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.