The rapid growth of small cells is driving cellular network toward randomness and heterogeneity. The multi-tier heterogeneous network (HetNet) addresses the massive connectivity demands of the emerging cellular networks. Cellular networks are usually modeled by placing each tier (e.g macro, pico and relay nodes) deterministically on a grid which ignores the spatial randomness of the nodes. Several works were idealized for not capturing the interference which is a major performance bottleneck. Overcoming such limitation by realistic models is much appreciated. Multi-tier relay cellular network is studied in this paper, In particular, we consider \({\mathscr {K}}\)-tier transmission modeled by factorial moment and stochastic geometry and compare it with a single-tier, traditional grid model and multi-antenna ultra-dense network (UDN) model to obtain tractable rate coverage and coverage probability. The locations of the relays, base stations, and users nodes are modeled as a Poisson Point Process. The results showed that the proposed model outperforms the traditional multi-antenna UDN model and its accuracy is confirmed to be similar to the traditional grid model. The obtained results from the proposed and comparable models demonstrate the effectiveness and analytical tractability to study the HetNet performance.

小细胞的快速增长正在推动蜂窝网络朝着随机性和异质性方向发展。多层异构网络（HetNet）满足了新兴蜂窝网络的巨大连接性需求。通常通过确定性地将每个层（例如，宏，微微和中继节点）放置在忽略节点空间随机性的网格上来对蜂窝网络进行建模。为了不捕获干扰（这是主要的性能瓶颈），对一些作品进行了理想化处理。人们非常赞赏通过现实模型克服这种限制。本文研究了多层中继蜂窝网络，特别是考虑了\（{\ mathscr {K}} \）通过阶乘矩和随机几何建模的分层传输，并将其与单层传统网格模型和多天线超密集网络（UDN）模型进行比较，以获得可控的速率覆盖率和覆盖率。中继站，基站和用户节点的位置被建模为泊松点过程。结果表明，所提出的模型优于传统的多天线UDN模型，其准确性与传统的网格模型相似。从提议的模型和可比较的模型中获得的结果证明了研究HetNet性能的有效性和分析性。