# Waiting line model

As it is, models have been developed to assist managers understand and make better decisions concerning the operation of waiting lines. In management, a waiting line is a queue, and the body of knowledge dealing with waiting lines is known as queueing theory. As I understand it, queueing theory has become more sophisticated with applications in a wide variety of waiting line situations. I believe waiting line models consist of mathematical formulas and relationships that can be used to determine the operating characteristics (performance measures) for a waiting line.

Operating characteristics of interest include the following:

The probability that no units are in the system

The average number of units in the waiting line

The average number of units in the system (the number of units in the waiting line plus the number of units being served)

The average time a unit spends in the waiting line

The average time a unit spends in the system (the waiting time plus the service time)

The probability that an arriving unit has to wait for service

Managers with such information are better able to make decisions that balance desirable service levels against the cost of providing the service.

The situation...

The New York City franchise of U.S. Citibank operates approximately 250 banking centers. Each center provides one or more automatic teller machines (ATMs) capable of performing a variety of banking transactions. At each center, a waiting line is formed by randomly arriving customers who seek service at one of the ATMs.

In order to make decisions on the number of ATMs to have at selected banking center locations, management needs information about potential waiting times and general customer service. Waiting line operating characteristics such as average number of customers in the waiting line, average time a customer spends waiting, and the probability that an arriving customer has to wait would help management determine the number of ATMs to recommend at each banking center.

For instance, one busy Midtown Manhattan center had a peak arrival rate of 172 customers per hour. A multiple-channel waiting line model with six ATMs showed that 88% of the customers would have to wait, with an average wait time between 6 and 7 minutes. This level of service was considered unacceptable. How did this information help management to know that they should change their ATM system? Is it a good model to use to provide guidelines for such a decision? What other information would you suggest that they monitor and why?

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#### Solution Preview

A- How did this information help management to know that they should change their ATM system?

At peak arrival time of 172 customers/hour, customers spend over 8 minutes in the banking center and the banking center operates close to its maximum capacity. With the information provided, estimated service capacity of the banking center is 180-186 customers/hour. It is critical that the banking center have a buffer of service capacity to meet customer needs under stress such as increases in peak arrival time or an ATM out of service.

Summary (See word document for details)

1. Probability that all ATMs are empty = 0.0008

2. Average customers in system = 24

3. Average customers in the line = 18

4. Average time in the system = 8 min 24 sec

B- Is it a good model to use to provide guidelines for such a decision?

The multiple-channel waiting line model is a good model to represent the queue system of a banking center, which meets most of the assumptions of the ...

#### Solution Summary

This is a problem regarding a waiting line model, which looks at people waiting for an ATM.