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Internodal Distance Distributions for Static and Mobile Nodes in 2D/3D Wireless Networks

Nicholas Vaiopoulos, Alexander Vavoulas, Harilaos G. Sandalidis, Konstantinos K. Delibasis, Dimitris Varoutas

TL;DR

This letter presents a unified analytical framework for internodal distance distributions in 2D and 3D wireless networks, with nodes confined to concentric circular or spherical regions, incorporating both geometric constraints and spatial effects introduced by mobility.

Abstract

This letter presents a unified analytical framework for internodal distance distributions in 2D and 3D wireless networks, with nodes confined to concentric circular or spherical regions. Four deployment scenarios are considered, covering all combinations of static (uniform) and mobile (random waypoint-based) nodes. For each scenario, closed-form expressions for the internodal distance probability density functions are derived, incorporating both geometric constraints and spatial effects introduced by mobility. Equal-radius cases are also addressed. Beta-distribution approximations and Monte Carlo simulations demonstrate the accuracy and validity of the analytical results.

Internodal Distance Distributions for Static and Mobile Nodes in 2D/3D Wireless Networks

TL;DR

This letter presents a unified analytical framework for internodal distance distributions in 2D and 3D wireless networks, with nodes confined to concentric circular or spherical regions, incorporating both geometric constraints and spatial effects introduced by mobility.

Abstract

This letter presents a unified analytical framework for internodal distance distributions in 2D and 3D wireless networks, with nodes confined to concentric circular or spherical regions. Four deployment scenarios are considered, covering all combinations of static (uniform) and mobile (random waypoint-based) nodes. For each scenario, closed-form expressions for the internodal distance probability density functions are derived, incorporating both geometric constraints and spatial effects introduced by mobility. Equal-radius cases are also addressed. Beta-distribution approximations and Monte Carlo simulations demonstrate the accuracy and validity of the analytical results.

Paper Structure

This paper contains 9 sections, 4 theorems, 12 equations, 2 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Model Assumptions
  3. Internodal Distance Distributions
  4. 2D Networks
  5. 3D Networks
  6. Approximation with Beta Distributions
  7. Concluding remarks
  8. Proof of Theorem 1
  9. Proof of Theorem 2

Key Result

Theorem 1

The PDF expression for the internodal distance distribution, $r$, in a 2D network with $R_{1}<R_{2}$, is given by where $k_{1}(r)=r^{2}+R_{1}^{2}-R_{2}^{2}$ and $k_{2}(r)=r^{2}-R_{1}^{2}+R_{2}^{2}$ are system dependent parameters and the polynomials $q_{1}(r)-q_{4}(r)$ are specified for each Scenario and tabulated in Table qTable.

Figures (2)

  • Figure 1: PDF for the 2D case with $\{R_1,R_2\}=\{1,2\}\text{m}$: analytical solutions (red), beta approximations (blue), and Monte Carlo results (histogram).
  • Figure 2: PDF plots for the 3D case with $\{R_1,R_2\}=\{1,2\}\text{m}$: analytical solutions (red), beta approximations (blue), and Monte Carlo results (histogram).

Theorems & Definitions (4)

  • Theorem 1
  • corollary 1
  • Theorem 2
  • corollary 2