  We report microscopic calculation of key $\beta$-decay properties for some of
the crucial waiting point species having neutron closed magic shells 50 and 82.
Our calculation bear astrophysical significance vis-\'{a}-vis speeding of the
$r$-process. The $\beta$-decay properties include half-lives, energy rates of
$\beta$-delayed neutrons and their emission probabilities, both under
terrestrial and stellar conditions. We perform a pn-QRPA calculation with a
separable multi-shell interaction and include both allowed and unique
first-forbidden (U1F) transitions in our calculation. We compare our results
with previous calculations and measured data. Our calculation is in good
agreement with the experimental data. For certain cases, we noted a significant
decrease in the half-life calculation with the contribution of U1F transitions.
This is in contradiction to the shell model study where only for $N$ = 126
waiting-point nuclei, the forbidden transitions were reported to significantly
reduce the calculated half-lives. Our model fulfills the Ikeda sum rule for
even-even cases. For odd-A cases the rule is violated by up to 15$\%$ for
$^{125}$Tc.
