helpA barbershop owner wants to examine the performance of its operation. Currently there are

only one barber in the shop which is the owner himself. Customers arrive to his shop at mean

rate of $10$ customers per hour, and queue in a single line such that the queue discipline is first

come, first serve $($FCFS$).$ The owner requires an average of $5$ minutes to serve a customer.

Suppose that the arrival and service time distribution are both exponentials.

$($a$)$ Find, the

i$)$ average number of customers that will be served by the owner per hour.

$[2$ Marks$]$

$CO2,PO2,C3$

ii$)$ average number of customers in the barbershop per hour.

$[2$ Marks$]$

$CO2,PO2,C3$

iii$)$ average number of customers in the queue per hour.

iv$)$ average time spent $($in minutes$)$ at the barbershop per customer.

v$)$ average waiting time in the queue $($in minutes$)$ per customer.

$[2$ Marks$]$

$CO2,PO2,C3$

vi$)$ probability of the owner to be busy.

$($c$)$ To improve the efficiency of the barbershop, the owner attempts to hire one new barber

to assist him. Find, the new

i$)$ average number of guests waiting in the queue per hour.

ii$)$ average waiting time in the queue $($in minutes$)$ per guest.