  The stationary isotropic Poisson line process was used to derive upper bounds
on mean excess network geodesic length in Aldous and Kendall [Adv. in Appl.
Probab. 40 (2008) 1-21]. The current paper presents a study of the geometry and
fluctuations of near-geodesics in the network generated by the line process.
The notion of a "Poissonian city" is introduced, in which connections between
pairs of nodes are made using simple "no-overshoot" paths based on the Poisson
line process. Asymptotics for geometric features and random variation in length
are computed for such near-geodesic paths; it is shown that they traverse the
network with an order of efficiency comparable to that of true network
geodesics. Mean characteristics and limiting behavior at the center are
computed for a natural network flow. Comparisons are drawn with similar network
flows in a city based on a comparable rectilinear grid. A concluding section
discusses several open problems.
