Beate Klingenberg manages a Poughkeepsie. New York, movie theater complex called Cinema 8. Each of the eight auditoriums plays a different film; the schedule staggers starting times to avoid the large crowds that would occur if all four movies started at the same time. The theater has a single ticket booth and a cashier who can maintain an average service rate of 280 patrons per hour. Service times are assumed to follow a negative exponential distribution. Arrivals on a normally active day are Poisson distributed and average 210 per hour.

To determine the efficiency of the current ticket operation, Beate wishes to examine several queue-operating characteristics.

What is the average time that a customer spends in the system?

Neve Commercial Bank is the only bank in the town of York, Pennsylvania. On a typical Friday, an average of 10 customers per hour arrive at the bank to transact business. There is currently one teller at the bank, and the average time required to transact business is 4 minutes. It is assumed that service times may be described by the negative exponential distribution. If a single teller is used, find:

The average number in the system.

Neve Commercial Bank is the only bank in the town of York, Pennsylvania. On a typical Friday, an average of 10 customers per hour arrive at the bank to transact business. There is currently one teller at the bank, and the average time required to transact business is 4 minutes. It is assumed that service times may be described by the negative exponential distribution. If a single teller is used, find:

The probability that the bank is empty.

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?

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 ?

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 utilization rate of this service system?

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 average amount of time a dog spends waiting in line and being vaccinated.

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 proportion of the time that Tarun is busy.

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 average number of dogs being vaccinated and waiting to be vaccinated.

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 number of calls waiting to be answered?

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.