Geodesics and flows in a Poissonian city
Kendall, Wilfrid S. (2009) Geodesics and flows in a Poissonian city. Working Paper. Coventry: University of Warwick. Centre for Research in Statistical Methodology. (Working papers).
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The stationary isotropic Poisson line network was used to derive upper bounds on mean excess network-geodesic length in Aldous and Kendall (2008). This new paper presents a study of the geometry and fluctuations of near-geodesics in such a network. 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 behaviour at the centre 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.
|Item Type:||Working or Discussion Paper (Working Paper)|
|Subjects:||Q Science > QA Mathematics|
|Divisions:||Faculty of Science > Statistics|
|Library of Congress Subject Headings (LCSH):||Geodesics (Mathematics), Poisson processes|
|Series Name:||Working papers|
|Publisher:||University of Warwick. Centre for Research in Statistical Methodology|
|Place of Publication:||Coventry|
|Number of Pages:||35|
|Status:||Not Peer Reviewed|
|Access rights to Published version:||Open Access|
|References:||Afimeimounga, H., W. Solomon, and I. Ziedins (2005). The Downs-Thomson paradox: existence, uniqueness and stability of user equilibria. Queueing Syst. 49(3-4), 321– 334. Aldous, D. J. and W. S. Kendall (2008, March). Short-length routes in low-cost networks via Poisson line patterns. Advances in Applied Probability 40(1), 1–21. Alsmeyer, G., A. Iksanov, and U. Roesler (2009). On Distributional Properties of Perpetuities. Journal of Theoretical Probability 22, 666–682. Ambartzumian, R. (1990). Factorization Calculus and Geometric Probability. Cambridge: Cambridge University Press. Baccelli, F., K. Tchoumatchenko, and S. Zuyev (2000). Markov paths on the Poisson- Delaunay graph with applications to routing in mobile networks. Advances in Applied Probability 32(1), 1–18. Baricz, Á. (2008). Mills’ ratio: monotonicity patterns and functional inequalities. J. Math. Anal. Appl. 340(2), 1362–1370. Bertoin, J. and M. Yor (2001). On subordinators, self-similar Markov processes and some factorizations of the exponential variable. Electronic Communications in Probability 6, 95–106 (electronic). Bertoin, J. and M. Yor (2002). On the entire moments of self-similar Markov processes and exponential functionals of Lévy processes. Annales de la Faculté des Sciences de Toulouse Mathématiques (Série 6) 11(1), 33–45. Bertoin, J. and M. Yor (2005). Exponential functionals of Lévy processes. Probability Surveys 2, 191–212 (electronic). Beskos, A. and G. O. Roberts (2005, November). Exact Simulation of Diffusions. The Annals of Applied Probability 15(4), 2422–2444. Birnbaum, Z. W. (1942). An inequality for Mill’s ratio. Annals of Mathematical Statistics 13, 245–246. Böröczky, K. J. and R. Schneider (2008). The mean width of circumscribed random polytopes. Submitted manuscript. Calvert, B., W. Solomon, and I. Ziedins (1997). Braess’s paradox in a queueing network with state-dependent routing. Journal of Applied Probability 34(1), 134–154. Davidson, R. (1974). Line-processes, roads, and fibres. In E. F. Harding and D. G. Kendall (Eds.), Stochastic geometry (a tribute to the memory of Rollo Davidson), pp. 248–251. London: Wiley. Dufresne, D. (1990). The distribution of a perpetuity, with applications to risk theory and pension funding. Scand. Actuar. J. 1-2 (1-2), 39–79. Goldie, C. M. and R. Grübel (1996). Perpetuities with thin tails. Advances in Applied Probability 28(2), 463–480. Hitczenko, P. and J. Wesolowski (2010). Perpetuities with thin tails, revisited. The Annals of Applied Probability To appear. Kellerer, H. (1992). Ergodic behaviour of affine recursions III: positive recurrence and null recurrence. Technical report, Math. Inst. Univ. München, Theresienstrasse 39, 8000 München, Germany. Kendall, W. S. (1997). On some weighted Boolean models. In D. Jeulin (Ed.), Advances in Theory and Applications of Random Sets, Singapore, pp. 105–120. World Scientific. Kendall, W. S. (2008). Networks and Poisson line patterns: fluctuation asymptotics. Oberwolfach Reports 5(4), 2670–2672. Littlewood, J. E. (1969). On the probability in the tail of a binomial distribution. Advances in Applied Probability 1, 43–72. McKay, B. D. (1989). On Littlewood’s estimate for the binomial distribution. Advances in Applied Probability 21(2), 475–478. Miles, R. E. (1964). Random polygons determined by random lines in a plane. Proc. Nat. Acad. Sci. U.S.A. 52, 901–907. Narasimhan, G. and M. Smid (2007). Geometric spanner networks. Cambridge: Cambridge University Press. Prömel, H. J. and A. Steger (2002). The Steiner tree problem. Advanced Lectures in Mathematics. Braunschweig: Friedr. Vieweg & Sohn. A tour through graphs, algorithms, and complexity. Rebolledo, R. (1980). Central limit theorems for local martingales. Z. Wahrsch. Verw. Gebiete 51(3), 269–286. Rényi, A. and R. Sulanke (1968). Zufällige konvexe Polygone in einem Ringgebiet. Zeitschrift für Wahrscheinlichkeitstheorie und Verwe Gebiete 9, 146–157. Sampford, M. R. (1953). Some inequalities on Mill’s ratio and related functions. Annals of Mathematical Statistics 24, 130–132. Santaló, L. A. (1976). Integral geometry and geometric probability. Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam. With a foreword by Mark Kac, Encyclopedia of Mathematics and its Applications, Vol. 1. Steele, J. M. (1997). Probability theory and combinatorial optimization, Volume 69 of CBMS-NSF Regional Conference Series in Applied Mathematics. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). Stoyan, D., W. S. Kendall, and J. Mecke (1995). Stochastic geometry and its applications (Second ed.). Chichester: John Wiley & Sons. (First edition in 1987 joint with Akademie Verlag, Berlin). Vervaat, W. (1979). On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Advances in Applied Probability 11(4), 750–783. Voss, F., C. Gloaguen, and V. Schmidt (2009). Scaling limits for shortest path lengths along the edges of stationary tessellations. Preprint, Dept Math, University of Ulm. Wardrop, J. G. (1952). Some theoretical aspects of road traffic research. Proceedings, Institute of Civil Engineers, Part II 1, 325–378. Whitt, W. (2007). Proofs of the martingale FCLT. Probability Surveys 4, 268–302. Yor, M. (1992). On some exponential functionals of Brownian motion. Adv. in Appl. Probab. 24(3), 509–531. Yukich, J. E. (1998). Probability theory of classical Euclidean optimization problems, Volume 1675 of Lecture Notes in Mathematics. Berlin: Springer-Verlag.|
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