Transcript Lecture07-CTMC-Part02-BIRTH

Al-Imam Mohammad Ibn Saud University
CS433
Modeling and Simulation
Lecture 11
Birth-Death Process
02 May 2009
Dr. Anis Koubâa
Birth-Death Chain
2


The birth-death process is a special case of Continuous-time
Markov process where the states represent the current size of a
population and where the transitions are limited to births and
deaths.
Birth-death processes have many application in demography,
queueing theory, or in biology, for example to study the evolution
of bacteria.
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Birth-Death Chain
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



A pure birth process is a birth-death process where μi = 0 for all i≥0
A pure death process is a birth-death process where λi = 0 for all i≥0
A (homogeneous) Poisson process is a pure birth process where λi = λ for all
A M/M/1 queue is a birth-death process used to describe customers in an
infinite queue.
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Birth-Death Chain
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λ0
0
μ1
λ1
λi-1
1
μi

Find the steady state probabilities

Similarly to the previous example,

λi
i
0
0
 0

  1  1 
1
1

Q
 0
2
  2  2 


And we solve

πQ  0
and
i  1
i 0
μi+1






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Example
5

The solution is obtained
0 0  11  0
0
 1   0
1
0 0   1  1  1  2 2  0

In general
 0 1
 2  
 1  2
 j 1 j 1    j   j   j   j 1 j 1  0


 0

  j 1
Making the sum equal to 1
   ...

0
j 1  
 0 1   
 1





...

j

1
1
j




 0 ... j 
 
  0
 1 ... j 1 
Solution exists if
 0 ... j 1 
S  1   
  
j 1  1 ... j 

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End of Chapter