A new theoretical approach for modeling the saturated single pass gain in a three-level fiber amplifier is presented, relevant to the behavior of rare-earth-doped silica fibers. A basic approximation considering the stimulated emission rate Ws(z,ν) as a dimensionless parameter S, independent of the spatial and frequency variables z and ν, allows to obtain analytic expressions for input and output pump, ASE and signal powers inside the fiber core. We show that these expressions only depend on the S parameter, which is determined by solving the photon balance equation, and S is shown to be fully representative of the saturation in the medium. The main result of the model is that the pump repartition P(z) takes a simple analytical form, which can be separated into two parts, below and above the saturated absorption length L0, which is a function of S. The first part 0〈z〈L0 is the saturated absorption where the pump distribution is linear and the second part L0〈z〈L is the non saturated absorption region with an exponentially decreasing pump distribution. Compared to other analytical models with the assumption of an averaged inversion population 〈N2(z)〉, we obtain a good description of the difference between the co- and counter-propagating ASE behaviors accounting for the fiber length and the pumping level. The model, which is well suited to longitudinal pumping, can also describe a side pumped fiber amplifier by simple adjustment of some of the model parameters.