Summary

Chapter 21 Summary: Cellular Network Theory

Key Points

• 1.

  Hexagonal cell model and frequency reuse: The co-channel reuse distance scales as $D = R\sqrt{3N}$, and the worst-case cell-edge SIR is $(3N)^{\alpha/2}/6$ with 6 first-tier interferers. Valid cluster sizes satisfy $N = i^2 + ij + j^2$, and the minimum $N$ for a given SIR target is $N_{\min} = (6\,\text{SIR}_{\min})^{2/\alpha}/3$. While modern systems use $N = 1$ with advanced interference management, the hexagonal model remains essential for understanding the fundamental trade-off between bandwidth per cell and interference protection.

• 2.

  Stochastic geometry and PPP: Modelling BS locations as a homogeneous PPP with intensity $\lambda$ yields a closed-form coverage probability $p_c(\tau) = 1/(1 + \rho(\tau, \alpha))$ that is, remarkably, independent of $\lambda$ in the interference-limited regime. For $\alpha = 4$: $p_c(\tau) = 1/(1 + \sqrt{\tau}\arctan\sqrt{\tau})$. This invariance arises because signal and interference scale identically with density — the foundation for understanding why densification improves capacity but not coverage.

• 3.

  Heterogeneous networks: Multi-tier HetNets with macro and small cells create asymmetric interference landscapes. Cell range expansion (CRE) bias offloads users to lightly loaded small cells at the cost of reduced SINR in the expanded region. The association probability to tier $k$ depends on the product $\lambda_k(B_k P_k)^{2/\alpha}$, providing a principled framework for load balancing. eICIC (Almost Blank Subframes) protects CRE-zone users.

• 4.

  Area spectral efficiency: The ASE $\mathcal{A} = \lambda \cdot C(\alpha)$ scales linearly with BS density, with the per-link rate constant $C(\alpha)$ depending only on the propagation environment. This linear scaling is the theoretical justification for ultra-dense networks and shows that each 2$\times$ densification doubles the area capacity. Practical limits include LOS/NLOS transitions, backhaul constraints, and pilot contamination.

• 5.

  Handover and mobility: The A3 event triggers handover when the target cell's RSRP exceeds the source by hysteresis $H_{\mathrm{hys}}$ for time-to-trigger $T_{\mathrm{TTT}}$. The handover rate scales as $2v/(\pi R)$, and the ping-pong rate is suppressed by both $H_{\mathrm{hys}}$ and $T_{\mathrm{TTT}}$ but at the cost of increased handover failure risk at high velocity. Mobility robustness optimisation (MRO) balances these competing requirements.

• 6.

  Sectorisation: Tri-sector antennas reduce the effective number of first-tier interferers from 6 to approximately 2, providing a practical SIR gain of $\sim$2.5 (4 dB). Combined with reduced reuse factor, sectorisation can yield 5$\times$ or greater capacity improvements. The gain is limited by finite front-to-back ratio and side-lobe levels.