  Poisson's equation plays a fundamental role as a tool for performance
evaluation and optimization of Markov chains. For continuous-time birth-death
chains with possibly unbounded transition and cost rates as addressed herein,
when analytical solutions are unavailable its numerical solution can in theory
be obtained by a simple forward recurrence. Yet, this may suffer from numerical
instability, which can hide the structure of exact solutions. This paper
presents three main contributions: (1) it establishes a structural result
(convexity of the relative cost function) under mild conditions on transition
and cost rates, which is relevant for proving structural properties of optimal
policies in Markov decision models; (2) it elucidates the root cause, extent
and prevalence of instability in numerical solutions by standard forward
recurrence; and (3) it presents a novel forward-backward recurrence scheme to
compute accurate numerical solutions. The results are applied to the accurate
evaluation of the bias and the asymptotic variance, and are illustrated in an
example.
