# 23 47 birth processes birth death processes

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Unformatted text preview: usy t ) 1 = λP0 (t ) − µP1 (t ) ì í î ì í î 0 = −λP0 (t ) + µP1 (t ) ~ Exp(m) ~ Exp(l) Given d π0(t) P0−tλπ+t) + µπ)(t= 1, P0 (t ) + (λ + µ)P0 (t ) = µ. that: = ( ) 0( P1 (t 1 ) dt µ t) π1(t) = P0 (t ) λπ0(t) − µπ1(+ = BY adding both sides of λ + µ the equations d dt Q= P0 (0) − −λ λ µ µ −µ −(λ+µ)t e λ+µ d λ λ ⇒ π0 (t) + π1 (t) = constant = 1 ⇒ (λπ1(t)t= 1 − π0 (t) ) dt (π0 (t) + π1 (t)) = 0 + P1 (0) − P1 (t ) = e− +µ d λ+ λ+µ π0 (t) + (λ + µ)π0 (t) = µµ ⇒ d (e(λ+µ)t π0(t)) = µe(λ+µ)t dt dt µ µ −(λ+µ)t π0(t) Solution = λEquilibrum+distribution + Deviation from the = +µ + (π0(0) − λ µ )e λ π1(t) = λ λ µ + ( (0) − λ+µ )e−(λ+ t equilibrium+withπ1exponentialµ)decay. 23 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Example 2. Pure birth process (Poisson process) PoissonProcess. Probabilities λi = λ µ =0 i i = 0, 1 , 2 , . . . πi (0) = 1 0 i=0 i&gt;0 Poisson Process birth probability per time unit is initially the population size • Birth probability per time unit is constant λ constant λ • The population size is initially 0 l 0 l 1 l l 2 l i-1 ... l i All states All states are transient d dt Equations i π (t) = −λπi (t) + λπi−1(t) d dt i&gt;0 &gt; π0(t) Pi (t−= −(tPi (t ) + λPi −1 (t ), i⇒ 0 π0(t) = e−λt = ) λπ0 λ ) P0 (t ) = −λP0 (t ) d λt (e πi(t)) dt = λπi−1(t)eλt ⇒ πi(t) = e−λtλ t 0 πi−1(t )eλ 24 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Poisson Process. Probabilities Equations Pi (t ) = −λPi (t ) + λPi −1 (t ), i &gt; 0 P0 (t ) = −λP0 (t ) ⇒ P0 (t ) = e−λt d λt [e Pi (t )] = λPi −1 (t )eλt ⇒ Pi (t ) = e−λt λ dt t P1 (t ) = e−λt λ t Pi −1 (t )eλt dt 0 e−λt eλt dt = e−λt (λt ) 0 (λt )i −λt e i! Number of births in interval (0, t ) ∼ Poisson(λt ). Recursively: Pi (t ) = 25 / 47 Example Processes Relationship toprocess Birth-Death 1. Pure death Markov Chains Birth Processes Linear Birth-Death Processes Examples λi = 0 Pure Death Process. i = 0, 1, 2, . . . Probabilities µ = iµ i πi(0) = all individuals have the same the Pure Death Process mortality rate µ • All the individuals have the same mortality rate µ 1 0 i= i= system starts • The population size is initially n 0 1 m 2 2m n-1 ... 3m (n-1) m Sta oth n nm State 0 is an absorbing state. The rest are transient. Equations d dt πn(t) = −nµπn(t) ⇒ P(t ) = −nµPn (t ) n d Pi (tdt =i(it) 1)µPii +t1)µπiPi (tt) − = 0, i (t) n − 1 ) π ( + = ( +1 ( ) − i µ +1( ), i iµπ . . . , d iµt dt (e πi (t)) = (i + 1)µπi+1 (t)eiµt ⇒ πn(t) i = 0, 1 , . . πi (t) 26 / 47(i = Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Pure Death Process. Probabilities Equations Pn (t ) = −nµPn...
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## This document was uploaded on 01/20/2014.

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