Birth-death processes with temporary birth and/or death halts

Birth-death processes, with temporary halts in birth/death, may help model many real-world problems in diverse fields, from energy harvesting to medical research. These are unstudied particular cases of birth-death processes influenced by the changes in an underlying random environment. Unlike many models in the literature, which study birth-death processes in random environments, the above models have some unique features, such as not possessing a product-form steady-state solution. This paper considers a birth-death process where both birth and death may have temporary halts. More precisely, we model the population size as a Markov chain operating in a random environment. If the environment state is 0, birth and death can occur; when it is 1, only death can occur, and when the state is 2, no birth and death can occur. We obtain the stability condition and steady-state distribution of the process. Some crucial system performance measures are computed. We also considered a finite version of the process and analyzed the evolution of the process in the transient time. Finally, we fitted real-world tumor growth data to the deterministic counterpart of the model to compare its performance with some of the existing models in the literature.