Transient analysis of a birth-death process with alternating rates.

A birth-death process on Z was proposed in [Conolly et al., 1997] to describe chain molecular diffusion. In that paper, the molecule is modeled as an infinitely long chain of atoms joined by links subjected to random shocks which cause the atoms to move and the molecule to diffuse. The shock mechanism is different according to whether the atom occupies an odd or an even position in the chain. Following the line of that model, we consider a birth-death process on the whole set of integers, characterized by a constant transition rate from even states and a possibly different rate from odd states. We determine the probability generating functions of even and odd states and then the transition probabilities of the process. Some features of the birth-death process confined to the nonnegative integers by the reflecting zero-state are also analyzed. In particular, making use of a Laplace transform approach, we obtain the expression of the zero-state probability.