18 47 birth processes birth death processes

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Unformatted text preview: th Processes Examples Birth-Death Processes For n = 0 P0 (t + h) = P0 (t )(1 − λ0 h) + P1 (t )µ1 h + o(h) ⇒ P0 (t ) = −λ0 P0 (t ) + µ1 P1 (t ) • If λ0 = 0, then E0 → E1 is impossible and E0 is an absorbing state. • If λ0 = 0, then P0 (t ) = µ1 P1 (t ) ≥ 0 and hence P0 (t ) increases monotonically. Note: limt →∞ P0 (t ) = P0 (∞) = Probability of being absorbed. 18 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Steady-state distribution P0 (t ) = −λ0 P0 (t ) + µ1 P1 (t ) Pn (t ) = −(λn + µn )Pn (t ) + λn−1 Pn−1 (t ) + µn+1 Pn+1 (t ) As t → ∞, Pn (t ) → Pn (limit ). Hence, P0 (t ) → 0 and Pn (t ) → 0. Therefore, 0 = −λ0 P0 + µ1 P1 λ0 ⇒P1 = P0 µ1 0 = −(λ1 + µ1 )P1 + λ0 P0 + µ2 P2 λ0 λ1 ⇒P2 = P0 µ1 µ2 λ0 λ1 λ2 ⇒P3 = P0 etc µ1 µ2 µ 2 19 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Steady-state distribution P1 = λ0 P0 ; µ1 P2 = λ0 λ1 P0 ; µ 1 µ2 P3 = λ0 λ1 λ2 P0 ; µ1 µ2 µ 2 P4 = . . . The dependence on the initial conditions has disappeared. ∞ After normalizing, i.e., Pn = 1: n =1 n−1 1 P0 = ∞ n −1 1+ n=1 i =0 ; λi µi +1 λi µ i +1 i =0 ∞ n −1 Pn = 1+ n=1 i =0 , n≥1 λi µi +1 20 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Steady-state distribution n−1 1 P0 = ∞ n −1 1+ n=1 i =0 ; i =0 ∞ n −1 Pn = λi µi +1 λi µ i +1 1+ n=1 i =0 , n≥1 λi µi +1 Ergodicity condition Pn > 0, for all n ≥ 0, i.e.,: ∞ n−1 n=1 i =0 λi <∞ µ i +1 21 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Example. A single server system / Birth-death processes J. Virtamo 38.3143 Queueing Theory Example 3. A single server system l 0 - constant arrival rate λ (Poisson arrivals) • constant arrival rate λ (Poisson - stopping rate of the service µ (exponential distribution arrivals) 1 m The states of the system • stopping rate of service µ 0 server free (exponential distribution) 1 server busy • states of the system: 0 (server 1 ì í î ì í î 0 free), 1 (server busy) ~ Exp(m) ~ Exp(l) d dt π0(t) = − λπ0(t) + µπ1(t) d dt π1(t) = P0 (t ) = −λP0 (t ) + µP1=t ) −λ Q( λπPt)(− µπ1λtP (t ) − µP (t ) 0( t ) = ( ) 0 1 1 BY adding both sides of the equations d dt (π0 (t) d + π1(t)) = 0 ⇒ π (t) + (λ + µ)π (t) = µ µ π0 (t) + π1 (t) = constant = 1 ⇒ d λ −µ ⇒ (e(λ+µ)t π (t)) = µe(λ+µ)t π1(t) = 1 − π0 (t 22 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains J. Virtamo Linear Birth-Death Processes 38.3143 Queueing Theory / Birth-death processes Example. 3. A single server system A single server system Example Examples 7 l 0 - constant arrival rate λ (Poisson arrivals) - stopping rate of the service µ (exponential distribution) 1 m The states of theP (t ) system 0 0 server free P1 ( 1 server b...
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This document was uploaded on 01/20/2014.

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