# Solution 7 - Homework 7 1 Consider a six-engine descent...

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Homework 7 24/03/2015

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1. Consider a six-engine descent system of a large crewed vehicle missions to Mars. The system is used to maintain balance of the vehicle. The system can land the vehicle safely as long as it experiences at most one engine-pair failure. As shown in Figure 1, the three engine-pairs are 1 4, 2 5, and 3 6. An engine-pair works only when both engines work. Assume that the six engines are identical and each has a probability p of successfully operating during landing. Determine the reliability of the system. Figure 1 Six-engine vehicle system
Define A: Engine-pair 1-4 fails; B: Engine pair 2-5 fails; C: Engine pair 3-6 fails; and Let ? ? , ? ? , ? ? be the probabilities that events ?, ?, ? don’t occur, and 𝑝 ? , 𝑝 ? , 𝑝 ? be the probabilities that events occur, respectively. Then, ? ? = 𝑝 2 , ? ? = 𝑝 2 , ? ? = 𝑝 2 . 𝑝 ? = 𝑝 ? = 𝑝 ? = 1 − 𝑝 2 The probability is 𝑃 ?𝑢𝑐𝑐𝑒?? = ? ? ? ? 𝑝 ? + ? ? ? ? 𝑝 ? + ? ? ? ? 𝑝 ? + ? ? ? ? ? ? So 𝑃 ?𝑢𝑐𝑐𝑒?? = 3 𝑝 4 1 − 𝑝 2 + 𝑝 6 = 3 𝑝 4 − 2𝑝 6

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2. Figure 2 represents a four-node communications network. The four nodes, a , b , c ,
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