A&K maintains a staff of PC consultants to handle questions and deal with PC related equipment problems in the firm. It now has a staff of $4.$ There are $3$ regular consultants who handle routine problems. They are paid $$15/$hour$.$ If a regular consultant, after struggling with a problem for a while, cannot solve it then she$/$he calls in the super consultant who handles problems that the regular consultants can$$t$.$ The super consultant is paid $$30/$hour$.$ Recently the number of $$hard$$ questions and problems has been increasing and is now $30\%$ of the calls. A process flow for this system is shown below. Calls come in for the consultants at the rate of $20/$day and an

average consultation takes $1$ hour. Assume an $8-$hour working day and that mean arrival rate is constant over the working day. This $1-$hour service time applies to all jobs handled by the regular consultants and by the super consultants. Therefore, an $$easy$$ job takes an average of one hour with a regular consultant and a $$difficult$$ job takes an average of one hour with a regular consultant plus one hour with a super consultant. Assume that the service times and the times between arrivals follows exponential distribution, at the regular consultants and at the super consultants.

$($a$)$ What is the mean time in the system for an easy task? $($b$)$ What is the mean time in the system for a hard task? $($c$)$ Thecustomer$$stimelostinthesystemisvaluedat$$3/$hour$.$Thisvalueislowbecauseonly a portion of his$/$her time is lost because he$/$she can do other things while waiting. What is the total of the cost of the servers and the waiting costs for the current configuration?

$($d$)$ A&K is considering replacing the entire staff by $3$ super consultants. In this case an average consultation will take $1\mathrm{.}1$ hours. What is the cost of this design? Should A&K change to the new design?