t h if n t n is nh oh then pn t h pn t

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Unformatted text preview: ity of birth in (t , t + h) if N (t ) = n is nλh + o(h). Then, Pn (t + h) = Pn (t )(1 − nλh − o(h)) + Pn−1 (t )((n − 1)λh + o(h)) 7 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Pure Birth Process (Yule-Furry Process) Assumptions: • Probability of splitting in (t , t + h): λh + o(h) • Probability of more than one split in (t , t + h): o(h) The probability of birth in (t , t + h) if N (t ) = n is nλh + o(h). Then, Pn (t + h) = Pn (t )(1 − nλh − o(h)) + Pn−1 (t )((n − 1)λh + o(h)) Pn (t + h) − Pn (t ) = −nλhPn (t ) + Pn−1 (t )(n − 1)λh + f (h), with f (h) ∈ o(h) Pn (t + h) − Pn (t ) f (h) = −nλPn (t ) + Pn−1 (t )(n − 1)λ + h h Let h → 0, Pn (t ) = −nλPn (t ) + (n − 1)λPn−1 (t ) Initial condition Pn0 (0) = P {N (0) = n0 } = 1 7 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Pure Birth Process (Yule-Furry Process) Probabilities are given by a set of ordinary differential equations. Pn (t ) = −nλPn (t ) + (n − 1)λPn−1 (t ) Pn0 (0) = P {N (0) = n0 } = 1 Solution Pn (t ) = where n k n−1 e−λn0 t (1 − e−λt )n−n0 n = n0 , n0 + 1, . . . n − n0 = n! . k !(n − k )! 8 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Pure Birth Process (Yule-Furry Process) Solution P n (t ) = n−1 e−λn0 t (1 − e−λt )n−n0 n = n0 , n0 + 1, . . . n − n0 Observation: The solution can be seen as a negative binomial distribution, i.e., probability of obtaining n0 successes in n trials. Suppose p =prob. of success and q = 1 − p =prob. of failure. Then, the probability that the first (n − 1) trials result in (n0 − 1) successes and (n − n0 ) failures followed by success on the nth trial is: n−1 pn0 −1 q n−n0 p = n − n0 n−1 pn0 q n−n0 ; n = n0 , n0 + 1, . . . n − n0 If p = e−λt and q = 1 − e−λt , both equations are the same. 9 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Pure Birth Process (Yule-Furry Process) • 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. 10 / 47 J. Virtamo Birth Processes Birth-Death Processes 38.3143 Queueing Theory / Processes processes Relationship to Markov Chains Linear Birth-Death Birth-death Examples Example 2. Pure birth process (Poisson process) Pure Birth Processes. Generalization 1 λi = λ i=0 i = 0, 1 , 2 , . . . πi (0) = µ =0 0 > • In a Yule-Furry process, for N (t ) = n the probability ofi a 0 i change during (t , t + h) depends on n. birth probability per time unit is constant λ (t , t + h) is independent of N (t ). initially the population size • In a Poisson process, the probability of a change during l 0 l 1 l l 2 ....
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