Multi-tier cellular networks able to handle multiple classes of base stations with varying transmission power and coverage area offer a promising approach to meet increasing traffic demands by proper balancing of the input load. This paper investigates the performance of buffered multi-tier networks with randomly placed transmission nodes. Under simplifying model assumptions of Rayleigh fading and unbounded attenuation, we propose an approach to derive the multi-tier coverage probability and Shannon capacity of typical cells. The goal is to study the trade-off between base station density and interference effects in multi-tier compared to single-tier networks by assessing the system performance under varying policies. We also use stochastic simulation to verify the theoretical results and visualize the system behavior under different parameter settings.