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

Abstract

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 { p_ij(t) } and { ̃p_ij(t) } are said to be similar if, for all i, j ∈ N, there are constants c_ij such that ̃p_ij(t) = c_ijp_ij(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 are also resolved in the more general setting in which the two processes are called similar if there are constants c_ij and ν such that ̃p_ij(t) = c_ije^νtp_ij(t) for all t ≥ 0.