## Journal of Applied Probability

Wilfrid S. Kendall

#### Abstract

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

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

Dates
First available in Project Euclid: 2 December 2014

https://projecteuclid.org/euclid.jap/1417528482

Digital Object Identifier
doi:10.1239/jap/1417528482

Mathematical Reviews number (MathSciNet)
MR3317365

Zentralblatt MATH identifier
1314.60045

#### Citation

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

#### References

• Aldous, D. J. and Bhamidi, S. (2010). Edge flows in the complete random-lengths network. Random Structures Algorithms 37, 271–311.
• Aldous, D. J. and Kendall, W. S. (2008). Short-length routes in low-cost networks via Poisson line patterns. Adv. Appl. Prob. 40, 1–21.
• Aldous, D. J., McDiarmid, C. and Scott, A. (2009). Uniform multicommodity flow through the complete graph with random edge-capacities. Operat. Res. Lett. 37, 299–302.
• Blanchet, J. H. and Sigman, K. (2011). On exact sampling of stochastic perpetuities. In New Frontiers in Applied Probability (J. Appl. Prob. Spec. Vol. 48A), eds P. Glynn, T. Mikosch and T. Rolski, Applied Probability Trust, Sheffield, pp. 165–182.
• Chiu, S. N., Stoyan, D., Kendall, W. S. and Mecke, J. (2013). Stochastic Geometry and Its Applications. John Wiley, Chichester.
• Fill, J. A. and Huber, M. L. (2010). Perfect simulation of Vervaat perpetuities. Electron. J. Prob. 15, 96–109.
• Kendall, W. S. (2011). Geodesics and flows in a Poissonian city. Ann. Appl. Prob. 21, 801–842.
• Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.
• Møller, J. (1999). Perfect simulation of conditionally specified models. J. R. Statist. Soc. B 61, 251–264.
• Roberts, G. O. and Tweedie, R. L. (1996). Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms. Biometrika 83, 95–110.
• Vervaat, W. (1979). On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv. Appl. Prob. 11, 750–783.
• Wilson, D. B. (2000). Layered multishift coupling for use in perfect sampling algorithms (with a primer on CFTP). In Monte Carlo Methods (Fields Inst. Commun. 26; Toronto, Ontario, 1998), ed. N. Madras, American Mathematical Society, Providence, RI, pp. 143–179.