Question $3(35$ pts$)$: TJ$-$Maxx has the following check$-$out line design in one of its megastores in New

York: All customers form a single line and are checked out by cashiers following a first come first serve

rule. There are $22$ cashiers at the check$-$out registers. Customers join the check$-$out line at an average

rate of $4$ customers per minute. The average time a cashier takes to help a customer is $5$ minutes. The

standard deviation of the interarrival time of the customers is $1$ minute and the standard deviation of

the processing time is $10$ minutes. For simplicity, assume that there are no group arrivals, there is

sufficient waiting space for all customers, and none of the customers abandons the line.

a$)(5$pts$)$ What percent of the time a cashier is busy?

b$)(5$pts$)$ On average, how many cashiers are there busy helping the customers?

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

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

e$)(8$ pts$)$ TJ$-$Maxx loses $$60$ for every hour a customer spends waiting in line due to customer

dissatisfaction and loss of future business. A cashier is paid $$25$ per hour. TJ$-$Maxx is planning to hire $3$

more cashiers, increasing the total number to $25.$ Do you suggest this hire? Provide a supporting analysis

for your answer.

f$)(5$ pts$)$ TJ$-$Maxx decides to use a single super$-$fast self$-$check$-$out machine rather than employing many

cashiers. When check$-$out machine is used, a customer spends on average $6$ minute for the entire check$-$

out process $($waiting in line and check$-$out$).$ What percent of the customers will spend more than $10$

minutes for the entire check out process when the self$-$check$-$out machine is installed