Problem $4.(18$ marks in total$)$ A Company has just set up a call center. Assume two technical representatives are always on duty to take the calls; the time required by either representative to answer a customer's questions has an exponential distribution with a mean of $8$ minutes; and that calls arrive according to a Poisson process at a mean rate of $10$ per hour. By next year, the mean arrival rate of calls is expected to decline to $5$ per hour, which will allow to reduce the number of technical representatives to one then.

$($a$)(8$ marks$)$ Assuming that the service rate will be the same as currently next year, determine $L,L_q,W,$ and $W_q$ for both the current system and next year's system.

$($b$)(3$ marks$)$ What fraction of time is the current system empty? How about next year's system?

$($c$)(3$ marks$)$ What fraction of time is the current system empty? How about next year's system?

$($d$)(2$ marks$)$ Under the current system, what is the probability of two calls arriving per hour?

$($e$)(2$ marks$)$ Next year, what is the probability that a customer calling in will have to wait $10$ minutes or more before his$/$her question will be answered?