 Sequential two-step waiting time

For independent exponential waits of rates $a$ and $b$, their sum has density $ab(e^{-bt}-e^{-at})/(a-b)$, obtained by convolution on $[0,t]$. The limit at $a=b$ is $a^{2}t{e}^{-at}$ and the mean is $a^{-1}+b^{-1}$.