[Reliability of three s - t networks] Consider s - t networks A through C shown.

Question

[Reliability of three s - t networks] Consider s - t networks A through C shown. For each network, suppose that each link fails with probability p, independently of the other links, and suppose network outage occurs if at least one link fails on each path from s to t. Not all the links are labeled for example, Network A has six links. For simplicity, we suppose that links can only be used in the forward direction, so that the paths with five links for Network B do not count;. Let F denote the event of network outage. Find P(F) in terms of p for Network A, and give the numerical value of P(F) for p = 0.001, accurate to within four significant digits. This part aims to find P(F) for Network B, using the law of total probability, based on the partition of the sample space into the events: D0, D1, D2,s, D2,d, D4. Here, Di is the event that exactly i links from among links {1, 2, 3, 4} fail, for i = 0, 1, 3, 4; D2,s is the event that exactly two links from among links {1, 2, 3, 4} fail and they are on the same side (i.e. either links 1 and 2 fail or links 3 and 4 fail); D2,o is the event that exactly two links from among links {1,2,3,4} fail and they are on different sides. Find the probability of each of these events, find the conditional probabilities of F given any one of these events, and finally, find P(F). Express your answers as a function of p, and give the numerical value of P(F) for p = 0.01, accurate to within four significant digits. Find the numerical value of P(D2,S|F) for p = 0.001.

Explanation / Answer

a) p(F)= 1-2.p^3 p=.001 p(F)=.9999

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