This paper develops a tractable framework for analyzing the downlink signal-to-interference-plus-noise ratio (SINR) in 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 different transmit powers, path loss exponents, spatial densities, and bias factors. The authors derive the outage probability of a typical user in the network or a specific tier, which is equivalent to the downlink SINR cumulative distribution function. They also derive the average ergodic rate and the minimum average user throughput, which measure spectral efficiency. The results are accurate for all SINRs and admit simple closed-form expressions in certain special cases. The paper observes that the number of BSs or tiers does not affect the outage probability or average ergodic rate in an interference-limited fully-loaded HCN with unbiased cell association. Biasing alters these metrics, with higher bias factors generally improving performance in lightly loaded networks but degrading it in fully-loaded networks. The paper provides a novel analytical model and results for SINR in HCNs with flexible cell association, contributing to a better understanding of how system parameters affect performance.This paper develops a tractable framework for analyzing the downlink signal-to-interference-plus-noise ratio (SINR) in 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 different transmit powers, path loss exponents, spatial densities, and bias factors. The authors derive the outage probability of a typical user in the network or a specific tier, which is equivalent to the downlink SINR cumulative distribution function. They also derive the average ergodic rate and the minimum average user throughput, which measure spectral efficiency. The results are accurate for all SINRs and admit simple closed-form expressions in certain special cases. The paper observes that the number of BSs or tiers does not affect the outage probability or average ergodic rate in an interference-limited fully-loaded HCN with unbiased cell association. Biasing alters these metrics, with higher bias factors generally improving performance in lightly loaded networks but degrading it in fully-loaded networks. The paper provides a novel analytical model and results for SINR in HCNs with flexible cell association, contributing to a better understanding of how system parameters affect performance.