Joint distribution of distances to the first and the second nearest facilities

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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