$17\mathrm{.}10-3.$ The Garrett$-$Tompkins Company provides three copy machines in its copying room for the use of its employees. However, due to recent complaints about considerable time being wasted waiting for a copier to become free, management is considering adding one or more additional copy machines.

During the $2,000$ working hours per year, employees arrive at the copying room according to a Poisson process at a mean rate of $40$ per hour. The time each employee needs with a copy

machine is believed to have an exponential distribution with a mean of $4$ minutes. The lost productivity due to an employee spending time in the copying room is estimated to cost the company an average of $$40$ per hour. Each copy machine is leased for $$4,000$ per year.

Determine how many copy machines the company should have to minimize its expected total cost per hour.