The asymptotic probability of the first degeneration for branching processes with immigration

Let $\eta(t)$ be the number of particles in a branching process with immigration at time $t$. The initial lifeperiod of the branching process with immigration equals $\tau$ if $\eta(0)=n>0$, $\eta(t)>0$ for all $t\in(0,\tau)$ and $\eta(\tau)=0$ (sample paths of the process, are supposed to be right continuous). We obtain asymptotic formulas for $Q_n=\mathbf P\{\tau<\infty\mid\eta(0)=n\}$ as $n\to\infty$.