Stationary freeway traffic flow modelled by a linear stochastic partial differential equation

Abstract

A stochastic model of stationary high density freeway traffic flow is presented. The high density assumption implies the choice of a macroscopic model. The assumption of stationarity allows us to linearize the model equations, but at the same time leads us to adding an appropriate stochastic term to the equation obtained from linearizing the original equations. The result is an infinite-dimensional stochastic differential equation describing the evolution of R( t,x). R( t,x) is the deviation of the density around some fixed value at time t and location x. R( t,x) is a stationary Gaussian process, if the initial value is suitably chosen. The process R( t,x) is determined by three parameters. Knowledge of these parameters may enable the traffic engineer to characterize the traffic stream as homogeneous or inhomogeneous. The model is tested with data measured on a Dutch freeway.