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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 BD 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.
 Fall '08
 Staff
 Computer Networks

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