  In this paper we develop a tractable framework for SINR analysis in downlink
heterogeneous cellular networks (HCNs) with flexible cell association policies.
The HCN is modeled as a multi-tier cellular network where each tier's base
stations (BSs) are randomly located and have a particular transmit power, path
loss exponent, spatial density, and bias towards admitting mobile users. For
example, as compared to macrocells, picocells would usually have lower transmit
power, higher path loss exponent (lower antennas), higher spatial density (many
picocells per macrocell), and a positive bias so that macrocell users are
actively encouraged to use the more lightly loaded picocells. In the present
paper we implicitly assume all base stations have full queues; future work
should relax this. For this model, we derive the outage probability of a
typical user in the whole network or a certain tier, which is equivalently the
downlink SINR cumulative distribution function. The results are accurate for
all SINRs, and their expressions admit quite simple closed-forms in some
plausible special cases. We also derive the \emph{average ergodic rate} of the
typical user, and the \emph{minimum average user throughput} -- the smallest
value among the average user throughputs supported by one cell in each tier. We
observe that neither the number of BSs or tiers changes the outage probability
or average ergodic rate in an interference-limited full-loaded HCN with
unbiased cell association (no biasing), and observe how biasing alters the
various metrics.
