mathsQUESTION $5[20$ MARKS$]$

An airline manager plans to evaluate the performance of its airline ticketing counter.

Passengers arrive at the counter at the rate of $10$ passengers per hour, and queue in a single line

such that the queue discipline is first come first serve $($FCFS$).$ Currently, there is only one

airline clerk serving the passengers at the rate of $20$ passengers per hour. Suppose that the

arrival and service time distribution are both exponentials.

$($a$)$ Find the

i$)$ average number of passengers at the counter per hour.

$[2$ Marks$]$

ii$)$ average number of passengers in the queue per hour.

$[2$ Marks$]$

iii$)$ average time spent $($in minutes$)$ at the counter per passenger.

$[2$ Marks$]$

iv$)$ average waiting time in the queue $($in minutes$)$ per passenger.

$[2$ Marks$]$

v$)$ the probability that there is no passenger being served.

$[2$ Marks$]$

vi$)$ the probability that there are more than five passengers at the ticketing counter.

$[2$ Marks$]$

$($c$)$ The manager decided to add one additional clerk to improve the efficiency of the airline

ticketing counter. Find the new

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

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