**Unformatted text preview: **+ (n − 1), (n + 1)n = (n + 1)2 − (n + 1)
34 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Mean of a Linear Birth-Death Process
∞ n2 Pn (t ) M (t ) = −(λ + µ)
n =1
∞ ∞
2 +λ (n + 1)2 Pn+1 (t ) + P1 (t ) (n − 1) Pn−1 (t ) + µ
n=1
∞ +λ n=1
∞ (n − 1)Pn−1 (t ) − µ
n=1 (n + 1)Pn+1 (t ) + P1 (t )
n =1 ∞ ⇒ M (t ) = λ ∞ nPn (t ) − µ
n=1 nPn (t ) = (λ − µ)M (t )
n=1 M (t ) = n0 e(λ−µ)t if Pn0 (0) = 1
35 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Mean of a Linear Birth-Death Process
M (t ) = n0 e(λ−µ)t
• If λ > µ then M (t ) → ∞
• If λ < µ then M (t ) → 0
∞ Similarly if M2 (t ) = n2 Pn (t ) one can show that: n=1 M2 (t ) = 2(λ − µ)M2 (t ) + (λ + µ)M (t )
and when λ > µ, the variance is:
n0 e2(λ−µ)t 1 − e(µ−λ)t λ+µ
λ−µ
36 / 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 37 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Linear Birth-Death Process. Example
Let X (t ) be the number of bacteria in a colony at instant t .
Evolution of the population is described by:
• the time that each of the individuals takes for division in two (binary ﬁssion), independently of the other bacteria
• the life time of each bacterium (also independent) Assume that:
• Time for division is exponentially dist. (rate λ)
• Life time is also exponentially dist. (rate µ) M (t ) = n0 e(λ−µ)t
• If λ > µ then the population tends to inﬁnity
• If λ < µ then the population tends to 0
38 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples A queueing system
s servers
µ K places
λ µ
µ µ • s servers
• K waiting places
• λ arrival rate (Poisson)
• µ Exp(µ) holding time (expectation 1/µ) Is it a birth-death process? 39 / 47 Birth Processes Birth-Death Processes s palvelinta Relationship to Markov Chains m
m
m K places
λ ì
í
î m K=5
s=4 Examples s servers
K waiting places λ arrival rate (Poisson) s servers
• µ
Exp(µ) holding time (expectation 1/µ
• K waiting places A queueing system
K odotuspaikkaa
l Linear Birth-Death Processes s servers
µ
µ • λ arrival rate (Poisson) µ • µ Exp(µ) holding time (expectation 1/µ)
The number of customers in system N is an appropriate state variable
µ Let “N =number of customers in the system” be the state variable.
• N determines uniquely the number of customers in waiting
- uniquelydeterminesthe number of customers in service andin service room and waiting room.
- after each arrival and departure the remaining service timservice
• After each arrival and departure the remaining es
of thetimes of the customers Exp(µ) distributed (memoryless)
customers in service are in service are Exp(µ) distributed
(memoryless).
0l 1l 2l 3l 4l 5l 6l 7l 8l 9 m 2m 3m 4m 4m 4m 4m 4m 4m 40 / 47 Birth Processes Bir...

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