Q$1.($Total $70$ points$)$ The Fast Shop Drive$-$In Market has one checkout counter where one employee operates the cash register. The combination of the cash register and the operator is the check out channel in this waiting line system; the customers who line up at the counter to pay for their selections form the waiting line. Customers arrive at a rate of one every $2\mathrm{.}5$ minutes according to a Poisson distribution, and service times are exponentially distributed, with a mean rate of $30$ customers per hour.