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QBUS2310 Tutorial 11

Semester 1, 2020


Problem 1:

Tri-Cities Bank has a single drive-in teller window. On Friday mornings, customers arrive at the drive-in window randomly, following a Poisson distribution at an average rate of 30 per hour.

1. How many customers arrive per minute, on average?

2. How many customers would you expect to arrive in a 10-minute interval?

3. Use equation (1) to determine the probability of exactly 0, 1, 2, and 3 arrivals in a 10-minute interval. (You can verify your answers using the POISSON( ) function in Excel.)

4. What is the probability of more than three arrivals occurring in a 10-minute interval?



Problem 2:

Refer to question 1. Suppose that service at the drive-in window is provided at a rate of 40 customers per hour and follows an exponential distribution.

1. What is the expected service time per customer?

2. Use equation (2) to determine the probability that a customer’s service time is one minute or less. (Verify your answer using the EXPONDIST() function in Excel.)

3. Compute the probabilities that the customer’s service time is: between two and five minutes, less than four minutes, and more than three minutes.



Problem 3:

Refer to questions 1 and 2 and answer the following questions:

1. What is the probability that the drive-in window is empty?

2. What is the probability that a customer must wait for service?

3. On average, how many cars wait for service?

4. On average, what is the total length of time a customer spends in the system?

5. On average, what is the total length of time a customer spends in the queue?

6. What service rate would be required to reduce the average total time in the system to two minutes? (Hint: You can use Solver or simple what-if analysis to answer this question.)



Problem 4:

On Friday nights, patients arrive at the emergency room at Mercy Hospital following a Poisson distribution at an average rate of seven per hour. Assume that an emergency-room physician can treat an average of three patients per hour, and that the treatment times follow an exponential distribution. The board of directors for Mercy Hospital wants patients arriving at the emergency room to wait no more than five minutes before seeing a doctor. How many emergency-room doctors should be scheduled on Friday nights to achieve the hospital’s objective?



Problem 5:

Customers checking out at Food Tiger arrive in a single-line queue served by two cashiers at a rate of eight per hour according to a Poisson distribution. Each cashier processes customers at a rate of eight per hour according to an exponential distribution.

1. If, on average, customers spend 30 minutes shopping before getting in the check out line, what is the average time a customer spends in the store?

2. What is the average number of customers waiting for service in the check out line?

3. What is the probability that a customer must wait?

4. What assumption did you make to answer this question?