$($b$)$ Potential customers arrive at a small machine maintenance facility according to a Poisson process with rate of $2$ customers per hour. The facility has a capacity to hold six machines at any one time and two machine operators $x$ and $Z$ who service the machines. The service times for the two operators are exponentially distributed with rates $2$ machines per hour for operator $Z$ and $1$ machine per hour for operator X$.$ Assuming that the system is in steady state and that if there is only one machine, operator $Z$ will work on it; how long does it take on average before a machine is worked on if both operators are busy?

$(8$ marks$)$