The first part of this paper presents a model of a subway network to evaluate a cost function that includes an operational cost and social costs measured in terms of passenger waiting times. We reformulate the model with a simple discrete event simulation model that considerably reduces the complexity of the simulation. The simplified model uses conditional expectations to filter out rapid dynamics, and it can be interpreted in terms of a subway network with "fluid" passenger levels. Because this network only sees train movements and no individual passengers are described, we call it the "ghost" model.

In the second part of the paper, we explore the benefits of using stochastic approximations to adjust the headways of different subway lines as the network is operating, thus learning passenger traffic patterns and adaptively seeking the best values. Our formulation of the ghost model is amenable for decentralized estimation of gradients of the cost function with respect to the control parameters (the line headways) and we use ersatz estimation methods to formulate a control scheme that uses minimal measurements and virtually no overhead.