The convection process in colloidal suspensions of solid particles is usually investigated by adopting either a homogeneous binary mixture (HM) or a particulate medium (PM) model. The present study aims only at comparing the instability threshold conditions for the onset of convection in the particle-dominated regime obtained through these two models. We consider a colloidal suspension of solid particles in a Rayleigh-Benard set up. A PM model is adopted which accounts for the effects of thermophoresis, sedimentation, and Brownian diffusion. We assume that the suspension layer is subjected to a vertical temperature gradient that is not thermally destabilizing as in the case of negligibly small coefficient of thermal expansion. Thus, we consider the idealized situation wherein the thermally induced buoyancy is absent and the focus is solely on the particle-dominated convection regime. We show that the coupled effects of thermo-migration, sedimentation, and Brownian diffusion give rise to a weak convection regime that sets in on the long particle diffusion time scale. A small wave number expansion is undertaken to determine the stability threshold conditions. When contrasted with those obtained through the HM model, the latter are found to have the same functional dependence on parameters as those describing the onset of Soret-driven convection for the case of a positive and large separation ratio. However, we show that a PM model, unlike the HM model, yields stability threshold conditions that can be mapped to corresponding experimental parameters such as the size of the particles and of the fluid cell.