model based on conservation laws building on previous work and (ii) a mean-field model of a series of infinite server queues, where each stage in the tandem models a segment of highway. The models define the ``highway-map''---a transformation of time-varying arrival rate functions according to which vehicles arrive at the highway to the corresponding departure rate functions of vehicles exiting the highway. The two models are shown to be equivalent in that they obtain the same highway-map. The cost of congestion for vehicles traversing the highway is the total extra time they spend on the highway due to congestion. This cost is shown to be equal to the ``d-bar'' distance between the input and the output rate measures of the highway-map. This fact is used to formulate a convex optimization problem for determining the optimal way to shift users from peak to off-peak hours using incentives so that congestion costs are lowered.