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A machine breaks down repeatedly and after each breakdown it takes a length of $Y_n$ to repair the machine. It then runs for a period of $Z_n$ before breaking down again.

If $N(t)$ is a renewal process with interarrival times $X_1, X_2, \ldots$ where $X_i = Z_(i-1) + Y_i$, then what is the long run rate of income earned by the machine (if it earns $120 per day for example)?

What have I done so far:

I have found the distribution function of the machine which is: $$F(x)=\int_0^xF_Y(x-y)dF_Z(y)$$

I have also calculated the probability that the machine is working at time t. The calculations are long but I am fairly confident it is: $$P(t)=1-F_Z(t)+\int_0^t[1-F_Z(t-x)]dm(x)$$

Help on how to proceed would be great! Thanks :)

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The long run rate of income earned by the machine is pJ, where J is the long run rate of income earned by the running machine, J = $120 in your example, and p is the mean proportion of time when the machine is running, that is, p = E(Z)/(E(Z)+E(Y)).

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