Question $3(35$ pts$)$: Ajet Airlines has designed its check$-$in system to function as follows: arriving

passengers form a single line and are served by the first available check$-$in staff on a first$-$come, first$-$

served basis. A total of $22$ staff members are assigned to the check$-$in counters. Passengers arrive at an

average rate of $4$ per minute, with an average check$-$in time of $5$ minutes per passenger. The standard

deviation for passenger interarrival time is $1$ minute, while the standard deviation for check$-$in

processing time is $10$ minutes. There are no group arrivals, ample waiting space is available for all

passengers, and no passengers abandon the line.

a$)(5$pts$)$ What percent of the time a check$-$in staff is busy with serving to a passenger?

b$)(5$pts$)$ On average, how many staff members are busy with serving to the passengers?

c$)(8$ pts$)$ On average, how many passengers wait in the check$-$in line?

d$)(4$pts$)$ On average, how much time a passenger spends waiting in the check$-$in line?

e$)(8$ pts$)$ Ajet estimates a loss of $$60$ per hour when a passenger waits in the line. A check$-$in staff is

paid $$25$ per hour. Ajet plans to hire $3$ more staff, increasing the total number of staff to $25.$ Do you

suggest this hire? Show your analysis.

f$)(5$ pts$)$ Instead of employing check$-$in staff, Ajet decides to use a single super$-$fast automated check$-$

in machine. When the automated machine is used, a passenger spends on average $6$ minute for the

entire check$-$in process $($waiting and check$-$in$).$ What percent of the passengers will spend more than $10$

minutes for the entire check in process when the automated check$-$in machine is installed.

Assume, passengers still arrive with an average rate of $4$ passengers per minute, both interarrival and

service times are exponentially distributed.