# The capacity of an s t ow network is the maximum ow

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Unformatted text preview: ∪ (F2 F4 ) just as it does for network D. Outage also occurs in network E if exactly three links fail and those links are links 1,5,4, or links 3,5,2. This covers all ways outage can occur in network E , and yields the outage probability given in Figure 2.12, where qi = 1 − pi for all i. For example, if pi = 0.001 for 1 ≤ i ≤ 5, then P (F ) = 0.000002001995002 ≈ 0.000002. The network outage probability is determined mainly by the probability that both links 1 and 3 fail or both links 2 and 4 fail. Thus, the outage probability of network E is about the same as for network D. Applying the union bound The union bound doesn’t apply and isn’t needed for network A. For network B, F = F1 ∪ F2 . The union bound for two events implies that P (F ) ≤ p1 + p2 . The numerical value is 0.002 if p1 = p2 = 0.001. For network C , F = F1 F3 ∪ F1 F4 ∪ F2 F3 ∪ F2 F4 so that the union bound for m = 4 yields P (F ) ≤ p1 p3 + p1 p4 + p2 p3 + p2 p4 = (p1 + p2 )(p3 + p4 ). This upper bound could also be obtained by ﬁrst upper bou...
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## This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.

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