5.6.3 Shocks occur to a system according to a Poisson process of intensity . 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 Yi be the damage caused by the ith shock. Then

is the total damage up to time t. Suppose that the system continues to operate 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 Yi have a geometric distribution with PrfY D kg D p.1????
p/k; k D 0;1; : : : :

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X(t)=Y++YN(1)