I investigate the effect of tube diameter D and red blood cell capillary number C a (i.e. the ratio of viscous to elastic forces) on platelet margination in blood flow at a parts per thousand 37 % tube haematocrit. The system is modelled as three-dimensional suspension of deformable red blood cells and nearly rigid platelets using a combination of the lattice-Boltzmann, immersed boundary and finite element methods. Results of simulations during the dynamics before the steady state has been reached show that a non-diffusive radial platelet transport facilitates margination. This non-diffusive effect is important near the edge of the cell-free layer, but only for C a > 0.2, when red blood cells are tank-treading. I also show that platelet trapping in the cell-free layer is reversible for C a a parts per thousand currency sign 0.2. Margination is essentially independent of C a only for the smallest investigated tube diameter (D = 10 mu m). Once platelets have reached the cell-free layer, they tend to slide rather than tumble. The tumbling rate is essentially independent of C a but increases with D. Strong confinement suppresses tumbling due to the relatively small cell-free layer thickness at a parts per thousand 37 % tube haematocrit.