Galton-Watson processes with random generating times

A new class of branching model, Galton-Watson processes with random generating times, is consider in this paper. Unlike the traditional case, we consider each particle can generate $k$ times with probability $h_k$ where $k$ can be infinite, and producing $m$ new particles with probability $p_m$ after each splitting. These are Galton-Waston processes with countably many types in which particles of type $i$ may only give descendants to type $i+1$ and type $1$. It is revealed that the extinction probability $q$ of such model is the smallest roots of the equation $s=h(g(s))$ in $[0,1]$, where $h(s)$ and $g(s)$ are p.g.f of $\{h_k,k=0,1,2,...\}$ and $\{p_k,k=0,1,2,...\}$ respectively. Moreover, the reciprocal of the analogue of Perron-Frobenius eigenvalue in infinity many types case is actually the extinction probability of a continuous-time branching process, from which the ergodic properties are discussed.