  Gamma-ray bursts (GRBs) are the most luminous astrophysical events observed
so far. They are conventionally classified into long and short ones depending
on their time duration, $T_{90}$. Because of the advantage their high redshifts
offer, many efforts have been made to apply GRBs to cosmology. The key to this
is to find correlations between some measurable properties of GRBs and the
energy or the luminosity of GRBs. These correlations are usually referred to as
luminosity relations and are helpful in understanding the GRBs themselves. In
this paper, we explored such correlations in the X-ray emission of GRBs. The
X-ray emission of GRBs observed by \emph{Swift} has the exponential functional
form in the prompt phase and relaxes to a power-law decay at time $T_p$. We
have assumed a linear relation between $\log L_{X, p}$ (with $L_{X, p}$ being
the X-ray luminosity at $T_p$) and $\log [T_p/(1+z)]$, but there is some
evidence for curvature in the data and the true relationship between $L_{X, p}$
and $T_p/(1+z)$ may be a broken power law. The limited GRB sample used in our
analysis is still not sufficient for us to conclude whether the break is real
or just an illusion caused by outliers. We considered both cases in our
analysis and discussed the implications of the luminosity relation, especially
on the time duration of GRBs and their classification.
