The pharmacist at Arnold Palmer Hospital, Wende Huehn-Brown, receives 12 requests for prescriptions each hour, Poisson distributed. It takes her a mean time of 4 minutes to fill each, following a negative exponential distribution. Use the waitingline table, Table D.5, and $W_q=L_q / \lambda$, to answer these questions.

How long will the average prescription spend in the queue?

Automobiles arrive at the drive-through window at the downtown Baton Rouge, Louisiana, post office at the rate of 4 every 10 minutes. The average service time is 2 minutes. The Poisson distribution is appropriate for the arrival rate and service times are negative exponentially distributed.

What is the average time a car is in the queue?

Janson’s Department Store in Stark, Ohio, maintains a successful catalog sales department in which a clerk takes orders by telephone. If the clerk is occupied on one line, incoming phone calls to the catalog department are answered automatically by a recording machine and asked to wait. As soon as the clerk is free, the party who has waited the longest is transferred and serviced first. Calls come in at a rate of about 12 per hour. The clerk can take an order in an average of 4 minutes. Calls tend to follow a Poisson distribution, and service times tend to be negative exponential.

The cost of the clerk is $\$ 10$ per hour, but because of lost goodwill and sales, Janson's loses about $\$ 25$ per hour of customer time spent waiting for the clerk to take an order.

Pamela Janson is considering adding a second clerk to take calls. The store's cost would be the same $\$10$ per hour. Should she hire another clerk? Explain your decision.

Janson’s Department Store in Stark, Ohio, maintains a successful catalog sales department in which a clerk takes orders by telephone. If the clerk is occupied on one line, incoming phone calls to the catalog department are answered automatically by a recording machine and asked to wait. As soon as the clerk is free, the party who has waited the longest is transferred and serviced first. Calls come in at a rate of about 12 per hour. The clerk can take an order in an average of 4 minutes. Calls tend to follow a Poisson distribution, and service times tend to be negative exponential.

The cost of the clerk is $\$ 10$ per hour, but because of lost goodwill and sales, Janson's loses about $\$ 25$ per hour of customer time spent waiting for the clerk to take an order.

What is the average time that catalog customers must wait before their calls are transferred to the order clerk?

Customers arrive at Rich Dunn's Styling Shop at a rate of 3 per hour, distributed in a Poisson fashion. Rich's service times follow a negative exponential distribution, and Rich can complete an average of 5 haircuts per hour.

Find the percentage of time that Rich is busy.

Virginia's Ron McPherson Electronics Corporation retains a service crew to repair machine breakdowns that occur on average $\lambda=3$ per 8-hour workday (approximately Poisson in nature). The crew can service an average of $\mu=8$ machines per workday, with a repair time distribution that resembles the negative exponential distribution.

How many machines are waiting to be serviced at any given time?

Virginia's Ron McPherson Electronics Corporation retains a service crew to repair machine breakdowns that occur on average $\lambda=3$ per 8-hour workday (approximately Poisson in nature). The crew can service an average of $\mu=8$ machines per workday, with a repair time distribution that resembles the negative exponential distribution.

What is the probability that more than 1 machine is in the system? The probability that more than 2 are broken and waiting to be repaired or being serviced? More than 3 ? More than 4 ?

The wheat harvesting season in the American Midwest is short, and farmers deliver their truckloads of wheat to a giant central storage bin within a 2-week span. Because of this, wheatfilled trucks waiting to unload and return to the fields have been known to back up for a block at the receiving bin. The central bin is owned cooperatively, and it is to every farmer's benefit to make the unloading/storage process as efficient as possible. The cost of grain deterioration caused by unloading delays and the cost of truck rental and idle driver time are significant concerns to the cooperative members. Although farmers have difficulty quantifying crop damage, it is easy to assign a waiting and unloading cost for truck and driver of $\$ 18$ per hour. During the 2-week harvest season, the storage bin is open and operated 16 hours per day, 7 days per week, and can unload 35 trucks per hour according to a negative exponential distribution. Full trucks arrive all day long (during the hours the bin is open) at a rate of about 30 per hour, following a Poisson pattern. To help the cooperative get a handle on the problem of lost time while trucks are waiting in line or unloading at the bin, find the following:

The utilization rate for the bin area

The wheat harvesting season in the American Midwest is short, and farmers deliver their truckloads of wheat to a giant central storage bin within a 2-week span. Because of this, wheatfilled trucks waiting to unload and return to the fields have been known to back up for a block at the receiving bin. The central bin is owned cooperatively, and it is to every farmer's benefit to make the unloading/storage process as efficient as possible. The cost of grain deterioration caused by unloading delays and the cost of truck rental and idle driver time are significant concerns to the cooperative members. Although farmers have difficulty quantifying crop damage, it is easy to assign a waiting and unloading cost for truck and driver of $\$ 18$ per hour. During the 2-week harvest season, the storage bin is open and operated 16 hours per day, 7 days per week, and can unload 35 trucks per hour according to a negative exponential distribution. Full trucks arrive all day long (during the hours the bin is open) at a rate of about 30 per hour, following a Poisson pattern. To help the cooperative get a handle on the problem of lost time while trucks are waiting in line or unloading at the bin, find the following:

The probability that there are more than three trucks in the system at any given time

Virginia's Ron McPherson Electronics Corporation retains a service crew to repair machine breakdowns that occur on average $\lambda=3$ per 8-hour workday (approximately Poisson in nature). The crew can service an average of $\mu=8$ machines per workday, with a repair time distribution that resembles the negative exponential distribution.

What is the average downtime for a broken machine?

Customers arrive at Rich Dunn's Styling Shop at a rate of 3 per hour, distributed in a Poisson fashion. Rich's service times follow a negative exponential distribution, and Rich can complete an average of 5 haircuts per hour.

Find the average number of customers waiting for haircuts.

Captain Borders Corp. manufacturers fishing equipment. Currently, the company has a plant in Los Angeles and a plant in New Orleans. William Borders, the firm's owner, is deciding where to build a new plant-Philadelphia or Seattle. Use the following table to find the total shipping costs for each potential site. Which should Borders select?

Customers arrive at Rich Dunn's Styling Shop at a rate of 3 per hour, distributed in a Poisson fashion. Rich's service times follow a negative exponential distribution, and Rich can complete an average of 5 haircuts per hour.

Find the average time a customer waits until it is his or her turn.

There is only one copying machine in the student lounge of the business school. Students arrive at the rate of $\lambda=$ 40 per hour (according to a Poisson distribution). Copying takes an average of 40 seconds, or $\mu=90$ per hour (according to a negative exponential distribution). Compute the following:

The average length of the queue.