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Explore birth-death processes in this lecture, studying population size changes, applications in demography, queueing theory, biology, and more. Learn about pure birth and pure death processes, Poisson processes, and M/M/1 queues.
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Al-Imam Mohammad Ibn Saud University CS433Modeling and SimulationLecture 11 Birth-Death Process Dr. Anis Koubâa 02 May 2009
Birth-Death Chain • The birth-death process is a special case of Continuous-time Markov process where the states represent the current size of a population and where the transitions are limited to births and deaths. • Birth-death processes have many application in demography, queueing theory, or in biology, for example to study the evolution of bacteria.
Birth-Death Chain • A pure birth process is a birth-death process where μi = 0 for all i≥0 • A pure death process is a birth-death process where λi = 0 for all i≥0 • A (homogeneous) Poisson process is a pure birth process where λi = λfor all • A M/M/1 queue is a birth-death process used to describe customers in an infinite queue.
Birth-Death Chain λ0 λ1 λi-1 λi 0 1 i μi μi+1 μ1 • Find the steady state probabilities • Similarly to the previous example, • And we solve and
Example Solution exists if • The solution is obtained • In general • Making the sum equal to 1
