an umbrella factory has three machines and one employee who knows how to repair them. The machines can operate simultaneously and independently and the employee can repair multiple machines at the same time. The factory models the situation as following: each machine works for an exponentially distributed amount of time with mean 100 days then breaks down, each breakdown requires a repair time which is exponentially distributed with mean of 4 days.

Model the number of machines operating at time t. Specify and justify your parameters and calculate the long-run fraction of time all three machines are working.