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A generic model is introduced to describe the lasing characteristics of continuous-wave circular and racetrack-shaped ring Raman lasers based on micro- and nano-scale silicon waveguides. This model explicitly takes into account the effective Raman gain values for forward and backward lasing in the ring resonator, the presence of a bus waveguide in which the Stokes laser radiation coupled out from the ring undergoes additional Raman amplification, and the spatial gain variations for different polarization states in the ring structure. I show numerically that ring lasers based on micro-scale waveguides generate unidirectional lasing in either the forward or backward direction because of an asymmetry in nonlinear losses, whereas those based on nanowires yield only backward lasing due to a non-reciprocity in effective gain. Furthermore, the model indicates that backward lasing can yield a significantly higher Stokes output at the bus waveguide facets than lasing in the forward direction. Finally, considering a TE-polarized pump input for a (100) grown silicon ring Raman laser, I demonstrate numerically that the polarization state of the Stokes lasing radiation strongly depends on whether micro-scale or nano-scale waveguides are used.