A Controllable Branching Process

The following model is considered in this paper. There is a population of cells, each of which lives for a unit of time, after which it either divides into two or dies. The probability of division $p(i)$ depends only on the size of the population $2i$. A study is made of the random variables $\tau_i$ – the time at which the population number deviates from given limits. It is proved that $\mathbf M\tau_i<\infty$ always. An asymptotic function of the distribution of the random variable $\tau_i/\mathbf M\tau_i$ is found with wide assumptions. The results of the calculation of $\mathbf M\tau_i$ on a computer are given for some particular cases.