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 $,,,$ and

L L W W

q 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 i