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We describe a finite-difference time-domain (FDTD) model of a long (edge-emitting) gain medium based on a quantum-dot (QD) in-a-well structure under the framework of the Maxwell-Schrödinger equations. The model includes the dynamic behavior of a QD gain medium including an excited state incorporated within carrier rate equations and considers the carrier density dependence of the refractive index. The model enables us also to calculate carrier diffusion effects, which, unlike in quantum well based structures, play an important role in QD devices, since carrier capture and escape processes modify the effective carrier diffusion length. We present results of basic static and dynamic lasers properties as well as of the interaction of a QD amplifier with short, 150 fs pulses. We identify four regimes of operation for the pulse-QD interaction, two of which are important: the linear-saturated regime and the Rabi-oscillation dominated regime. The latter leads to Rabi floppings with a period shorter than the pulse itself. The model can be easily employed for any complicated process such as four-wave mixing, saturable absorption, semiconductor pulse laser sources, etc.