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

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