It has recently been observed that the SIR distributions of a variety of cellular network models and transmission techniques look very similar in shape. As a result, they are well approximated by a simple horizontal shift (or gain) of the distribution of the most tractable model, the Poisson point process (PPP). To study and explain this behavior, this paper focuses on general single-tier network models with nearest-base station association and studies the asymptotic gain both at 0 and at infinity. We show that the gain at 0 is determined by the so-called mean interference-to-signal ratio (MISR) between the PPP and the network model under consideration, while the gain at infinity is determined by the expected fading-to-interference ratio (EFIR). The analysis of the MISR is based on a novel type of point process, the so-called relative distance process, which is a one-dimensional point process on the unit interval [0,1] that fully determines the SIR. A comparison of the gains at 0 and infinity shows that the gain at 0 indeed provides an excellent approximation for the entire SIR distribution. Moreover, the gain is mostly a function of the network geometry and barely depends on the path loss exponent and the fading. The results are illustrated using several examples of repulsive point processes.