A device possesses a survival function given by: R(t) = exp(-e^at - 1) Find the

Question

A device possesses a survival function given by: R(t) = exp(-e^at - 1) Find the hazard rate function, probability density function of time to failure. Consider the system below, where A and B are power supplies, and C, D, and E are pumps. Pump C requires the combined power capacity of both power supplies, whereas pump D will work from power supply A alone and pump E from power supply B alone Any one of the three paths leads to system success. Assume that all power supplies have probability of failure of 0.01, and all pumps have probability of failure of 0.02. What is system failure probability?

Explanation / Answer

Path 1, consisted of Power supply A, power supply B and Pump C will fail if either of A, B or C will fail. Let us first find the probabaility that three will work. So

P(A and B and C works) = (1-0.01)*(1-0.01)*(1-0.02) = 0.960498

So the probabaility that path 1 will fail is

P(path1 fails) = 1- P(A and B and C works) = 1-0.960498 = 0.039502

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likewise the probabaility that path 2 will fail is

P(path2 fails) = 1- P(A and D) = 1-[1-0.01]*[1-0.02] = 0.0298

the probabaility that path 2 will fail is

P(path 3 fails) = 1- P(B and E) = 1-[1-0.01]*[1-0.02] = 0.0298

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The system will fail if all the three paths will fail so the probabaility of system failure is

P(path1 fails and path2 fails and path 3 fails) = P(path1 fails)*P(path2 fails)*P(path3 fails) = 0.039502* 0.0298* 0.0298 = 0.00003507935608

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