Birth–death process

The birth–death process is a special case of continuous-time Markov process where the state transitions are of only two types: 'births', which increase the state variable by one and 'deaths', which decrease the state by one. The model's name comes from a common application, the use of such models to represent the current size of a population where the transitions are literal births and deaths. Birth–death processes have many applications in demography, queueing theory, performance engineering, epidemiology and biology. They may be used, for example, to study the evolution of bacteria, the number of people with a disease within a population, or the number of customers in line at the supermarket. When a birth occurs, the process goes from state n to n + 1. When a death occurs, the process goes from state n to state n − 1. The process is specified by birth rates { λ i } i = 0 … ∞ {displaystyle {lambda _{i}}_{i=0dots infty }} and death rates { μ i } i = 1 … ∞ {displaystyle {mu _{i}}_{i=1dots infty }} . A pure birth process is a birth–death process where μ i = 0 {displaystyle mu _{i}=0} for all i ≥ 0 {displaystyle igeq 0} . A pure death process is a birth–death process where λ i = 0 {displaystyle lambda _{i}=0} for all i ≥ 0 {displaystyle igeq 0} . A (homogeneous) Poisson process is a pure birth process where λ i = λ {displaystyle lambda _{i}=lambda } for all i ≥ 0 {displaystyle igeq 0} M/M/1 model and M/M/c model, both used in queueing theory, are birth–death processes used to describe customers in an infinite queue. In queueing theory the birth–death process is the most fundamental example of a queueing model, the M/M/C/K/ ∞ {displaystyle infty } /FIFO (in complete Kendall's notation) queue. This is a queue with Poisson arrivals, drawn from an infinite population, and C servers with exponentially distributed service time with K places in the queue. Despite the assumption of an infinite population this model is a good model for various telecommunication systems.