This article examines the kth nearest neighbor distance for three regular point patterns: square, triangular, and hexagonal lattices. The probability density functions of the kth nearest distance and the average kth nearest distances are theoretically derived for k 5 1, 2, . . ., 7. As an application of the kth nearest distance, we consider a facility location problem with closing of facilities. The problem is to find the optimal regular pattern that minimizes the average distance to the nearest open facility. Assuming that facilities are closed independently and at random, we show that the triangular lattice is optimal if at least 68% of facilities are open by comparing the upper and lower bounds of the average distances.

This paper develops a simple analytical model for determining the hierarchical system of road networks. The model is based on a grid road network where roads are classified into three types according to road widths and travel speeds. We derive the optimal ratios of road areas that minimize the average and maximum travel time. Minimizing the average travel time provides an efficient solution, whereas minimizing the maximum travel time provides an equitable solution. Both of the solutions are expressed in terms of road widths and travel speeds. As an application of the grid network model, we evaluate the hierarchical system of the road network of Tokyo.

This paper derives the joint distribution of the distances to the first and the second nearest points for regular and random patterns. Distance is measured as the Euclidean and the rectilinear distances on a continuous plane. The joint distribution extends the kth nearest distance distribution of previous works. The kth nearest distance distribution only shows how the distance to the kth nearest point is distributed, whereas the joint distribution provides the relationship between the distances. An application of the joint distribution can be found in a facility location problem with non-closest facility service where the distance to the second nearest facility is also important. The joint distribution that allows us to examine the first and the second nearest distances simultaneously is useful for evaluating the reliability of facility location when some of the existing facilities are closed. The joint distribution of the road network distances is also obtained to confirm that the model on a continuous plane can be applied to actual road networks.

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