We consider an age-dependent model which not only allows the generating function to be age-dependent, but also allows the parent to reproduce several times during its lifetime. By using the notion of $\vee$-space-time harmonic functions, we study the behavior of $Z_t$, the number of particles alive at time $t$, in the supercritical case. In particular we obtain results which are analogous to the classically known results for the Bellman-Harris model; in fact, we obtain convergence in probability.