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

With the fast development of reconfigurable intelligent surface (RIS), the network topology becomes more complex and varied, which makes the network design and analysis extremely challenging. Most of the current works adopt the binary system stochastic geometric, missing the coupling relationships between the direct and reflected paths caused by RISs. In this paper, we first define the typical triangle which consists of a base station (BS), a RIS and a user equipment (UE) as the basic ternary network unit in a RIS-assisted ultra-dense network (UDN). In addition, we extend the Campbell's theorem to the ternary system and present the ternary probability generating functional (PGFL) of the stochastic geometry. Based on the ternary stochastic geometry theory, we derive and analyze the coverage probability, area spectral efficiency (ASE), area energy efficiency (AEE) and energy coverage efficiency (ECE) of the RIS-assisted UDN system. Simulation results show that the RISs can improve the system performances, especially for the UE who has a high signal to interference plus noise ratio (SINR), as if the introduced RIS brings in Matthew effect. This phenomenon of RIS is appealing for guiding the design of complex networks.