# h is 0t et the probability of more than one change

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Unformatted text preview: .. l i-1 l i All states Generalization = −λπ (t) + λπ (t) d π (t) i>0 dt i i i −1 • Assume that for N (t ) = n the probability of a new change d nπ() = to dt +01tin (t , t−λπ)0(t)λn h + o(h). + h is ⇒ π0(t) = e−λt • The probability of more than one change is o (h). d (eλt πi(t)) = λπi−1(t)eλt ⇒ πi(t) = e−λtλ t πi−1(t/ )eλ 11 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Pure Birth Processes. Generalization Generalization • Assume that for N (t ) = n the probability of a new change to n + 1 in (t , t + h) is λn h + o(h). • The probability of more than one change is o (h). Then, Pn (t + h) = Pn (t )(1 − λn h) + Pn−1 (t )λn−1 h + o(h), n = 0 P0 (t + h) = P0 (t )(1 − λ0 h) + o(h) ⇒Pn (t ) = −λn Pn (t ) + λn−1 Pn−1 (t ) P0 (t ) = −λ0 Pn (t ) Equations can be solved recursively with P0 (t ) = P0 (0)e−λ0 t 12 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Pure Birth Process. Generalization Let the initial condition be Pn0 (0) = 1. The resulting equations are: Pn (t ) = −λn Pn (t ) + λn−1 Pn−1 (t ), n > n0 Pn0 (t ) = −λn0 Pn0 (t ) Yule-Furry processes assumed λn = nλ 13 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Outline 1 Birth Processes 2 Birth-Death Processes 3 Relationship to Markov Chains 4 Linear Birth-Death Processes 5 Examples 14 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Birth-Death Processes Notation • 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 → . . . J. Virtamo 38.3143 Queueing Theory / Birth-death processes • Birth-Death Processes consider transitions En → En−1 as well as En → E +1 if n ≥ 1. If n = 0, only (continued) The time-dependent nsolution of a BD process E0 → E1 is allowed. l0 0 l1 1 m1 l2 2 m2 m3 li l i-1 i ... mi l i+1 i+1 m i+1 m i+2 The equations component wise 15 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Birth-Death Processes 16 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Birth-Death Processes Assumptions If the process at time t is in En , then during (t , t + h): • Transition En → En+1 has probability λn h + o (h) • Transition En → En−1 has probability µn h + o (h) • Probability that more than 1 change occurs = o (h). Pn (t + h) = Pn (t )(1 − λn h − µn h) + Pn−1 (t )(λn−1 h) + Pn+1 (t )(µn+1 h) + o(h) Time evolution of the probabilities ⇒ Pn (t ) = −(λn + µn )Pn (t ) + λn−1 Pn−1 (t ) + µn+1 Pn+1 (t ) 17 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Dea...
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## This document was uploaded on 01/20/2014.

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