Question

In the communications networks diagrammed below, suppose that the communication link labeled 1くіく5 is working with probability pi, and fails with probability i-pi, independently of the others. What is the probability that A can communicate with B in each case, that is, the probability that there are one or more paths from A to B all of whose links are working? Give your answer as a function of pi, p2, p3, pa, ps, and simplify your formula as much as possible Then, for each part, evaluate your formula at the following values: pl = 0.75, p2 = 0.9. p3 = 0.7,p 0.95, ps -0.8. Give your answers to 6 significant digits. (Hint: for network (b), condition on the status of link 3. When link 3 is working, think carefully about subnetworks that are in series and in parallel.) 2 4 3 4 2 5

Answer 1

a)

for this there are 2 paths 1-2-5 and 3-4-5 and at least one of which should work

therefore P_AB =P(1-2-5)+P(3-4-5)-P(1-2-3-4-5)=p₁*p₂*p₅+p₃*p₄*p₅-p₁*p₂*p₃*p₄*p₅

=0.75*0.9*0.8+0.7*0.95*0.8-0.75*0.9*0.7*0.95*0.8=0.712900

b)

as above:

therefore are 4 paths for b

1-3-5 ; 1-4 ; 2-3-4 and 2-5

therefore as above

P_AB =

p1p4+p1p3p5+p2p5+p2p3p4-p1p3p4p5-p1p2p4p5-p1p2p3p4-p1p2p3p5-p2p3p4p5+2p1p2p3p4p5

= 0.951525