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 (p∼ij (t)) are said to be similar if, for all i, j ϵ N, there are constants cij such that p∼ij(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. These issues arealso resolved in the more general setting in which the two processes are called similar if there are constants cij and ν suchthat p∼ij(t) = cijeνtpij (t) for all t ≥ 0. © 2000 Applied Probability Trust.