## Abstract

The ability of small-cell wireless networks to self-organize is crucial for improving capacity and performance in modern communication networks. This paper considers one of the most basic questions: what is the expected distance to a cell's nearest neighbour in a spatially distributed network? We analyse a model problem in the asymptotic limit of large total received signal and compare the accuracy of different heuristics. We also analytically consider the effects of fading. Our analysis shows that the most naive heuristic systematically underestimates the distance to the nearest node; this is substantially corrected in cases of interest by inclusion of the next-order asymptotic term. We illustrate our theoretical results explicitly or several combinations of signal and path loss parameters and show that our theory is well supported by numerical simulations.

Original language | English |
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Pages (from-to) | 188-219 |

Number of pages | 32 |

Journal | IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications) |

Volume | 86 |

Issue number | 1 |

Early online date | 30 Dec 2020 |

DOIs | |

Publication status | Published - 28 Feb 2021 |

### Bibliographical note

Funding Information:Engineering and Physical Sciences Research Council through the Centre for Doctoral Training in Statistical Applied Mathematics at Bath (EP/L015684/1); British Telecommunications plc.

## Keywords

- Asymptotic approximation
- Distance estimation
- Femtocells
- Rayleigh fading
- Self-organizing network
- Wireless networks

## ASJC Scopus subject areas

- Applied Mathematics