It is known that the transport capacity of a dense wireless ad hoc network with n nodes scales like √n. We show that the transport capacity divided by √n approaches a nonrandom limit with probability 1 when the nodes are uniformly distributed on the unit square. To show the existence of the limit, we prove that the transport capacity under the protocol model is a subadditive Euclidean functional and use the machinery of subadditive functions in the spirit of Steele.

Consider the basic location problem in which k locations from among n given points X1,…,Xn are to be chosen so as to minimize the sum M(k; X1,…,Xn) of the distances of each point to the nearest location. It is assumed that no location can serve more than a fixed finite number D of points. When the Xi, i ≥ 1, are i.i.d. random variables with values in [0,1]d and when k = ⌈n/(D+1)⌉ we show that

where α := α(D,d) is a positive constant, f is the density of the absolutely continuous part of the law of X1, and c.c. denotes complete convergence.

Let

where Xi, 1 ≦ i ≦ n, are i.i.d. and uniformly distributed in [0, 1]2. It is proved that Mn ∽ cn1–p/2 a.s. for 1 ≦ p <2. This result is motivated by recent developments in the theory of algorithms and the theory of subadditive processes as well as by a well-known problem of H. Steinhaus.