Pure Birth Processes in Industrial Engineering
Birth-death processes, Yule-Furry models, and pure birth processes are discussed in the context of industrial engineering. The study of evolutionary processes, population dynamics, and radioactive transmutations are explored through mathematical models and examples. The concept of state transitions and steady-state distributions in pure birth processes is also examined.
- Industrial Engineering
- Birth-death Processes
- Yule-Furry Models
- Evolutionary Process
- Population Dynamics
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BIRTH BIRTH- -DEATH PROCESS DEATH PROCESS Prepared by A.Y T B LG N 131119 Department of Industrial Eng.
CONTENTS CONTENTS BIRTH PROCESS BIRTH-DEATH PROCESS RELATIONSHIP TO MARKOV CHAINS LINEAR BIRTH-DEATH PROCESS
BIRTH PROCESS BIRTH PROCESS
Pure Pure Birth Birth Process Process ( (Yule Yule- -Furry Furry Process Process) ) Example: Consider cells which reproduce according to the following rules: A cell present at time t has probability h + o(h) of splitting in two in the interval (t, t + h) This probability is independent of age Events betweeen different cells are independent.
Yule studied this process in connection with the theory of evolution, i.e., population consists of the species within a genus and creation of a new element is due to mutations. This approach neglects the probability of species dying out and size of species. Furry used the same model for radioactive transmutations.
Pure Pure Birth Birth Processes Processes - - Generalization Generalization In a Yule-Furry process, for N(t) = n the probability of a change during (t, t + h) depends on n. In a Poisson process, the probability of a change during (t, t + h) is independent of N(t).
BIRTH BIRTH- -DEATH PROCESS DEATH PROCESS
Pure Birth process: If n transitions take place during (0, t), we may refer to the process as being in state En. Changes in the pure birth process: En En+1 En+2 . . . Birth-Death Processes consider transitions En En 1as well as En En+1 if n 1. If n = 0, only E0 E1is allowed.
Steady Steady- -state state distribution distribution P 0 0 (t) = 0P0(t) + 1P1(t) P 0 n (t) = ( n + n)Pn(t) + n 1Pn 1(t) + n+1Pn+1(t) As t , Pn(t) Pn(limit). Hence, P 0 0 (t) 0 and P 0 n (t) 0. Therefore, 0 = 0P0 + 1P1 P1 = 0 1 P0 0 = ( 1 + 1)P1 + 0P0 + 2P2 P2 = 0 1 1 2 P0 P3 = 0 1 2 1 2 2 P0 etc.
Embedded Markov chain of the process. For t , define: P(En+1|En) = Prob. of transition En En+1 = Prob. of going to En+1 conditional on being in En Define P(En 1|En) similarly. Then; The same conditional probabilities hold if it is given that a transition will take place in (t, t + h) conditional on being in E.
LINEAR LINEAR BIRTH BIRTH- -DEATH DEATH PROCESS PROCESS
Linear Birth-Death Process n = n n = n P 0 0 (t) = P1(t) P 0 n (t) = ( + )nPn(t) + (n 1)Pn 1(t) + (n + 1)Pn+1(t)
REFERENCES REFERENCES Queueing Theory / Birth-death process. Winston, Wayne L. Queueing Theory / Birth-death processes. J. Vitano Performance modelling and evaluation. Birth-death processes. J. Campos Discrete State Stochastic Processes J. Baik
