Summary: Let \(N(t)\) be an exponentially ergodic birth-death process on the state space \(\{0,1,2,\dots\}\) governed by the parameters \(\{\lambda_ n,\mu_ n\}\), where \(\mu_ 0=0\), such that \(\lambda_ n=\lambda\) and \(\mu_ n=\mu\) for all \(n\geq N\), \(N\geq 1\), with \(\lambda<\mu\). We develop an algorithm to determine the decay parameter of such a specialized exponentially ergodic birth-death process, based on E. van Doorn’s representation [J. Approximation Theory 51, 254-266 (1987; Zbl 0636.42023)] of eigenvalues of sign-symmetric tridiagonal matrices. The decay parameter is important since it is indicative of the speed of convergence of ergodicity. Some comparability results for the decay parameters are given, followed by the discussion for the decay parameter of a birth-death process governed by the parameters such that \(\lim_{n\to\infty} \lambda_ n=\lambda\) and \(\lim_{n\to\infty} \mu_ n=\mu\). The algorithm is also shown to be a useful tool to determine the quasi-stationary distribution, i.e. the limiting distribution conditioned to stay in \(\{1,2,\dots\}\), of such specialized birth-death processes.