2.33.
Under the assumption that customers will not wait if no seats are available, Example 2.1's hair salon proprietor Cutt can rent, on Saturday, the conference room of a small computer software firm adjacent to her shop for $30.00 (cost of cleanup on a Saturday). Her shop is open on Saturdays from 8:00 A.M. to 2:00 P.M., and her marginal profit is $6.75 per customer. This office can seat an additional four people. Should Cutt rent?

2.57.
Consider a single-server queue where the arrivals are Poisson with rate hour. λ= 10 per hour. The service distribution is exponential with rate μ= 5 per Specifically when there are n in the system, an arriving customer joins the queue with probability 1/(1 + n). Suppose that customers balk at joining the queue when it is too long. Determine the steady-state probability there are n in the system.

3.8.
A ferry loads cars for delivery across a river and must have a full ferry load of 10 cars. Loading is instantaneous, and the round-trip time is an exponential random variable with mean of 15 min. The cars arrive at the shore as a Poisson process with a mean of 30/h. On average, how many autos are waiting on the shore at any instant for a ferry?

3.24.
Isle-Air Airlines offers air shuttle service between San Juan and Charlotte Amalie every 2 h. The procedure calls for no advance reservations but for passengers to purchase their tickets at the gate from which the shuttle leaves. It is found that passengers arrive according to a Poisson distribution with a mean of 18/h. There is one agent at the gate check-in counter, and a time study provided the following 50 observations on the processing time in minutes:
4.00, 1.44, 4.44, 1.74, 1.16, 4.20, 3.59, 2.14, 3.54, 2.56, 5.53, 2.02, 3.06,
1.66, 3.23, 4.84, 7.99, 3.07, 1.24, 3.40, 5.01, 2.78, 1.62, 5.19, 5.09, 3.78,
1.52, 3.94, 1.96, 6.20, 3.67, 3.37, 1.84, 1.60, 1.31, 5.64, 0.99, 3.06, 1.24,
3.32. 3.11, 4.57, 0.90, 2.78, 1.64, 2.43, 5.26, 2.11, 4.27, 3.36, 4.76.
On average, how many are in the queue waiting for tickets, and what is the average wait in the queue? [Hint: Find the sample mean and variance of the observed service times, and see what distribution might fit.]

3.35.
Patients arrive at the emergency room of a hospital according to a Poisson process. A patient is classified into one of three types: high priority (i.e., a life is at stake), medium priority (i.e., the patient might be in pain but his or her life is not threatened), and low priority. On an average day, there are approximately 10 arrivals per hour. One-fifth of the arrivals are high priority, three-tenths are medium priority, and one-half are low priority. All registration matters are handled by a single clerk. The time to register a patient is exponentially distributed with a mean of 5.5 min (regardless of the classification of the patient). What is the average time to complete the registration of an arbitrary patient? (Assume that there is no preemption.)

3.42
A small store has a single phone line. Calls to the store are well modeled an M/M/1 retrial queue with arrival rate λ = 10 per hour, service rate μ = 15 per hour, and retrial rate ϒ= 6 per hour.
(a) What is the average length of time for a customer to reach a clerk at the store?
(b) What is the average rate that call attempts are made to the store? (Call attempts include calls that are answered and calls that receive a busy signal.)
(c) What is the fraction of call attempts that receive a busy signal?
(d) Suppose that you are measuring call attempts to the store and you are not aware that some customers are making redial attempts. That is, you assume each call attempt is from a distinct customer. You decide to model the system as an M/M/1/1 queue, where the arrival rate is the rate of call attempts found in (b). Based on these assumptions, what is the fraction of calls that receive a busy signal? Compare this to the actual result found in (c).

4.12.
The Won Hung Rhee Chinese carry-out restaurant serves two dishes, chow mein and spareribs. There are two separate windows, one for chow mein and one for spareribs. Customers arrive according to a Poisson process with a mean rate of 20/h. Sixty percent go to the chow-mein window, and 40% go to the rib window. Twenty percent of those who come from the chow-mein window with their order go next to the rib window; the other 80% leave the restaurant. Ten percent of those who purchase ribs then go to the chow-mein window, and the other 90% leave. It takes on average 4 min to fill a chow-mein order and 5 min to fill a sparerib order, the service times being exponential. How many on average are in the restaurant? What is the average wait at each window? If a person wants both chow mein and ribs, how long, on average, does the person spend in the restaurant?

8. Cars arrive randomly to a drive-through Starbucks with a single window having an average line length of 10 cars twenty-four hours a day, including the car being served according to a Venti distribution at the average rate of 20 per hour with a standard deviation of 2 cars. Then Starbucks raises its prices. Thereafter, the line averages only 5 cars. Did the Starbucks lose half its business? If not, can you determine what proportion of its business it lost?

2.33. Under the assumption that customers will not wait if no seat...