The authors consider a class of nonhomogeneous birth-death processes on the nonnegative integers with birth and death rates of the form \(\lambda_n(t)= \lambda_n a(t)\) and \(\mu_n(t)= \mu_nb(t)\). Their main objective is to study the speed of convergence of such processes when convergence occurs in some sense. By employing a technique based on the concept of logarithmic norm of a linear operator, the authors obtain bounds for the decay function of the process, which is a generalization of the decay parameter (or rate of convergence) of a homogeneous birth-death process. The results are applied to several Markovian queueing models.