Lecture 12

# We essentially get a band around the diagonal with

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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 state-transition 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 C-K equation. For the birth-and-death process, we can rewrite the C-K equation (from (4)) as p t = , +   p t +  1 p p 0t = ,0 p 0 t + 1 p 1t: 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 non-zero. Then, the Markov chain is irreducible (means every state can be reached from every other state by chaining), it can be shown that in steady-state lim p t =  t!1 ij j exists and are independent of the initial state i. Revisiting C-K 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 ,00...
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