In fact the above is nothing but q 0 specialized for

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Unformatted text preview: + 1 1: 10 These are also known as balance equations. In fact, the above is nothing but Q = 0, specialized for the B-D model. The nice thing is that this linear system of equations is easily solvable analytically. From (10), we have For j  1 =  0  0 : 1 =1 from (9), we have 0= which implies [using (10)], ,1 + 1  1 + + 2 2 2 2 = 1 1 + 1 1 , 00 = 1 1: Thus,  2 = 1 1 = 01 0:   2 Generalizing, we get  j CS 522, v 0.94, d.medhi, W’99 0 0 = 12 0 1 ::: 1  1 2 ::: 0 j, j j  1: 11 10 Recall that all the probabilities must sum to 1, i.e., 0 + 1 + 2 + ::: = 1: which, using the previous relation, gives 0 1 + 0 =1 + 0 1 =1 2  + :::::: = 1 Thus, as long as the sum is convergent, we can calculate 0 , and thus, all the other probabilities, j , due to (11). This is important since this probabilities help us determine something about the system behavior as we will see soon. X Aside: for the system to converge, we need the condition 0 1    j ,1 1 2    j : 1 [A finite note is that the balance equation can be easily written by looking...
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This document was uploaded on 03/19/2014 for the course CS 6030 at Western Michigan.

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