**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).
.
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.
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.
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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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