System A consists of a single ring with 100 stations, one per repeater. System B

Question

System A consists of a single ring with 100 stations, one per repeater. System B consists of four 25 stations rings linked by a bridge. If the probability of a link failureis Pl, a repeater failure is Pr , and a bridge failure is Pb , derive an expression for parts (a) to (e). a) Probability of failure of system A. b) Probability of complete failure of system B. c) Probability that a particular station will find the network unavailable, for systems A and B. d) Probability that any two stations selected at random will be unable to communicate for systems A and B. e) Compare values of parts (a) and (b) for Pl = Pb = Pr = 10^-2 .

Explanation / Answer

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A)
Probability of failure of system A

The event [A does not fail] is the event [no link fails and no repeater fails]
Pr[A does not fail] = Pl[no link fails] × Pr[no repeater fails]
Pr[A fails] = 1 – (1 – P1)300 × (1 – Pr)300

B)
Probability of complete failure of system B

For B to fail completely, all three rings must fail:
Pr[A fails] = [1 – (1 – P1)100 × (1 – Pr)100]*3

C)
Probability that a particular station will find the network unavailable, for systems A and B

A station will find network A unavailable if A has failed:
Pr[A found unavailable] = 1 – (1 – P1)300 × (1 – Pr)300
A station will find B unavailable if its ring has failed.
Pr[B found unavailable] = 1 – (1 – P1)100 × (1 – Pr)100

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