On a class of birth-death processes with time-varying intensity functions

•Special class of birth-death chains.•Time-inhomogeneous unrestricted birth-death chains.•Time-inhomogeneous restricted birth-death processes.•Time-inhomogeneous double-ended chains.•Composition method.•Transient probabilities.•Reflecting boundary.•Symmetric birth-death processes with respect to zero state.•Conditional moments•First-passage time problems.•First-passage time densities.•Proportional intensity functions.•Periodic intensity functions•Numerical computations.•Flexibility in the shape of the distributions. In this paper, we investigate on a class of time-inhomogeneous birth-death chains obtained by applying the composition method to two time-inhomogeneous double-ended chains. Then, we consider the corresponding restricted birth-death process, with zero reflecting boundary. Finally, starting from the restricted process, we construct a time-inhomogeneous BD chain symmetric with respect to zero-state. We obtain closed form expressions for the transition probabilities and for the conditional moments; furthermore, the first-passage-time problem is also taken in consideration. Finally, various numerical computations are performed for periodic intensity functions.