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**Unformatted text preview: **ity of birth in (t , t + h) if N (t ) = n is nλh + o(h).
Then,
Pn (t + h) = Pn (t )(1 − nλh − o(h)) + Pn−1 (t )((n − 1)λh + o(h)) 7 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Pure Birth Process (Yule-Furry Process)
Assumptions:
• Probability of splitting in (t , t + h): λh + o(h)
• Probability of more than one split in (t , t + h): o(h) The probability of birth in (t , t + h) if N (t ) = n is nλh + o(h).
Then,
Pn (t + h) = Pn (t )(1 − nλh − o(h)) + Pn−1 (t )((n − 1)λh + o(h))
Pn (t + h) − Pn (t ) = −nλhPn (t ) + Pn−1 (t )(n − 1)λh + f (h), with f (h) ∈ o(h)
Pn (t + h) − Pn (t )
f (h)
= −nλPn (t ) + Pn−1 (t )(n − 1)λ +
h
h Let h → 0,
Pn (t ) = −nλPn (t ) + (n − 1)λPn−1 (t )
Initial condition Pn0 (0) = P {N (0) = n0 } = 1
7 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Pure Birth Process (Yule-Furry Process)
Probabilities are given by a set of ordinary differential
equations.
Pn (t ) = −nλPn (t ) + (n − 1)λPn−1 (t )
Pn0 (0) = P {N (0) = n0 } = 1
Solution
Pn (t ) =
where n
k n−1
e−λn0 t (1 − e−λt )n−n0 n = n0 , n0 + 1, . . .
n − n0
= n!
.
k !(n − k )!
8 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Pure Birth Process (Yule-Furry Process)
Solution
P n (t ) = n−1
e−λn0 t (1 − e−λt )n−n0 n = n0 , n0 + 1, . . .
n − n0 Observation: The solution can be seen as a negative binomial
distribution, i.e., probability of obtaining n0 successes in n trials.
Suppose p =prob. of success and q = 1 − p =prob. of failure.
Then, the probability that the ﬁrst (n − 1) trials result in (n0 − 1)
successes and (n − n0 ) failures followed by success on the nth
trial is:
n−1
pn0 −1 q n−n0 p =
n − n0 n−1
pn0 q n−n0 ; n = n0 , n0 + 1, . . .
n − n0 If p = e−λt and q = 1 − e−λt , both equations are the same.
9 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Pure Birth Process (Yule-Furry Process) • Yule studied this process in connection with the theory of
evolution, i.e., population consists of
the species within a genus and
creation of a new element is due to
mutations.
• This approach neglects the probability of species dying out and
size of species.
• Furry used the same model for radioactive transmutations.
10 / 47 J. Virtamo
Birth Processes
Birth-Death Processes 38.3143 Queueing Theory / Processes
processes
Relationship to Markov Chains
Linear Birth-Death Birth-death Examples Example 2. Pure birth process (Poisson process) Pure Birth Processes. Generalization 1
λi = λ
i=0
i = 0, 1 , 2 , . . .
πi (0) = µ =0
0
>
• In a Yule-Furry process, for N (t ) = n the probability ofi a 0
i change during (t , t + h) depends on n. birth probability per time unit is
constant λ
(t , t + h) is independent of N (t ). initially the population size • In a Poisson process, the probability of a change during l
0 l
1 l l
2 ....

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