Matrix Analytic Methods in Branching processes

We examine the question of solving the extinction probability of a particular class of continuous-time multi-type branching processes, named Markovian binary trees (MBT). The extinction probability is the minimal nonnegative solution of a xed point equation that turns out to be quadratic, which makes its resolution particularly clear. We analyze rst two linear algorithms to compute the extinction probability of an MBT, of which one is new, and, we propose a quadratic algorithm arising from Newton's iteration method for xed-point equations. Finally, we add a catastrophe process to the initial MBT, and we analyze the resulting system. The extinction probability turns out to be much more di cult to compute; we use a G/M/1-type Markovian process approach to approximate this probability.