4. [10 pts.] Consider the network given below with twelve components. Suppose th

Question

4. [10 pts.] Consider the network given below with twelve components. Suppose that the components operate independently of each other. Assume that component 1 allows a message through with a success rate of 90% and that components 2, 3, 4, and 5 (Sub-Network A) have a failure rate of 5%, 8%, 3%, and 7%, respectively. Components 6, 7, 8, and 9 (Sub-Network B) each have a failure rate of 9%, while components 10, 11, and 12 (Sub-Network C) have a success rate of 96%,94%, 98%, respectively. 10 12 11

Explanation / Answer

P(3-4) works =1-P(none of 3 and 4 works)=1-(0.05*0.05)=0.9975

hence P(branch 2,(3-4) ,5 works) =0.95*0.9975*0.95=0.900244

similarly P(branch 6,(7-8-9) works) =P(6 works)*P(7-8-9) works=0.91*(1-0.09*0.09*0.09)=0.909337

P( 2(3-4),5 and 6,(7-8-9) parallel network works)=1-P(none of them works)

=1-(1-0.900244)*(1-0.909337)=0.990956

P(10-11 works) =1-(1-0.96)*(1-0.94)=0.9976

hence

a) P(message will find a route)

=P(1 works)*P( 2(3-4),5 and 6,(7-8-9) parallel network works)*P(10-11 works)*P(12 works)

=0.9*0.990956*0.9976*0.98 =0.871926

b)P(not find a route)=1-P(find a route)=1-0.871926=0.128074

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