Alternating renewal process

Question

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, ... 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:

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:

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

Answer 1

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)).