  We have numerically investigated the mean-field dynamics of the the $p$-spin
interaction spin glass model with p=3 using an efficient method of integrating
the dynamic equations. We find a new time scale associated with the onset of
the breakdown of the fluctuation-dissipation theorem in the intermediate time
regime. We also find that the off-equilibrium relaxation exhibits a sub-aging
behavior in the intermediate times and crosses over to a simple aging in the
asymptotic regime.
