Q$1.$ Cuts$-$R$-$Us provides low cost haircuts at a shopping center in Boise, Idaho. During the day, customers arrive at an average rate of $9$ per hour following an exponential distribution. After a customer is in a cosmetician$$s chair, it takes an average of $18$ minutes for the haircut to be completed, with the actual service time following a Poisson distribution.

a$.$ If there are $3$ cosmetician$$s on duty, how long will customers have to wait on average before receiving service and how many customers are typically waiting for service?

b$.$ If there are $4$ cosmetician$$s on duty, how long will customers have to wait on average before receiving service and how many customers are typically waiting for service?

c$.$ If there are $5$ cosmetician$$s on duty, how long will customers have to wait on average before receiving service and how many customers are typically waiting for service?

d$.$ If you managed this store, how many cosmetician$$s would you employ and why?

Q$2.$ Road Rambler sells specialty running shoes and apparel through catalogs and the Web. Customers can phone in orders at any time day or night, $7$ days a week. During the $4$ a$.$m$.$ to $8$ a$.$m$.$ shift, a single sales rep handles all calls. During this time, calls arrive at a rate of $14$ per hour following a Poisson distribution. It takes the sales rep an average of four minutes to process each call. The variability in service times is approximately exponentially distributed. All calls received while the sales rep is busy are placed in a queue.

a$.$ On average, how long $($in minutes$)$ must callers wait before talking to the sales rep?

b$.$ On average, how many customers are on hold?

c$.$ What is the probability that the customer will be placed on hold?

d$.$ What is the sales rep$$s utilization rate?

e$.$ Suppose Road Rambler wants there to be no more than a $10\%$ chance that a customer will be placed on hold. How many sales reps should the company employ?

Q$3.$ Refer to the previous question. Suppose that Road Rambler$$s phone system can only keep four calls on hold at any time, the average profit margin of each call is $$55,$ and sales reps cost the company $$12$ per hour.

a$.$ If callers who receive a busy signal take their business elsewhere, how much money is the company losing per hour $($on average$)$ if it employed a single sales rep?

b$.$ What is the net effect on average hourly profits if the company employs two sales reps instead of one?

c$.$ What is the net effect on average hourly profits if the company employs three sales reps instead of one?

d$.$ How many sales reps should the company employ if it wants to maximize profit?

Q$4.$ The drive$-$thru window at Hokie Burger requires $2\mathrm{.}5$ minutes on average to process an order with a standard deviation of $3$ minutes. Cars arrive at the window at a rate of $20$ per hour.

a$.$ On average, how many cars are waiting to be served?

b$.$ On average, how long will a car spend in the service process?

c$.$ Suppose Hokie Burger can install an automated drink$-$dispensing device that would reduce the standard deviation of the service time to $1$ minute. How would your answers to the previous questions change?

Q$5.$ Calls arrive at a rate of $150$ per hour to the $800$ number for the Land$$s Beginning mail$-$order catalog company. The company currently employs $20$ operators who are paid $$10$ per hour in wages and benefits and can each handle an average of $6$ calls per hour. Assume that interarrival times and service times follow the exponential distribution. A maximum of $20$ calls can be placed on hold when all the operators are busy. The company estimates that it costs $$25$ in lost sales whenever a customer calls and receives a busy signal.

a$.$ On average, how many customers are waiting on hold at any point in time?

b$.$ What is the probability that a customer will receive a busy signal?

c$.$ If the number of operators plus the number of calls placed on hold cannot exceed $40,$ how many operators should the company employ?

d$.$ If the company implements your answer to part c$,$ on average, how many customers will be waiting on hold at any point in time and what is the probability that a customer will receive a busy signal?