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Cellular, Cell-less, and Everything in Between: A Unified Framework for Utility Region Analysis in Wireless Networks

Renato Luis Garrido Cavalcante, Tomasz Piotrowski, Slawomir Stanczak

TL;DR

The paper develops a unified framework for analyzing wireless utility regions using SINR and achievable rate metrics in modern architectures such as cell-less and massive MIMO. It recasts user utilities as $U_n({\boldsymbol{p}})=\frac{p_n}{t_n({\boldsymbol{p}})}$ with standard interference mappings and shows a fundamental link between conditional eigenpairs and the nonlinear spectral radius $\rho(T_{\|\cdot\|})$, enabling compact, geometry-based representations of feasible regions. This framework yields practical convexity certificates via inverse Z-matrices, facilitates tractable convex reformulations for sum-rate optimization, and introduces channel-compatibility concepts to gauge when time-sharing is unnecessary. Counterexamples and simulations demonstrate the framework’s breadth, guiding design choices for interference management in next-generation networks while highlighting when rate-based formulations are advantageous over SINR-based ones.

Abstract

We introduce a unified framework for analyzing utility regions of wireless networks, with a focus on signal-to-interference-plus-noise-ratio (SINR) and achievable rate regions. The framework provides valuable insights into interference patterns of modern network architectures, including extremely large MIMO and cell-less networks. A central contribution is a simple characterization of feasible utility regions using the concept of spectral radius of nonlinear mappings. This characterization provides a powerful mathematical tool for wireless system design and analysis. For example, it allows us to generalize existing characterizations of the weak Pareto boundary using compact notation. It also allows us to derive tractable sufficient conditions for the identification of convex utility regions. This property is particularly important because, on the weak Pareto boundary, it guarantees that time sharing (or user grouping) cannot simultaneously improve the utilities of all users. Beyond geometrical insights, these sufficient conditions have two key implications. First, they identify a family of (weighted) sum-rate maximization problems that are inherently convex, thus paving the way for the development of efficient, provably optimal solvers for this family. Second, they provide justification for formulating sum-rate maximization problems directly in terms of achievable rates, rather than SINR levels. Our theoretical insights also motivate an alternative to the concept of favorable propagation in the massive MIMO literature -- one that explicitly accounts for self-interference and the beamforming strategy.

Cellular, Cell-less, and Everything in Between: A Unified Framework for Utility Region Analysis in Wireless Networks

TL;DR

The paper develops a unified framework for analyzing wireless utility regions using SINR and achievable rate metrics in modern architectures such as cell-less and massive MIMO. It recasts user utilities as with standard interference mappings and shows a fundamental link between conditional eigenpairs and the nonlinear spectral radius , enabling compact, geometry-based representations of feasible regions. This framework yields practical convexity certificates via inverse Z-matrices, facilitates tractable convex reformulations for sum-rate optimization, and introduces channel-compatibility concepts to gauge when time-sharing is unnecessary. Counterexamples and simulations demonstrate the framework’s breadth, guiding design choices for interference management in next-generation networks while highlighting when rate-based formulations are advantageous over SINR-based ones.

Abstract

We introduce a unified framework for analyzing utility regions of wireless networks, with a focus on signal-to-interference-plus-noise-ratio (SINR) and achievable rate regions. The framework provides valuable insights into interference patterns of modern network architectures, including extremely large MIMO and cell-less networks. A central contribution is a simple characterization of feasible utility regions using the concept of spectral radius of nonlinear mappings. This characterization provides a powerful mathematical tool for wireless system design and analysis. For example, it allows us to generalize existing characterizations of the weak Pareto boundary using compact notation. It also allows us to derive tractable sufficient conditions for the identification of convex utility regions. This property is particularly important because, on the weak Pareto boundary, it guarantees that time sharing (or user grouping) cannot simultaneously improve the utilities of all users. Beyond geometrical insights, these sufficient conditions have two key implications. First, they identify a family of (weighted) sum-rate maximization problems that are inherently convex, thus paving the way for the development of efficient, provably optimal solvers for this family. Second, they provide justification for formulating sum-rate maximization problems directly in terms of achievable rates, rather than SINR levels. Our theoretical insights also motivate an alternative to the concept of favorable propagation in the massive MIMO literature -- one that explicitly accounts for self-interference and the beamforming strategy.

Paper Structure

This paper contains 19 sections, 10 theorems, 46 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Time sharing in resource allocation
  3. Main contributions
  4. Relation to existing studies
  5. Mathematical framework
  6. Notation and standard definitions
  7. Relating the conditional eigenvalue of standard interference mappings to the spectral radius of general interference mappings
  8. Applications to wireless networks
  9. Interference models
  10. Achievable rate and SINR regions in the absence of power constraints
  11. Achievable rate and SINR regions with power constraints
  12. Discussion
  13. Impact of self-interference on the convexity of the feasible rate and SINR regions
  14. Users offering compatible channels
  15. Numerical aspects
  16. ...and 4 more sections

Key Result

Lemma 1

Let $\mathcal{Y}$ be a nonempty set. Suppose that, for each $y\in\mathcal{Y}$, $h_{y}:{\mathbb R}_+^N\to{\mathbb R}_+$ is a general interference function and ${u}_y\in{\mathbb R}_{++}$ is a positive scalar. Assume that there exists $\delta>0$ satisfying $(\forall y\in\mathcal{Y})~ {u}_{y}\ge {\delta In addition, given a monotone norm $\|\cdot\|$ in ${\mathbb R}^N$, define the function $t_{\|\cdot\

Figures (3)

  • Figure 1: Utility for two users. Points A and B represent the utilities achieved with the power vectors ${\boldsymbol{p}}_1=(p_\mathrm{max},0)$ and ${\boldsymbol{p}}_2=(0,~p_\mathrm{max})$, respectively. Point C indicates the max-min fair utility point achieved with concurrent transmission, and Point D indicates the max-min fair utility point achieved through time sharing. Any utility below the line segment connecting points A and B (the convex hull of $\mathcal{S}_\mathcal{P}$) is achievable through time sharing.
  • Figure 2: Sample points on the weak Pareto boundary when the conditions of Corollary \ref{['cor.withpc']} are satisfied.
  • Figure 3: Sample points on the weak Pareto boundary when the conditions of Corollary \ref{['cor.withpc']} are not satisfied.

Theorems & Definitions (41)

  • Definition 1
  • Definition 2
  • Definition 3
  • Example 1
  • Remark 1
  • Lemma 1
  • proof
  • Definition 4
  • Example 2
  • Proposition 1
  • ...and 31 more