Coalescence times for critical Galton-Watson processes with immigration

Let $X^I_n$ be the coalescence time of two particles picked at random from the $n$th generation of a critical Galton-Watson process with immigration, and let $A^I_n$ be the coalescence time of the whole population in the $n$th generation. In this paper, we study the limiting behaviors of $X^I_n$ and $A^I_n$ as $n\to\infty$.