Families of birth - death processes with similar time - dependent behaviour

We consider birth-death processes taking values in N ≡ {0, 1, . . .}, but allow the death rate in state 0 to be positive, so that escape from N is possible. Two such processes with transition functions {pij(t)} and {p̃ij(t)} are said to be similar if, for all i, j ∈ N , there are constants cij such that p̃ij(t) = cijpij(t) for all t ≥ 0. We determine conditions on the birth and death rates of a birth-death process for the process to be a member of a family of similar processes, and we identify the members of such a family.