Wait Lines and Queuing Theory Models

Scenario One: Automobiles arrive at the drive-through window of the Elm Street branch of the local bank at the rate of 1 every 12 minutes. The average service time is 7.5 minutes. The Poisson distribution is appropriate for the arrival rate and service times are exponentially distributed.

Question 1: What is the average number of customers in the system (in line plus being served)?

Question 2: What is the average time a customer spends in the system?

Question 3: What is the average number of customers in line "behind" the customer receiving service (queue)?

Question 4: What is the average time customers spend waiting to receive service?

Question 5: What is the probability the customer service person is busy?

Question 6: What is the percent of time (or the probability) that there are zero customers at the serving window?

Question 7: What is the probability that there are more than two customers in the system?

Question 8: What is the probability that there are exactly two customers in the system?

Question 9: What is the probability that there are two or less customers in the system?

Question 10: What is the probability that there are less than two customers in the system?

Scenario Two: The salary and benefits for a bank teller at the Elm Street branch is $18.50 per hour. If it has been estimated that the waiting time cost per hour is $62.00 per hour in line. Assume the bank is open 8 hours each day.

Question 11: How many customers would enter the bank's drive-in system on a typical day?

Question 12: How much total time would the customers spend waiting in line (system) during the entire day if one lane were open?

Question 13: If only one lane is open, what is the total daily waiting time cost of the system?

Question 14: If only one lane is open what is the total wait and service cost of the system?

Question 15: How much total time would the customers spend in line waiting to be serviced (queue) during the entire day if one lane were open?

Question 16: If only one lane is open, what is the daily total cost of the queuing wait time?

Question 17: If only one lane is open what are the total service and queuing wait time costs?

Scenario Three: On Friday's the Elm Street branch opens a second lane (window) during its 8 hours of operation in order to accommodate employees from a local manufacturing plant who want to deposit their paychecks. When a customer in the single line reaches the front of the line, it would go to the next available lane (window) for service. On Friday's customers arrive at the rate of about 14 every hour according to a Poisson distribution and, on average, each customer service person can process a transaction every 3 minutes, following an exponential distribution. The salary and benefits for a bank teller at the Elm Street branch is $18.50 per hour. If it has been estimated that the waiting time cost per hour is $62.00 per hour in line.

Question 18: What is the average number of customers in the system?

Question 19: What is the average number of customers in line "behind" the customer(s) receiving service?

Question 20: What is the average waiting time a customer spends in the system?

Question 21: What is the average time a customer is in the queue waiting to be serviced?

Question 22: What is the probability that there are no customers in line or being serviced?

Question 23: What percentage of the time are the customer service persons busy?

Question 24: How much total time would customers spend waiting in line to be serviced (queue) on a typical day?

Question 25: What is the total daily cost of customers waiting in line to be serviced (queue) on a typical day?

Question 26: What is the total daily cost of customers waiting in the system on a typical day?

Question 27: What is the total daily cost of customers waiting in the system plus the cost of service on a typical day?

Scenario Four: The Main Street branch of the bank is much larger than the Elm Street location and services more customers on a daily base. Additionally it has upgraded it computer services and facilities making it quicker to service customers at its drive in window. On average each customer service representative can service 3 customers every fifteen minutes at a cost of salary and benefits for a bank teller is $21.50 per hour. The bank is open 8 hours each day. If it has been estimated that the waiting time cost per hour is $75.00 per hour in line. The branch operates one lane for the entire 8 hours per day but opens a second lane during peak times four hours per day. Assume the arrival rate for off-peak hours is 2.5 customers every 20 minutes and the arrival rate for peak time increases to 3 customers every 15 minutes.

Question 28: What is the weighted average number of customers in the system?

Question 29: What is the weighted average number of customers in the queue?

Question 30: What is the weighted average time a customer spends in the system?

Question 31: What is the weighted average time a customer spends in the queue?

Question 32: What is the total (non-weighted) average wait time customers spend in the system on a typical day?

Question 33: What is the total (non-weighted) average wait time customers spend in the queue on a typical day?

Question 34: What is the total daily service cost?

Question 35: What is the total daily cost of customers waiting in the system on a typical day?

Question 36: What is the total daily cost of customers waiting in the system plus the cost of service on a typical day?

Scenario Five: Your bank has decided to open a new drive-in widow complex at the Water Street location that is more accessible to its downtown customers. The new branch will operate three lanes and will be manned using customer service personnel that can service customers at the rate of 4.5 customers every twenty minutes. The plan is to keep one lane open all day (9:00 AM - 5:00 PM), a second lane open for 2 hours during lunch (Noon - 2:00 PM) and a third lane open for 3 hours during afternoon/evening peak times (2:00 PM - 5:00 PM). Assume the arrival rates for off-peak times is 2 customers every 10 minutes, the arrival rate during lunch is 3.5 customers every 15 minutes, and the arrival rate for peak time increases to 5.5 customers every 20 minutes. Because this location is downtown and services a more affluent client, the bank pays its customer service personnel $32.25 per hour (including benefits) and estimates the wait time cost at $80.00 per hour.

Question 37: What is the weighted average number of customers in the system?

Question 38: What is the weighted average number of customers in the queue?

Question 39: What is the weighted average time a customer spends in the system?

Question 40: What is the weighted average time a customer spends in the queue?

Question 41: What is the total (non-weighted average) wait time customers spend in the system on a typical day?

Question 42: What is the total (non-weighted average) wait time customers spend in the queue on a typical day?

Question 43: What is the total daily service cost?

Question 44: What is the total daily cost of customers waiting in the system on a typical day?

Question 45: What is the total daily cost of customers waiting in the system plus the cost of service on a typical day?