  Relaxation processes in supercooled liquids are known to exhibit interesting
as well as complex behavior. One of the hallmarks of this relaxation process
observed in the measured auto correlation function is occurrence of multiple
steps of relaxation. The shorter time relaxation is known as the
$\beta$-relaxation which is believed to be due to the motion of particles in
the cage formed by their neighbors. One the other hand longer time relaxation,
the $\alpha$-relaxation is believed to be the main relaxation process in the
liquids. The timescales of these two relaxations processes dramatically
separate out with supercooling. In spite of decades of researches, it is still
not clearly known how these relaxation processes are related to each other. In
this work we show that, there is a possible universal relation between short
time $\beta$-relaxation and the long time $\alpha$-relaxation. This relation is
found to be quite robust across many different model systems. Finally we show
that length scale obtained from the finite size scaling analysis of $\beta$
timescale is same as that of length scale associated with the dynamic
heterogeneity in both two and three dimensions.
