Scale height of silicate dust in a unit of that of the gas (top panels) and midplane dust-to-gas ratio (bottom panels) for different models of H_d (Case of F_p/g = 0.3 and τ_s,p = 0.1 at 3 au). Here, Drift-BKR and Diff-BKR for pebbles are neglected, whereas Drift-BKR and Diff-BKR (K = 0 or 1) for silicate dust are included. Left two panels: cases of α_acc = 10⁻² and α_Dr = α_Dz = 10⁻⁴ with K = 0 or 1, respectively. Right two panels: cases of α_acc = 10⁻² and α_Dr = α_Dz = 10⁻² with K = 0 or 1, respectively. The gray lines are the case of Ida & Guillot (2016). The blue lines are the case of Schoonenberg & Ormel (2017). The red lines are the case of Hyodo et al. (2019) and the results show runaway pile-up when α_acc = 10⁻² and α_Dr = α_Dz = 10⁻⁴ with K = 1. The black lines are the case of this work (Eq. (36)). In the cases of the gray lines (i.e., Ida & Guillot 2016), pile-ups occur in a runaway fashion for the left two panels, and the time-evolutions are shown for t = 1 × 10⁴ yr, t = 5 × 10⁴ yr, and t = 1.5 × 10⁵ yr. The other cases reach a steady-state within t = 5 × 10⁵ yr. Here, the recycling of water vapor and silicate dust onto pebbles are not included to focus on the different models of H_d.