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

Distance to the kth nearest facility is important when facilities are closed Emergency facilities !! Hospitals !! Fire stations !! Refuges Facilities are closed because of accidents and disasters Analytical expressions for the kth nearest distance have been obtained Grid Random Grid Random For finding the relationship between the distances, we need the joint distribution Distribution of the kth nearest distance shows how the distance is distributed Purpose To derive the joint distribution of the distances to the first and the second nearest facilities !! Facilities are represented as points of regular and random patterns !! Customers are uniformly distributed over the study region