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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online