The Knudsen compressor can be applied as either a vacuum pump or compressor for gases. Earlier investigations have indicated that there are several interesting potential applications of the Knudsen compressor because it has no moving parts and requires no lubricants or supplementary working fluids. However, its energy efficiency tends to be low, so that careful optimization is necessary. An important aspect of the optimization is an understanding of the Knudsen compressor's operating characteristics in the transitional flow regime of rarefied gas dynamics. This article presents a formulation of Knudsen compressor operation in transitional, rarefied flow. In certain simplified but meaningful situations the formulation provides essentially analytical results for the sensitivity of key performance indicators, such as the energy use and device volume per unit of upflow, to changes in operating and geometric parameters. A numerical study of more complicated situations, using the most general form of the formulation developed here, is substantiated by the analytical investigation. The numerical results also extend the understanding of the Knudsen compressor's performance characteristics to conditions that cannot be addressed by the simplified analytical form. Specifically, minimization of the device's volume per unit of upflow is found when the entire cascade operates in transitional flow, which can only be studied using the complete formulation. The results make clear that operation in the transitional flow regime can lead to very significant (factor of 5 to 10) reductions of energy use and device volume for a given task.