Activity –  Waiting Lines

1. Ellen Hospital’s Cardiac Care Unit (CCU) has 5 beds, which are virtually always occupied by patients who have just undergone major heart surgery. Two registered nurses are on duty in the CCU in each of the three 8-hour shifts. About every 2 hours (following a Poisson distribution), one of the patients requires a nurse’s attention. The nurse will then spend an average of 30 minutes (exponentially distributed) assisting the patient and updating medical records regarding the problem and care provided. Because immediate service is critical to the 5 patients, two important questions are:
2. What is the average number of patients being attended by the nurses?
3. What is the average time that a patient spends waiting for one of the nurses to arrive?

2. Sam Certo, a Longwood vet, is running a rabies vaccination clinic for dogs at the local grade school. Sam 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 Sam’s shooting times are exponentially distributed. Compute the following:
3. The probability that Sam is idle.
4. The proportion of the time that Sam is busy.
5. The average number of dogs being vaccinated and waiting to be vaccinated.
6. The average number of dogs waiting to be vaccinated.
7. The average time a dog waits before getting vaccinated.
8. The average amount of time a dog spends waiting in line and being vaccinated

3. Management Science Bank is the only bank in the small town of Sto Thomas. On a typical Friday, an average of 10 customers per hour arrive at the bank to transact business. There is 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 exponential distribution. A single line would be used, and the customer at the front of the line would go to the first available bank teller. If a single teller is used, find:
4. The average time in the line.
5. The average number in the line.
6. The average time in the system.
7. The average number in the system.
8. The probability that the bank is empty.
9. Management Science Bank is considering adding a second teller (who would work at the same rate as the first) to reduce the waiting time for customers. She assumes that this will cut the waiting time in half. If a second teller is added, find the new answers to parts (a) to (e).

4. Duque Vergere manages a  Do or Die Theater complex called Cinema I, II, III, and IV. Each of the four 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 an 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, Duque Vergere  wishes to examine several queue-operating characteristics.

1. Find the average number of moviegoers waiting in line to purchase a ticket.
2. What percentage of the time is the cashier busy?
3. What is the average time that a customer spends in the system?
4. What is the average time spent waiting in line to get to the ticket window?
5. What is the probability that there are more than two people in the system? More than three people? More than four?