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Neodymium-doped crystalline and glass laser media are widely used to produce picosecond and short nanosecond pulses of high peak power. Most often, laser generation occurs between the Stark sublevels of the transition 7F3/2→4I11/2, which is often addressed as a classic 4-level scheme. For the most efficient conversion of pump energy into output radiation of a laser amplifier, it must operate near the saturation condition, which tends to equalize the upper and lower laser levels populations and make the medium more transparent. A sufficiently rapid relaxation of the lower laser level population, in turn, contributes to the partial recovery of the population difference and the amplification properties of the medium. Thus, the generation and amplification of short pulses, comparable in duration or shorter than the lifetime of the lower laser level, is a substantially non-stationary problem. Based on the available data on the lifetime values of the lower laser level in various neodymium-doped crystalline and glass matrices, one can conclude that the values differ very significantly (for example, in the cases of popular media as Nd:YAG, Nd:YLF, Nd:glass, etc.). Thus, the value of the lifetime of the lower laser level should be adequately taken into account when creating efficient, especially two-pass, schemes for amplifying picosecond and short nanosecond pulses. Previously, we proposed and experimentally implemented an approach for direct measurement of the lifetime of the lower laser level in Nd:YAG, based on recording the dynamics of partial gain recovery of a probe pulse following with a variable delay after a saturating pulse. In the present work, we develop an approach to detailed modeling of efficient amplification schemes based on neodymium-doped crystalline and glass media operating under saturation conditions. In the framework of theoretical modeling, we modify the well-known model of laser pulse amplification taking into account finite relaxation rate value of the lower laser level. Based on the experimental data and calculations, we obtained a more accurate estimation 60±20 ps of the lower level relaxation time in Nd:YAG and then analyzed the operation of a picosecond pulse amplifier scheme based on sequential two-pass diode-pumped amplifiers. This provides on the output 25 ps pulses with an energy of 5 mJ at a fundamental wavelength with a repetition rate of 1 kHz.