A birth-death process is
completely determined by its set of rates if and only if this set satisfies a certain
condition C, say. If for a set of rates R the condition C is not fulfilled, then the
problem arises of characterizing all birth-death processes which have rate set R (the
indeterminate rate problem associated with R). We show that the characterization
may be effected by means of the decay parameter, and we determine the set of
possible values for the decay parameter in terms of R. A fundamental role in our
analysis is played by a duality concept for rate sets, which, if the pertinent rate sets
satisfy C, obviously leads to a duality concept for birth-death processes. The latter
can be stated in a form which suggests the possibility of extension in the context of
indeterminate rate problems. This, however, is shown to be only partially
true.
