Analytical investigation on the performance of a two stage twin rotary compressor for $CO_2$ heat pump water heater system has been carried out. A computer simulation program was made based on analytical models for gas compression in control volumes, leakages among neighboring volumes, and dynamics of moving elements of the compressor. Calculated cooling capacity, compressor input, and COP were well compared to those of experiments over the compressor speeds tested. For the operating condition of suction pressure of 3 MPa, and discharge pressure of 9 MPa, and compressor inlet temperature of $35^{circ}C$, the compressor efficiency was calculated to be 80.2%: volumetric, adiabatic, and mechanical efficiencies were 88.3%, 93.2%, and 92.7%, respectively. For the present compressor model, volumetric and adiabatic efficiencies of the second stage cylinder were lower by about $6{sim}7%$ than those of the first stage mainly due to the smaller discharge port at the second stage. Parametric study on the discharge port size showed that the compressor performance could be improved by 3.5% just by increasing the discharge port diameter by 20%.