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This paper studies a stochastic model that describes the evolution of vehicledensities in a road network. It is consistent with the class of (deterministic)kinematic wave models, which describe traffic flows on the basis ofconservation laws that incorporate the macroscopic fundamental diagram (afunctional relationship between vehicle density and flow). Our setup is capableof handling multiple types of vehicle densities, with general macroscopicfundamental diagrams, on a network with arbitrary topology.  Interpreting our system as a spatial population process, we derive, under anatural scaling, fluid and diffusion limits. More specifically, the vehicledensity process can be approximated with a suitable Gaussian process, whichyield accurate normal approximations to the joint (in the spatial and temporalsense) vehicle density process. The corresponding means and variances can becomputed efficiently. Along the same lines, we develop an approximation to thevehicles' travel-time distribution between any given origin and destinationpair. Finally, we present a series of numerical experiments that demonstratethe accuracy of the approximations and illustrate the usefulness of theresults.

A diffusion-based analysis of a multi-class road traffic network