the time dependent solution of a bd process

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Unformatted text preview: (t ) Pi (t ) = (i + 1)µPi +1 (t ) − i µPi (t ), i = 0, . . . , n − 1 ⇒ Pn (t ) = e−nµt d i µt [e Pi (t )] = (i +1)µPi +1 (t )ei µt ⇒ Pi (t ) = (i +1)e−i µt µ dt t Pi +1 (t )ei µt dt 0 t Pn−1 (t ) = ne−(n−1)µt µ e−nµt e(n−1)µt dt = ne−(n−1)µt (1 − e−µt ) 0 n (e−µt )i (1 − e−µt )n−i i Binomial distribution: The survival probability at time t is e−µt independent of others. Recursively: Pi (t ) = 27 / 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 28 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Relation to CTMC Infinitesimal generator matrix: −λ0 λ0 0 ... ... ... µ1 −(λ1 + µ1 ) λ1 0 ... ... 0 µ2 −(λ2 + µ2 ) λ2 0 ... Q= . J. Virtamo 38.3143 Queueing Theory / Birth-death processes . 0 µ3 −(λ3 + µ3 ) λ3 0 . . . . . The time-dependent solution of .a BD process (continued). . . . . . . . . . . . . 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 29 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Relation to DTMC 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 inEn Define P (En−1 |En ) similarly. Then P (En+1 |En ) P (En+1 |En ) = λn , P (En−1 |En ) µn λn µn , P (En−1 |En ) = λn + µn λn + µn The same conditional probabilities hold if it is given that a transition will take place in (t , t + h) conditional on being in En . 30 / 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 31 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Linear Birth-Death Processes Linear Birth-Death Process • λn = nλ • µn = n µ ⇒P0 (t ) = µP1 (t ) Pn (t ) = −(λ + µ)nPn (t ) + λ(n − 1)Pn−1 (t ) + µ(n + 1)Pn+1 (t ) Steady state behavior is characterized by: lim P0 (t ) = 0 ⇒ P1 (∞) = 0 t →∞ Similarly as t → ∞ Pn (∞) = 0 32 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Linear Birth-Death Processes Steady state behavior is characterized by: lim P0 (t ) = 0 ⇒ P1 (∞) = 0 t →∞ Similarly as t → ∞ Pn (∞) = 0 Two cases can happen: • If P0 (∞) = 1 ⇒ the probability of ultimate extinction is 1. • If P0 (∞) = P0 < 1, the relations P1 = P2 = P3 . . . = 0 imply with probability 1 − P0 that the population can increase without bounds. The population must either die out or increase indefinitely. 33 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Mean of a Linear Birth-Death Process Pn (t ) = −(λ + µ)nPn (t ) + λ(n − 1)Pn−1 (t ) + µ(n + 1)Pn+1 (t ) ∞ nPn (t ) Define Mean by M (t ) = n =1 ∞ nPn (t ), then: and consider M (t ) = n=1 ∞ ∞ 2 M (t ) = −(λ + µ) n Pn (t ) + λ n=1 (n − 1)nPn−1 (t ) n=1 ∞ +µ (n + 1)nPn+1 (t ) n=1 Write (n − 1)n = (n − 1)2...
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