  Heterogeneous cellular networks (HCNs) constitute a necessary step in the
evolution of cellular networks. In this paper, we apply the
signal-to-interference ratio (SIR) meta distribution framework for a refined
SIR performance analysis of HCNs, focusing on K-tier heterogeneous cellular
networks based on the homogeneous independent Poisson point process (HIP)
model, with range expansion bias (offloading bias) in each tier. Expressions
for the b-th moment of the conditional success probability for both the entire
network and each tier are derived, based on which the exact meta distributions
and the beta approximations are evaluated and compared. Key performance metrics
including the mean success probability, the variance of the conditional success
probability, the mean local delay and the asymptotic SIR gains of each tier are
obtained. The results show that the biases are detrimental to the overall mean
success probability of the whole network and that the b-th moment curve (versus
the SIR threshold) of the conditional success probability of each tier can be
excellently approximated by the horizontal shifted versions of the first moment
curve of the single-tier PPP network. We also provide lower bounds for the
region of the active probabilities of the base stations to keep the mean local
delay of each tier finite.
