The operation of quantum dot lasers epitaxially grown on silicon is investigated through a quantum-corrected Poisson-drift-diffusion model. This in-house developed simulation framework completes the traditional rate equation approach, which models the intersubband transitions involved into simultaneous ground-state and excited-state lasing, with a physics-based description of carrier transport and electrostatic effects. The code is applied to look into some of the most relevant mechanisms affecting the lasing operation. We analyze the impact of threading dislocations on non-radiative recombination and laser threshold current. We demonstrate that asymmetric carrier transport in the barrier explains the ground-state power quenching above the excited-state lasing threshold. Finally, we study p-type modulation doping and its benefits/contraindications. The observation of an optimum doping level, minimizing the ground-state lasing threshold current, stems from the reduction of the electron density, which counteracts the benefits from the expected increase of the hole density. This reduction is due to electrostatic effects hindering electron injection.