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Fall 2010 HW06 - University of Illinois Fall 2010 ECE 313...

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University of Illinois Fall 2010 ECE 313: Problem Set 6: Solutions Reliability and CDFs 1. [Reliability of three s - t networks] (a) Network outage occurs if and only if at least one link on the upper s - t path fails and at least one link on the lower s - t path fails. The probability that at least one link on the upper path fails is one minus the probability that no links on the path fail, or 1 - (1 - p ) 3 = 3 p - 3 p 2 + p 3 . The same is true for the lower path, and the states of the two paths are independent. So P ( F ) = (3 p - 3 p 2 + p 3 ) 2 . For p = 0 . 001 , P ( F ) 0 . 0000089820 . (b) P ( D i ) = ( 4 i ) p i (1 - p ) 4 - i for i = 0 , 1 , 3 , 4, because the number of links from among { 1 , 2 , 3 , 4 } that fail has the binomial distribution with parameters 4 and p. P ( D 2 ,s ) = 2 p 2 (1 - p ) 2 because D 2 ,s is the union of two disjoint events, each having probability p 2 (1 - p ) 2 . The first of these events is that links 1 and 2 fail and links 3 and 4 do not fail, and the second of these events is that links 3 and 4 fail and links 1 and 2 do not fail. Similarly, P ( D 2 ,d ) = 4 p 2 (1 - p ) 2 , because D 2 ,d is the
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