The modified Frantz–Nodvik equation considering the pulse repetition rate (PRR) for continuous-wave end-pumped picosecond laser amplifier is theoretically developed for the first time by analogizing Q-switch theory and solving rate equations. Based on the modified Frantz–Nodvik equation, a simple finite-element slice model is established to simulate the output characteristics of the end-pumped high-repetition-rate picosecond amplifier. Moreover, the validity of the theory and model is well verified by the experimental results of a two-staged diode-end-pumped Nd  :  YVO₄ amplifiers using a picosecond fiber laser with adjustable PRRs as the seed source. The agreement between the experimental and theoretical results illustrates that the as-developed pulse amplification theory and model are a powerful tool for designing and optimizing adjustable-high-repetition-rate solid-state picosecond laser amplifiers.