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 :)