Many random phenomena can be characterized in a state manner and with changes of states defining the nature of the phenomena. For the case where the states are integers, the state represents the population of a set of entities, and the population changes in integer steps according to births and deaths, a birth–death random process is defined.
The usual manner for characterizing a birth–death random process is via a sequence of differential equations that define the change in state probability with time. Results for important cases—constant birth rate and zero death rate processes, state-dependent birth rate and zero death rate processes, zero birth rate and constant death rate processes, and constant birth and constant death rate processes—are discussed. A constant birth and constant death process underpins the simplest queueing system—the M/M/1 queueing system (a first-come–first-serve queue with waiting and service times).
Fundamental to birth–death random processes are the following definitions.