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Traffic flow densities in large transport networks

Part of: Special processes Limit theorems Operations research and management science

Published online by Cambridge University Press:  17 November 2017

Christian Hirsch ,
Benedikt Jahnel ,
Paul Keeler  and
Robert I. A. Patterson
Christian Hirsch*
Affiliation:
Ludwig-Maximilians-Universität München
Benedikt Jahnel*
Affiliation:
Weierstrass Institute
Paul Keeler*
Affiliation:
Weierstrass Institute
Robert I. A. Patterson*
Affiliation:
Weierstrass Institute
*
* Postal address: Mathematisches Institut, Ludwig-Maximilians-Universität München, Theresienstraße 39, 80333 München, Germany. Email address: [email protected]
** Postal address: Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany.
** Postal address: Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany.
** Postal address: Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany.
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Abstract

We consider transport networks with nodes scattered at random in a large domain. At certain local rates, the nodes generate traffic flows according to some navigation scheme in a given direction. In the thermodynamic limit of a growing domain, we present an asymptotic formula expressing the local traffic flow density at any given location in the domain in terms of three fundamental characteristics of the underlying network: the spatial intensity of the nodes together with their traffic generation rates, and of the links induced by the navigation. This formula holds for a general class of navigations satisfying a link-density and a sub-ballisticity condition. As a specific example, we verify these conditions for navigations arising from a directed spanning tree on a Poisson point process with inhomogeneous intensity function.

Keywords

Traffic densityrouteingnavigationtransport networksub-ballisticity

MSC classification

Primary: 60K30: Applications (congestion, allocation, storage, traffic, etc.) 60F15: Strong theorems 90B20: Traffic problems
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 4 , December 2017 , pp. 1091 - 1115
Copyright
Copyright © Applied Probability Trust 2017 

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References

[1] Baccelli, F. and Bordenave, C. (2007). The radial spanning tree of a Poisson point process. Ann. Appl. Prob. 17, 305–359. CrossRefGoogle Scholar
[2] Baccelli, F., Tchoumatchenko, K. and Zuyev, S. (2000). Markov paths on the Poisson–Delaunay graph with applications to routing in mobile networks. Adv. Appl. Prob. 32, 1–18. CrossRefGoogle Scholar
[3] Bhatt, A. G. and Roy, R. (2004). On a random directed spanning tree. Adv. Appl. Prob. 36, 19–42. CrossRefGoogle Scholar
[4] Bonichon, N. and Marckert, J.-F. (2011). Asymptotics of geometrical navigation on a random set of points in the plane. Adv. Appl. Prob. 43, 899–942. CrossRefGoogle Scholar
[5] Bordenave, C. (2008). Navigation on a Poisson point process. Ann. Appl. Prob. 18, 708–746. CrossRefGoogle Scholar
[6] Bramson, M. and Durrett, R. (eds) (1999). Perplexing Problems in Probability. Birkhäuser, Boston, MA. CrossRefGoogle Scholar
[7] Chiu, S. N., Stoyan, D., Kendall, W. S. and Mecke, J. (2013). Stochastic Geometry and Its Applications, 3rd edn. John Wiley, Chichester. CrossRefGoogle Scholar
[8] Federer, H. (1969). Geometric Measure Theory. Springer, New York. Google Scholar
[9] Ferrari, P. A., Landim, C. and Thorisson, H. (2004). Poisson trees, succession lines and coalescing random walks. Ann. Inst. H. Poincaré Prob. Statist. 40, 141–152. CrossRefGoogle Scholar
[10] Howard, C. D. and Newman, C. M. (1997). Euclidean models of first-passage percolation. Prob. Theory Relat. Fields 108, 153–170. CrossRefGoogle Scholar
[11] Howard, C. D. and Newman, C. M. (2001). Geodesics and spanning trees for Euclidean first-passage percolation. Ann. Prob. 29, 577–623. CrossRefGoogle Scholar
[12] Keeler, H. P. and Taylor, P. G. (2010). A stochastic analysis of a greedy routing scheme in sensor networks. SIAM J. Appl. Math. 70, 2214–2238. CrossRefGoogle Scholar
[13] Keeler, H. P. and Taylor, P. G. (2012). Random transmission radii in greedy routing models for ad hoc sensor networks. SIAM J. Appl. Math. 72, 535–557. CrossRefGoogle Scholar
[14] Licea, C. and Newman, C. M. (1996). Geodesics in two-dimensional first-passage percolation. Ann. Prob. 24, 399–410. CrossRefGoogle Scholar
[15] Liggett, T. M., Schonmann, R. H. and Stacey, A. M. (1997). Domination by product measures. Ann. Prob. 25, 71–95. CrossRefGoogle Scholar
[16] Mao, G. and Anderson, B. D. O. (2014). Capacity of large wireless networks with generally distributed nodes. IEEE Trans. Wireless Commun. 13, 1678–1691. CrossRefGoogle Scholar
[17] Newman, C. M. and Piza, M. S. T. (1995). Divergence of shape fluctuations in two dimensions. Ann. Prob. 23, 977–1005. Google Scholar
[18] Penrose, M. D. (2003). Random Geometric Graphs. Oxford University Press. CrossRefGoogle Scholar
[19] Penrose, M. D. and Wade, A. R. (2006). On the total length of the random minimal directed spanning tree. Adv. Appl. Prob. 38, 336–372. CrossRefGoogle Scholar
[20] Penrose, M. D. and Wade, A. R. (2010). Limit theorems for random spatial drainage networks. Adv. Appl. Prob. 42, 659–688. CrossRefGoogle Scholar
[21] Penrose, M. D. and Yukich, J. E. (2002). Limit theory for random sequential packing and deposition. Ann. Appl. Prob. 12, 272–301. CrossRefGoogle Scholar
[22] Penrose, M. D. and Yukich, J. E. (2003). Weak laws of large numbers in geometric probability. Ann. Appl. Prob. 13, 277–303. CrossRefGoogle Scholar
[23] Rodolakis, G. (2013). Information theoretic cut-set bounds on the capacity of Poisson wireless networks. In 2013 IEEE International Symposium on Information Theory, IEEE, New York, pp. 1451–1455. CrossRefGoogle Scholar
[24] Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin. CrossRefGoogle Scholar
[25] Takagi, H. and Kleinrock, L. (1984). Optimal transmission ranges for randomly distributed packet radio terminals. IEEE Trans. Commun. 32, 246–257. CrossRefGoogle Scholar