This paper presents a comprehensive downlink SINR analysis for heterogeneous cellular networks (HCNs) with flexible cell association. The HCN is modeled as a multi-tier network where each tier's base stations (BSs) are randomly located with specific transmit power, path loss exponent, spatial density, and bias towards admitting users. The study derives the outage probability of a typical user, which is equivalent to the downlink SINR cumulative distribution function. It also computes the average ergodic rate and minimum average user throughput. The results show that in interference-limited full-loaded HCNs with unbiased cell association, the outage probability and average ergodic rate are not affected by the number of BSs or tiers. Biasing, however, significantly impacts these metrics. The analysis considers both the general case and specific scenarios, such as interference-limited networks with equal path loss exponents. The results demonstrate that the average ergodic rate is independent of BS transmit power, density, and the number of tiers in unbiased networks, but is worsened by biasing. The study also shows that adding more BSs increases network throughput without affecting SINR statistics, as interference and signal power increase proportionally. The findings provide a tractable framework for analyzing HCNs with flexible cell association, which is essential for optimizing network performance.This paper presents a comprehensive downlink SINR analysis for heterogeneous cellular networks (HCNs) with flexible cell association. The HCN is modeled as a multi-tier network where each tier's base stations (BSs) are randomly located with specific transmit power, path loss exponent, spatial density, and bias towards admitting users. The study derives the outage probability of a typical user, which is equivalent to the downlink SINR cumulative distribution function. It also computes the average ergodic rate and minimum average user throughput. The results show that in interference-limited full-loaded HCNs with unbiased cell association, the outage probability and average ergodic rate are not affected by the number of BSs or tiers. Biasing, however, significantly impacts these metrics. The analysis considers both the general case and specific scenarios, such as interference-limited networks with equal path loss exponents. The results demonstrate that the average ergodic rate is independent of BS transmit power, density, and the number of tiers in unbiased networks, but is worsened by biasing. The study also shows that adding more BSs increases network throughput without affecting SINR statistics, as interference and signal power increase proportionally. The findings provide a tractable framework for analyzing HCNs with flexible cell association, which is essential for optimizing network performance.