Ternary Stochastic Geometry Theory for Performance Analysis of RIS-Assisted UDN

TL;DR

This work develops a ternary stochastic geometry framework to analyze RIS-assisted ultra-dense networks by introducing a dual-coordinate, typical BRU triangle as the fundamental unit. It extends Campbell's theorem and PGFL to the ternary setting and derives approximate closed-form expressions for key metrics, including $P_c$, $ ext{ASE}$, $ ext{AEE}$, and $ ext{ECE}$, accounting for direct and cascaded RIS paths with RIS of $Q$ elements. The model relies on three independent PPPs with intensities $ abla_n$, $ abla_m$, and $ abla_u$, with thinning yielding the active BS intensity $ abla'_{UN}$ and RIS-beamforming behavior depending on RIS location. Simulations validate significant RIS gains and reveal a Matthew-like effect across cell-center and cell-edge UEs, offering deployment and configuration insights that balance spectral efficiency and energy consumption in RIS-enhanced networks.

Abstract

Currently, network topology becomes increasingly complex with the increased number of various network nodes, bringing in the challenge of network design and analysis. Most of the current studies are deduced based on the binary system stochastic geometry, overlooking the coupling and collaboration among nodes. This limitation makes it difficult to accurately analyze network systems, such as reconfigurable intelligent surface (RIS) assisted ultra-dense network (UDN). To address this issue, we propose a dual coordinate system analysis method, by using dual observation points and their established coordinates. The concept of a typical triangle that consists of a base station (BS), a RIS, and a user equipment (UE) is defined as the fundamental unit of analysis for ternary stochastic geometry. This triangle comprises the base station, the RIS, and the user equipment (UE). Furthermore, we extend Campbell's theorem and propose an approximate probability generating function for ternary stochastic geometry. Utilizing the theoretical framework of ternary stochastic geometry, we derive and analyze performance metrics of a RIS-assisted UDN system, such as coverage probability, area spectral efficiency, area energy efficiency, and energy coverage efficiency. Simulation results show that RIS can significantly enhance system performance, particularly for UEs with high signal-to-interference-plus-noise ratios, exhibiting a phenomenon similar to the Matthew effect.

Figures (11)

• Figure 1: The voronoi diagram of a RIS-assisted UDN, where $N=15$, $M=20$ and $U=20$.
• Figure 2: Signal transmission diagram of RIS-assisted UDN, where $D_1$,$D_2$, $I_1$ and $I_2$ respectively stand for the direct signal, the reflected signal, the direct interference and the reflection interference during the communication between the $u$th UE and the $n$th BS.
• Figure 3: Ternary systems differ from binary systems in previous literature in terms of unit modeling.
• Figure 4: The geometric relationship of the $\triangle {\rm BRU}$, where the $n$th BS acts as the central point of a circle with a cell radius of $\sqrt{{x_b}/({\pi\lambda'_{un}})}$, effectively partitioning the RIS into two distinct regions - $m\in \Psi_m^M$ (the interior of the cell) and $m \notin \Psi_m^M$ (the exterior of the cell).
• Figure 5: Diagram of four reflection cases in the network, where "solid line" represents the transmission link and "dashed line" represents the interrupted link. (a) full connection; (b) the interrupted link, i.e., ${B \nrightarrow R}$; (c) the interrupted link, i.e., ${R \nrightarrow U}$; (d) the interrupted link, i.e., ${B \nrightarrow R \nrightarrow U}$.

Theorems & Definitions (5)

• Proposition 1: Coexistence of observation points
• proof
• Definition 1
• Lemma 1: Campbell's theorem of ternary stochastic geometry
• Lemma 2: approximate PGFL analysis of ternary stochastic geometry