Flow Dependent Traffic Assignment on a Circular City

Consider a circular central business district of a city with n evenly spaced radial streets and m circular streets. We assume that during the evening rush hour driver origins are all inside the city and exits at the circumference with a joint distribution that depends only upon the relative angle between original and final radii, and the radial coordinate of the origin. Travel time on any directed link is assumed to be a function of the flow along that directed link and the radial coordinate of the link. For a flow of Q vehicles per hour leaving the city, our problem is to find an assignment of drivers to paths such that every driver follows a path of minimum travel time between his origin and exit. This extends some work of Smeed who considers a similar model with no flow dependence of the travel time and a uniform distribution of origins. It is first shown that this circularly symmetric system gives rise to a circularly symmetric optimal flow pattern. This flow pattern can be generated by assigning each driver to a path that uses at most one of the circular streets. Eventually the assignment is expressed explicitly as a function of 2( m - 1) unknown parameters that satisfy a system of equations and/or inequalities. The general solution of these equations were not found, but solutions were obtained for a number of special cases that showed the effect of congestion at the center and the effect of a belt expressway. Differential equations were also obtained for n (rightarrow) (infinity) and m (rightarrow) (infinity).