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For maximum signal-to-interference (SIR) ratio cell association, we investigate the coverage probability of HetNets under Nakagami fading. Prompting serious analytical complexities, Nakagami-type fading, however, describes several important wireless environments of 4G/5G standards, including multi-antenna systems. Adopting tools of stochastic geometry, we provide a number of closed-form approximations for the coverage probability, which have been missing in the literature. Our analysis covers integer and noninteger Nakagami, Rician, and <inline-formula> <tex-math notation="LaTeX">\kappa -\mu</tex-math></inline-formula> shadowed fading distributions, and also multiuser zero-forcing beamforming in the downlink. Furthermore, the analysis of this paper incorporates the traits of AWGN, partially loaded systems, and bounded path-loss function, which are often overlooked in studying HetNets. The result are easy to compute, preserve acceptable accuracy, and explicitly demonstrate the impact of fading parameters, density of BSs, SIR thresholds, loads, and path-loss model. We reveal important insights regarding the practice of densification in conjunction with SIR thresholds, BS loads, and path-loss model. It is observed that from a network-level perspective, a bounded path-loss model can be grasped via a reduction of the density of BSs granting association as well as interference, for which the former becomes dominant in dense configurations.