We study, in the dynamically evolving QGP fireball formed in relativistic
heavy ion collisions at RHIC and LHC, the growth of strangeness yield toward
and beyond the chemical equilibrium. We account for the contribution of the
direct strangeness production and evaluate the thermal-QCD strangeness
production mechanisms. The specific yield of strangeness per entropy, s/S, is
the primary target variable. We explore the effect of collision impact
parameter, i.e., fireball size, on kinetic strangeness chemical equilibration
in QGP. Insights gained in study the RHIC data with regard to the dynamics of
the fireball are applied to the study strangeness production at the LHC. We use
these results and consider the strange hadron relative particle yields at RHIC
and LHC in a systematic fashion. We consider both the dependence on s/S and
directly participant number dependence.Comment: 21 pages, 13 figures, PRC in press. Strangeness production recomputed
with K-factor K=1.7. Particle yields recomputed with SHARE 2.