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Unformatted text preview: oes the matrix Q look like? We essentially get a band around the diagonal with one
element on each side (tridiagonal matrix); all of the other elements are zero. 0 ,0
0
0
0
1
Q=B 1
@ 0 ,1+ 1 ,2+ 2 02
2
::: ::: ::: ::: 0 1
::: 0 C
::: 0 A
::: ::: ::: Pictorially, the statetransition diagram is as follows: CS 522, v 0.94, d.medhi, W’99 9 λ0 λ2 λ1 0 1 3 2 µ1 µ2 •• • µ3 Now, let’s go back to CK equation. For the birthanddeath process, we can rewrite the CK equation
(from (4)) as p t = , + p t + 1 p
p 0t = ,0 p 0 t + 1 p 1t:
0 j ij 0 j ij j, i i t 1 + i j, +1 p
j ij t for j +1 1 8 i Suppose that j ’s and i ’s are nonzero. Then, the Markov chain is irreducible (means every state can
be reached from every other state by chaining), it can be shown that in steadystate lim p t = t!1 ij j exists and are independent of the initial state i.
Revisiting CK equation above and using the fact that the derivative of a constant is zero, we have
0= , j +
j j + 1 0= j, 1+ j, +1 +1
j j for j 1 9 ,00...
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 Fall '08
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 Computer Networks

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