On a multi-type critical age-dependent branching process

Abstract

We will consider a branching process with m > 1 distinguishable particle types as follows. At time 0, one newly born cell of type i is born ( i = 1, 2, ···, m ). Cell type i lives a random lifetime with continuous distribution function G i ( t ), G i (0+) = 0. At the end of its life, cell i is replaced by j 1 new cells of type 1, j 2 new cells of type 2, ···, j m new cells of type m with probability , and we define the generating functions for i = 1,···, m , where and . Each new daughter cell proceeds independently of the state of the system, with each cell type j governed by G j (t ) and h j (s) .