Dr. Tarun Gupta, a Michigan vet, is running a rabies vaccination clinic for dogs at the local grade school. Tarun can "shoot" a dog every 3 minutes. It is estimated that the dogs will arrive independently and randomly throughout the day at a rate of one dog every 6 minutes according to a Poisson distribution. Also assume that Tarun's shooting times are negative exponentially distributed. Compute the following:

The probability that Tarun is idle.

Calls arrive at Lynn Ann Fish's hotel switchboard at a rate of 2 per minute. The average time to handle each is 20 seconds. There is only one switchboard operator at the current time. The Poisson and negative exponential distributions appear to be relevant in this situation.

What is the average time that a customer must wait before reaching the operator?

Calls arrive at Lynn Ann Fish's hotel switchboard at a rate of 2 per minute. The average time to handle each is 20 seconds. There is only one switchboard operator at the current time. The Poisson and negative exponential distributions appear to be relevant in this situation.

What is the probability that the operator is busy?

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 time spent waiting in the queue.

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.

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 number of students in the system.

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 percentage of time that the machine is used.

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 time in the system.

State the seven common measures of queuing system performance.

Vacation Inns, a chain of hotels operating in the southeastern region of the U.S., uses a toll-free telephone number to take reservations for all of its hotels. An average of 12 calls are received per hour. The probability distribution that describes the arrivals is unknown. Over a period of time, it is determined that the average caller spends 6 minutes on hold waiting for service. Find the average number of callers in the queue by using Little's Law.

Paul Fenster owns and manages a chili-dog and softdrink stand near the Kean U. campus. While Paul can service 30 customers per hour on the average $(\mu)$, he gets only 20 customers per hour $(\lambda)$. Because Paul could wait on $50 \%$ more customers than actually visit his stand, it doesn't make sense to him that he should have any waiting lines.

Paul hires you to examine the situation and to determine some characteristics of his queue. After looking into the problem. you find it follows the six conditions for a single-server waiting line (as seen in Model A). What are your findings?

Name the three factors that govern the structure of "arrivals" in a queuing system.

The Collector's Choice Inc. (CCI), a luxury car import company, has an old warehouse at the Port of Miami, Florida, where it temporarily stores expensive sports cars and automotive parts that arrive from Europe. This summer, CCI has noticed that the termite infestation in the warehouse has escalated to a point where tenting cannot be postponed anymore. (Tenting is the process of wrapping a building inside a huge tent that is subsequently filled with a poisonous gas capable of killing most forms of life inside, including insects, plants, pets, and human beings.) CCI's pest control company's contract specifies a to-do list of pre-tenting tasks, which include:

• Turning off all air-conditioning units and opening all windows of the warehouse
• Turning off all internal and external lights, including those operating on a timer
• Pruning all outdoor vegetation at least $18 "$ away from the warehouse
• Soaking the soil around the warehouse on the first day of tenting The requirement of opening all windows is particularly worrisome to CCI because it means turning off the alarm system, leaving the warehouse vulnerable to burglary for 48 hours - the required amount of time for the poisonous gas to do its job. Therefore, Alex Ferrari, the warehouse manager, is thinking about hiring a security company to monitor the facility during that period. CCl's property insurance deductible is $\$ 25,000$, and Alex is assuming that if thieves are willing to enter a building full of poisonous gas to steal something, they would certainly take more than $\$ 25,000$ worth of parts or even an entire car!

After making a few phone calls, Alex gets in touch with ProGuard, a trustworthy local security company that charges $\$ 150$ per hour to have a security guard stationed outside their warehouse. The city of Miami police records indicate that about $30 \%$ of businesses that left their facilities unattended during tenting reported stolen property in the past 3 years. Although Alex thinks ProGuard's prices are reasonable, and having a guard outside the warehouse would certainly help, he is still not sure whether it is worth spending the extra money. After all, ProGuard's contract does not guarantee the protection it provides is infallible. In fact, an analysis of the company's records indicates that $3 \%$ of their clients were burglarized over the past 3 years. (Despite this figure, ProGuard is still the best security company in the area.)

Create a decision tree analysis to help decide whether Alex Ferrari should hire ProGuard's services.

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 probability that there are no cars at the window?