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The redshift-space distortion (RSD) in the observed distribution of galaxies is known as a powerful probe of cosmology. Observations of large-scale RSD have given tight constraints on the linear growth rate of the large-scale structures in the universe. On the other hand, the small-scale RSD, caused by galaxy random motions inside clusters, has not been much used in cosmology, but also has cosmological information because universes with different cosmological parameters have different halo mass functions and virialized velocities. We focus on the projected correlation function $w(r_p)$ and the multipole moments $ξ_l$ on small scales ($1.4$ to $30\ h^{-1}\rm{Mpc}$). Using simulated galaxy samples generated from a physically motivated most bound particle (MBP)-galaxy correspondence scheme in the Multiverse Simulation, we examine the dependence of the small-scale RSD on the cosmological matter density parameter $Ω_m$, the satellite velocity bias with respect to MBPs, $b_v^s$, and the merger-time-scale parameter $α$. We find that $α=1.5$ gives an excellent fit to the $w(r_p)$ and $ξ_l$ measured from the SDSS-KIAS value added galaxy catalog. We also define the ``strength'' of Fingers-of-God as the ratio of the parallel and perpendicular size of the contour in the two-point correlation function set by a specific threshold value and show that the strength parameter helps constraining $(Ω_m, b_v^s, α)$ by breaking the degeneracy among them. The resulting parameter values from all measurements are $(Ω_m,b_v^s)=(0.272\pm0.013,0.982\pm0.040)$, indicating a slight reduction of satellite galaxy velocity relative to the MBP. However, considering that the average MBP speed inside haloes is $0.94$ times the dark matter velocity dispersion, the main drivers behind the galaxy velocity bias are gravitational interactions, rather than baryonic effects.