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We consider birth-death processes taking values in 
but allow the death rate in state 0 to be positive, so that escape from 
is possible. Two such processes with transition functions 
are said to be similar if, for all 
there are constants cij such that 
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 ν such that 
for all t ≥ 0.
