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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: [email protected]/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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