Applied Mathematics and Computation 379 (2020) 125255 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc On a class of birth-death processes with time-varying intensity functions Virginia Giorno a , Amelia G. Nobile a,∗ Dipartimento di Informatica, Università degli Studi di Salerno, Via Giovanni Paolo II n. 132, Fisciano (SA) 84084, Italy a r t i c l e i n f o Article history: Received 12 December 2019 Revised 13 March 2020 Accepted 22 March 2020 Available online 13 April 2020 MSC: 60J80 60J28 65C50 Keywords: Inhomogeneous birth-death chain Transient distributions First-passage time densities Periodic intensity functions a b s t r a c t 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. © 2020 Elsevier Inc. All rights reserved. 1. Introduction and background Continuous-time birth-death (BD) chains have been extensively used in various fields, such as physics, chemistry, engi- neering, biology, genetics, ecology, population growth dynamics and queueing systems. In some instances, unrestricted continuous-time BD chains are taken in account (cf. Conolly [1], Di Crescenzo and Mar- tinucci [2], Hongler and Parthasarathy [3], Pruitt [4], Tarabia and El-Baz [5]). In particular, double-ended BD chains can be employed to model several real systems, as the taxi-passenger queue (cf., for instance, Dobbie [6], Sharma and Nair [7], Tara- bia [8], Conolly et al. [9]). Moreover, BD chains on the nonnegative integers are considered to model the states of molecules in chemical and physical models (cf., Buonocore et al. [10], Conolly et al. [11]), to describe population growth (see, Crawford and Suchard [12], Ricciardi [13]), to analyze the evolution of queueing systems (cf., Giorno et al. [14,15], Lenin et al. [16], Medhi [17], Sharma [18]). BD chains are also used to model systems subject to various types of disasters (see, Dharmaraja et al. [20], Di Crescenzo et al. [21], Economou and Fakinos [22,23], Giorno and Nobile [24], Giorno et al. [25], Kapodistria et al. [26]). Moreover, various diffusion processes have been derived as continuous approximations of BD chains (cf., Abundo [19], Dharmaraja et al. [20], Di Crescenzo et al. [21,27,28]). The determination of first-passage time (FPT) probability density function (pdf) and of its moments is of interest in applications. Unfortunately, closed form solutions for the transition probabilities and for the FPT densities are available ∗ Corresponding author. E-mail addresses: giorno@unisa.it (V. Giorno), nobile@unisa.it (A.G. Nobile). https://doi.org/10.1016/j.amc.2020.125255 0096-3003/© 2020 Elsevier Inc. All rights reserved.