that a fan-out network is still optimal. First, the link costs are separable, and the total network cost is the sum of the individual costs of all the links. When the costs of some links depend on the flow of other links, the solutions can become more compli- cated. Each facility, each link, each flow determines its own cost independent of the other links. If a transport vendor decided to offer a package deal, a single price for a bundle of transport, then the costs would not be separable. However, let us assume that even a package deal still has separate costs for transport within the package. Second, the link costs have economy of scale, decreasing unit costs with increasing flow. We call a function f(x) concave when it increases less as x increases. The technical definition of concave is that a function f(x) is concave if and only if flxx + (1 - X)y] > Xf(x) + (1 - X)f(y), where l is between 0 and 1.6 The effect of concave cost functions is to concentrate usage on a smaller number of links. 7 The concave assumption is not 6A linear function is considered concave for this discussion. When the cost functions are convex, the solutions tend to "spread the pain." We might expect such solutions in environmental impact problems where all the pollution in one place is more than twice as bad as half the pollution in each of two places. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. Capacity Calculations for Cellular Networks Capacity and Quality Principles quite true in telephone transport because 1.5 DS-1 links costs more than 1.5 times the cost of a single DS-1. Probably the cheapest way to get 1.5 DS-1 links is to pay for 2. However, for larger quantities of transport, we think the concave assumption is reasonable. The optimal solution to a network flow formulation with linear constraints and a sep- arable concave cost function is a fan-out ne