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We present a comprehensive model of a tunneling injection quantum dot laser. The spatially resolved model incorporates homogeneous and inhomogeneous gain broadenings, both of which are found to be important. The tunneling injection improves dynamical properties as a result of an efficient carrier injection into the laser state and also because of an effective lowering of the carrier temperature. The lower temperature yields lower carrier densities, which reduces scattering and hence narrows the homogeneous linewidth, improving the modulation response. The inhomogeneous broadening is naturally found to deteriorate the modulation response.