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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...

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