Earlier two-dimensional (radial and poloidal angle), analytically tractable ion kinetic models of the scrape-off layer (SOL) in which a steady state is achieved by balancing the streaming loss of ions to the divertor target plates with the radial diffusion of ions from the core are unable to distinguish between limited and diverted plasmas. The model presented here removes this limitation while stilI remaining amenable to a similar Wiener-Hopf solution procedure. To phenomenologically model ion recycling, the boundary conditions employed at the divertor plates allow for partial reflection. The diffusion into the private flux region and the extended divertor channels (all of normalized length d along the magnetic field), as well as the rest of the SOL, is evaluated. The SOL is shown to be asymmetric about the separatrix because ions from the core must stream by the X point before diffusing into the private flux region.