Journal of Applied Probability

Return to the Poissonian city

Wilfrid S. Kendall


Consider the following random spatial network: in a large disk, construct a network using a stationary and isotropic Poisson line process of unit intensity. Connect pairs of points using the network, with initial/final segments of the connecting path formed by travelling off the network in the opposite direction to that of the destination/source. Suppose further that connections are established using `near geodesics', constructed between pairs of points using the perimeter of the cell containing these two points and formed using only the Poisson lines not separating them. If each pair of points generates an infinitesimal amount of traffic divided equally between the two connecting near geodesics, and if the Poisson line pattern is conditioned to contain a line through the centre, then what can be said about the total flow through the centre? In Kendall (2011) it was shown that a scaled version of this flow has asymptotic distribution given by the 4-volume of a region in 4-space, constructed using an improper anisotropic Poisson line process in an infinite planar strip. Here we construct a more amenable representation in terms of two `seminal curves' defined by the improper Poisson line process, and establish results which produce a framework for effective simulation from this distribution up to an L1 error which tends to 0 with increasing computational effort.

Article information

J. Appl. Probab., Volume 51A (2014), 297-309.

First available in Project Euclid: 2 December 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 90B15: Network models, stochastic

Improper anisotropic Poisson line process mark distribution point process Poisson line process Poissonian city network Mecke-Slivnyak theorem seminal curve spatial network traffic flow


Kendall, Wilfrid S. Return to the Poissonian city. J. Appl. Probab. 51A (2014), 297--309. doi:10.1239/jap/1417528482.

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