The often-studied problem known as Kramers' problem, in the general area of rarefied-gas dynamics, is investigated in terms of a linearized, variable collision frequency model of the Boltzmann equation. A convenient change of variables is used to reduce the general case considered to a canonical form that is well suited for analysis by analytical and/or numerical methods. While the general formulation developed is valid for an unspecified collision frequency, a recently developed version of the discrete-ordinates method is used to compute the viscous-slip coefficient and the velocity defect in the Knudsen layer for three specific cases: the classical BGK model, the Williams model (the collision frequency is proportional to the magnitude of the velocity) and the rigid-sphere model.