Constructing transient birth–death processes by means of suitable transformations

For a birth–death process N(t) with a reflecting state at 0 we propose a method able to construct a new birth–death process M(t) defined on the same state-space. The birth and death rates of M(t) depend on the rates of N(t) and on the probability law of the process N(t) evaluated at an exponentially distributed random time. Under a suitable assumption we obtain the conditional probabilities, the mean of the process, and the Laplace transforms of the downward first-passage-time densities of M(t). We also discuss the connection between the proposed method and the notion of ν-similarity, as well as a relation between the distribution of M(t) and the steady-state probabilities of N(t) subject to catastrophes governed by a Poisson process. We investigate new processes constructed from (i) a birth–death process with constant rates, and (ii) a linear immigration-death process. Various numerical computations are performed to illustrate the obtained results.