Al-Imam Mohammad Ibn Saud University CS433 Modeling and Simulation Lecture 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 = λ for all A M/M/1 queue is a birth-death process used to describe customers in an infinite queue.

Birth-Death Chain 1 i λ0 λ1 λi-1 λi μ1 μi μi+1 1 i λ1 λi-1 λi μ1 μi μi+1 Find the steady state probabilities Similarly to the previous example, And we solve and

Example The solution is obtained In general Making the sum equal to 1 Solution exists if

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