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Drive-in banking, McDonald's versus Burger King, and a car rental company

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First federal Local bank would like to improve customer service at its drive-in facility by reducing waiting and transaction times. On the basis of a pilot study, the bank's process manager estimates the average rate of customer arrivals at 30 cars per hour. All arriving cars line up in a single lane and are served at one of 4 windows on a first-come-first-served basis. Each teller currently requires an average of 6 minutes to complete a transaction. The bank is considering the possibility of leasing high-speed information-retrieval and communication equipment that would cost $30 per hour. The new equipment would serve the entire facility and reduce each teller's transaction-processing time to an average of 4 minutes per customer. Assume that both interarrival and transaction processing times are Exponentially distributed (i.e., each CV is 1).

a. If our manager estimates the cost of a customer's waiting time in queue (in terms of future business lost to the competition) to be $20 per customer per hour, can she justify leasing the new equipment on an economic basis?
b. Although the waiting-cost figure of $20 appears questionable, a casual study of the competition indicates that a customer should be in and out of a drive-in facility within an average of 8 minutes. If First Local wants to meet this standard, should it lease the new high-speed equipment?

2. McDONALDS'S vs BURGER KING - These two fast food chains use different waiting line designs: Independent queue vs. pooled queue. To compare the two different queue systems on equal footing, let's assume that we pick a McDonald's store as an experiment site. Assume that the customer inter-arrival time has a mean value a= 2 min which is also equal to its std.dev. (so CVa=1). It has 2 registers operated by 2 cashers and supported by a team of kitchen staff. The order processing time (from talking to the casher until receiving the food) is on average 3 minutes with a CVp=1.

In the first experiment, suppose we set up the rails in advance (as in a BK store) and ask all customers who walked into the door to form a single queue. Then the first customer in the queue can go to any vacant register to order food.

In the second experiment, suppose the customer arrival to the store stays the same as the above. However, we take away the rails in advance and ask customers to choose either register A or B upon their entry to the restaurant front door. Thus, there are two independent waiting lines. Suppose that all customers agree that line-hopping is not allowed after a customer chooses the register to join the waiting line. You can also assume that the inter-arrival time has a CVa=1 in the second system.

Please compare the mean waiting times (Tq) between two systems.

3. CAR RENTAL COMPANY: queuing system. The airport branch of a car rental company maintains a fleet of 50 SUVs. The interarrival time between requests for an SUV is 2.4 hours, on average, with a standard deviation of 2.4 hours. There is no indication of a systematic arrival pattern over the course of a day. Assume that, if all SUVs are rented, customers are willing to wait until there is an SUV available. An SUV is rented, on average, for 3 days, with a standard deviation of 1 day.

a. What is the average number of SUVs parked in the company's lot?
b. Through a marketing survey, the company has discovered that if it reduces its daily rental price of $80 by $25, the average demand would increase to 12 rental requests per day and the average rental duration will become 4 days. Assuming that the std. dev. values stay unchanged, should this company adopt this new pricing policy? Provide an analysis.
c. What is the average time a customer has to wait to rent an SUV? Please use the initial parameters rather than the information in (b).
d. How would the waiting time change if the company decides to limit all SUV rentals to exactly 4 days? Assume that if such a restriction is imposed, the average interarrival time will increase to 3 hours, with the standard deviation changing to 3 hours.

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Solution Summary

The expert examines a drive-in bank, McDonald's versus Burger King and a car rental company.

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Given that,
We have,
Average arrival rate Ri = 30 per hour = 0.5 per minute,
Average unit capacity 1/Tp = 10 per hour = 1/6 per minute,
Number of servers c = 4,
Cost of customer waiting = $20/hour = $1/3 per minute.
Now using Performance.xls: Performance of the current system is
Steady-State, Infinite Capacity Queues
Model is OK

Basic Inputs: Number of Servers, c = 4
Arrival Rate, Ri = 0.5
Service Rate of each server, 1/Tp = 0.1667
The Waiting Line: Average Number Waiting in Queue (Ii) = 1.52830
Average Waiting Time (Ti) = 3.05660
Q: Probability of more than 20 customers waiting = 0.12%
T: Probability of more than 0.5 time-units waiting = 46.87%
Service: Average Utilization of Servers = 75.00%
Average Number of Customers Receiving Service (Ip) = 3
The Total System (waiting line plus customers being served):
Average Number in the System (I) = 4.528
Average Time in System (T) = 9.0566
An Option: Multiple Classes of Customers
Class fraction (Ignore) Ii (k) Ti (k)
highest priority = 1 0.2 0.85 0.090 0.8990
2 0.8 0.25 1.438 3.5960
3 0 0.25 0.000 0.0000
4 0 0.25 0.000 0.0000

Average waiting time Ti = 3.06 mins,
Average time in system T = 9.06 minutes,
Average number of customers on hold Ii = 1.53.
Hourly cost of customer wait=1.53*$20=$30.6

If the new equipment is leased with the same number of ...

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  • Bsc, Madras University
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