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

We consider birth-death processes taking values in ${\cal N} \equiv \{0,1,\ldots\}$, but allow the death rate in state $0$ to be positive, so that escape from ${\cal N}$ is possible. Two such processes with transition functions $\{p_{ij}(t)\}$ and $\{{\tilde p}_{ij}(t)\}$ are said to be {\it similar} if, for all $i,j \in {\cal N},$ there are constants $c_{ij}$ such that ${\tilde p}_{ij}(t) = c_{ij}p_{ij}(t)$ for all $t \geq 0.$ 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.