Skip to main content
SpringerLink
Log in

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

  • Original Article
  • Published:
Journal of Geographical Systems Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  • Aras N, Orbay M, Altinel I (2008) Efficient heuristics for the rectilinear distance capacitated multi-facility Weber problem. J Oper Res Soc 59:64–79

    Article  Google Scholar 

  • Berman O, Krass D, Menezes M (2007) Facility reliability issues in network p-median problems: strategic centralization and co-location effects. Oper Res 55:332–350

    Article  Google Scholar 

  • Clark P, Evans F (1954) Distance to nearest neighbor as a measure of spatial relationships in populations. Ecology 35:85–90

    Article  Google Scholar 

  • Cressie N (1993) Statistics for spatial data. Wiley, New York

    Google Scholar 

  • Dacey M (1968) Two-dimensional random point patterns: a review and an interpretation. Papers of the Regional Science Association 13:41–55

    Article  Google Scholar 

  • Daskin M (1983) A maximum expected covering location model: formulation, properties and heuristic solution. Transportation Sci 17:48–70

    Article  Google Scholar 

  • Drezner Z (1987) Heuristic solution methods for two location problems with unreliable facilities. J Oper Res Soc 38:509–514

    Google Scholar 

  • Francis R, McGinnes F, White J (1992) Facility layout and location: an analytical approach. Prentice Hall, Upper Saddle River

    Google Scholar 

  • Getis A (1983) Second-order analysis of point patterns: the case of chicago as a multi-center urban region. Prof Geogr 35:73–80

    Article  Google Scholar 

  • He F, Duncan R (2000) Density-dependent effects on tree survival in an old-growth douglas fir forest. J Ecol 88:676–688

    Article  Google Scholar 

  • Holgate P (1965a) The distance from a random point to the nearest point of a closely packed lattice. Biometrika 52:261–263

    Google Scholar 

  • Holgate P (1965b) Some new tests of randomness. J Ecol 53:261–266

    Article  Google Scholar 

  • Illian J, Penttinen A, Stoyan H, Stoyan D (2008) Statistical analysis and modeling of spatial point patterns. Wiley, Chichester

    Google Scholar 

  • Jones A, Langford I, Bentham G (1996) The application of K-function analysis to the geographical distribution of road traffic accident outcomes in Norfolk, England. Soc Sci Med 42:879–885

    Article  Google Scholar 

  • Koshizuka T (1985) On the relation between the density of urban facilities and the distance to the nearest facility from a point in a given area. J City Plann Inst Jpn 20:85–90. (in Japanese)

    Google Scholar 

  • Krause E (1987) Taxicab geometry: an adventure in non-euclidean geometry. Dover, New York

    Google Scholar 

  • Larson R (1974) A hypercube queuing model for facility location and redistricting in urban emergency services. Comput Oper Res 1:67–95

    Article  Google Scholar 

  • Larson R (1975) Approximating the performance of urban emergency service systems. Oper Res 23:845–868

    Article  Google Scholar 

  • Larson R, Odoni A (1981) Urban operations research. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  • Lee S (2001) On solving unreliable planar location problems. Comput Oper Res 28:329–344

    Article  Google Scholar 

  • Marianov V, ReVelle C (1994) The queueing probabilistic location set covering problem and some extensions. Socio-Econ Plann Sci 28:167–178

    Article  Google Scholar 

  • Marianov V, ReVelle C (1996) The queueing maximal availability location problem: a model for the siting of emergency vehicles. Eur J Oper Res 93:110–120

    Article  Google Scholar 

  • Miyagawa M (2008) Analysis of facility location using ordered rectilinear distance in regular point patterns. FORMA 23:89–95

    Google Scholar 

  • Miyagawa M (2009a) Order distance in regular point patterns. Geogr Anal 41:252–262

    Article  Google Scholar 

  • Miyagawa M (2009b) Rectilinear distance in rotated regular point patterns. FORMA 24:111–116

    Google Scholar 

  • Murray A, O’Kelly M (2002) Assessing representation error in point-based coverage modeling. J Geogr Syst 4:171–191

    Article  Google Scholar 

  • Okabe A, Yamada I (2001) The K-function method on a network and its computational implementation. Geogr Anal 33:271–290

    Article  Google Scholar 

  • Okabe A, Yomono H, Kitamura M (1995) Statistical analysis of the distribution of points on a network. Geogr Anal 27:152–175

    Article  Google Scholar 

  • Okabe A, Boots B, Sugihara K, Chiu S (2000) Spatial tessellations: concepts and applications of voronoi diagrams. Wiley, Chichester

    Google Scholar 

  • O’Kelly M (2009) Rectilinear minimax hub location problems. J Geogr Syst 11:227–241

    Article  Google Scholar 

  • Persson O (1964) Distance methods: the use of distance measurements in the estimation of seedling density and open space frequency. Studia Forestalia Suecica 15:1–68

    Google Scholar 

  • Pielou E (1959) The use of point-to-plant distances in the study of the pattern of plant populations. J Ecol 47:607–613

    Article  Google Scholar 

  • Pirkul H (1989) The uncapacitated facility location problem with primary and secondary facility requirements. IIE Trans 21:337–348

    Article  Google Scholar 

  • ReVelle C, Hogan K (1989) The maximum availability location problem. Transportation Sci 23:192–200

    Article  Google Scholar 

  • Ripley B (1976) The second-order analysis of stationary point processes. J Appl Probab 13:255–266

    Article  Google Scholar 

  • Skellam J (1952) Studies in statistical ecology: spatial pattern. Biometrika 39:346–362

    Google Scholar 

  • Snyder L (2006) Facility location under uncertainty: a review. IIE Trans 38:537–554

    Article  Google Scholar 

  • Snyder L, Daskin M (2005) Reliability models for facility location: the expected failure cost case. Transportation Sci 39:400–416

    Article  Google Scholar 

  • Sorensen P, Church R (2010) Integrating expected coverage and local reliability for emergency medical services location problems. Socio-Econ Plann Sci 44:8–18

    Article  Google Scholar 

  • Thompson H (1956) Distribution of distance to nth neighbour in a population of randomly distributed individuals. Ecology 37:391–394

    Article  Google Scholar 

  • Vaughan R (1987) Urban spatial traffic patterns. Pion, London

    Google Scholar 

  • Weaver J, Church R (1985) A median location model with nonclosest facility service. Transportation Sci 19:58–74

    Article  Google Scholar 

  • Yamada I, Thill J (2004) Comparison of planar and network K-functions in traffic accident analysis. J Transport Geogr 12:149–158

    Article  Google Scholar 

Download references

Acknowledgments

This work was supported by Grant-in-Aid for Young Scientists (B) (22710140). I wish to thank three anonymous reviewers and the editor for their helpful comments.

Author information

Authors and Affiliations

  1. Department of Ecosocial System Engineering, University of Yamanashi, 4-3-11 Takeda, Kofu, Yamanashi, 400-8511, Japan

    Masashi Miyagawa

Authors
  1. Masashi Miyagawa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Masashi Miyagawa.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Miyagawa, M. Joint distribution of distances to the first and the second nearest facilities. J Geogr Syst 14, 209–222 (2012). https://doi.org/10.1007/s10109-010-0143-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10109-010-0143-3

Keywords

JEL Classification

Search

Navigation