Travel distance/time distribution with a transportation network

In this paper we develop a simple model of travel distance/time distribution in a rectangular or a rectangular parallelepiped city with rectilinear distance. Sometimes we bypass on arterial transportation axes such as a freeway or a rapid train, so that we can save time. Starting from two typical travel distance distributions on a line segment, we derive travel distance/time distributions in two- or three-dimensional urban space, which could describe such a behavior as detour on transportation axes, in a mathematically explicit form. We show that higher dimension of urban space becomes, larger proportion of travel distance around mean is brought, and that travel distance distribution with a typical configuration of transportation axes can be obtained by convoluting lower dimensional travel distance distributions which correspond to each axis of urban space. As an implication in urban context, we find out that each of transportation network configurations has its superiority or inferiority with respect to urban density, that optimal form of a compact city is influenced by transportation network, and that there is a limitation in reducing mean travel time by installation of rapid transportation networks. Results of the present paper enable us to systematize effects of designing transportation network on travel distance/time in urban space.