Families of birth–death processses 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 ∈ 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 are also resolved in the more general setting in which the two processes are called similar if there are constants cij and v such that p∼ij(t) = cij e^vt pij(t) for all t ≥ 0.