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Spatial transportation networks with transfer costs: asymptotic optimality of hub-and-spoke models

Published online by Cambridge University Press:  01 September 2008

DAVID J. ALDOUS
DAVID J. ALDOUS*
Affiliation:
Department of Statistics, 367 Evans Hall # 3860, U.C. Berkeley, CA 94720, U.S.A. e-mail: aldous@stat.berkeley.eduwww.stat.berkeley.edu/users/aldous
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Abstract

Consider networks on n vertices at average density 1 per unit area. We seek a network that minimizes total length subject to some constraint on journey times, averaged over source-destination pairs. Suppose journey times depend on both route-length and number of hops. Then for the constraint corresponding to an average of 3 hops, the length of the optimal network scales as n13/10. Alternatively, constraining the average number of hops to be 2 forces the network length to grow slightly faster than order n3/2. Finally, if we require the network length to be O(n) then the mean number of hops grows as order log log n. Each result is an upper bound in the worst case (of vertex positions), and a lower bound under randomness or equidistribution assumptions. The upper bounds arise in simple hub and spoke models, which are therefore optimal in an order of magnitude sense.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 145 , Issue 2 , September 2008 , pp. 471 - 487
Copyright
Copyright © Cambridge Philosophical Society 2008

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References

REFERENCES

[1]Adler, N. and Berechman, J.. Evaluating optimal multi-hub networks in a deregulated aviation market with an application to Western Europe. Transportation Res. A 35 (2001), 373–390.Google Scholar
[2]Aldous, D. J. and Kendall, W. S.. Short-length routes in low-cost networks via Poisson line patterns. http://front.math.ucdavis.edu/math.PR/0701140 (2007).Google Scholar
[3]Barbour, A. D., Holst, L. and Janson, S.. Poisson Approximation (Clarendon Press, 1992).CrossRefGoogle Scholar
[4]Bryan, D. L. and O'Kelly, M. E.. Hub-and-spoke networks in air transportation: An analytical review. J. Regional Science 39 (1999), 275–295.Google Scholar
[5]Fejes Tóth, L.. Regular Figures (A Pergamon Press Book. The Macmillan Co., 1964).Google Scholar
[6]Baccelli, F., Klein, M., Lebourges, M. and Zuyev, S.. Stochastic geometry and architecture of communication networks. Telecommunication Systems 7 (1997), 209–227.Google Scholar
[7]Gastner, M. T. and Newman, M. E. J.. Optimal design of spatial distribution networks. Physical Review E 74 (2006), 016117.Google Scholar
[8]Gendron, B., Crainic, T. and Frangioni, A.. Multicommodity capacitated network design. In Sansó, B. and Soriano, P., editors. Telecommunications Network Planning. (Kluwer, 1998),pages 1–19.Google Scholar
[9]Gruber, P. M.. In many cases optimal configurations are almost regular hexagonal. Rend. Circ. Mat. Palermo (2) Suppl. part II(65) (2000), 121–145. III International Conference in Stochastic Geometry, Convex Bodies and Empirical Measures (Mazara del Vallo, 1999).Google Scholar
[10]Morgan, F. and Bolton, R.. Hexagonal economic regions solve the location problem. Amer. Math. Monthly 109 (2) (2002), 165–172.Google Scholar
[11]Wikipedia. Spoke-hub distribution paradigm — wikipedia, the free encyclopedia (2006) [Online; accessed 25-September-2006].Google Scholar
[12]Yang, H. and Bell, M. G. H.. Models and algorithms for road network design: A review and some new developments. Transport Review 18 (1998), 257–278.Google Scholar
[13]Yoon, M. G., Yoon, D. Y. and Han, S. H.. A dual based heuristic for a hub-spoke air cargo network design problem with hop-count constraint. Scientiae Mathematicae Japonicae 64 (2006), 77–90.Google Scholar
[14]Zuyev, S., Desnogues, P. and Rakotoarisoa, H.. Simulations of large telecommunication networks based on probabilistic modeling. J. Electron. Imaging 6 (1997), 68–77.Google Scholar