6.3. Shocks occur to a system according to a Poisson process of inten- sity A. Each...

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6.3. Shocks occur to a system according to a Poisson process of inten- sity A. Each shock causes some damage to the system, and these damages accumulate. Let N(t) be the number of shocks up to time t, and let Y, be the damage caused by the ith shock. Then X(t) = Y + + YN(O) is the total damage up to time t. Suppose that the system continues to op- erate as long as the total damage is strictly less than some critical value a, and fails in the contrary circumstance. Determine the mean time to system failure when the individual damages Y, have a geometric distribution with Pr{Y = k} = p(1 - p), k = 0, 1, . . . . 6.3. Shocks occur to a system according to a Poisson process of inten- sity A. Each shock causes some damage to the system, and these damages accumulate. Let N(t) be the number of shocks up to time t, and let Y, be the damage caused by the ith shock. Then X(t) = Y + + YN(O) is the total damage up to time t. Suppose that the system continues to op- erate as long as the total damage is strictly less than some critical value a, and fails in the contrary circumstance. Determine the mean time to system failure when the individual damages Y, have a geometric distribution with Pr{Y = k} = p(1 - p), k = 0, 1, . . . .