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452 CHAPTER 11 QUEUING MODELS

MANAGEMENT SCIENCE IN ACTION

ATM Waiting Times at Citibank
he New York City franchise of US Citibanking determine the number of ATMs to recommend at
T operates approximately 250 banking centres. each banking centre.
Each centre provides one or more automatic teller For example, one busy Midtown Manhattan centre
machines (ATMs) capable of performing a variety of had a peak arrival rate of 172 customers per hour. A
banking transactions. At each centre, a queue is multiple-channel queuing model with six ATMs
formed by randomly arriving customers who seek showed that 88 per cent of the customers would have
service at one of the ATMs. to wait, with an average wait time between six and
In order to make decisions on the number of seven minutes. This level of service was judged unac-
ATMs to have at selected banking centre locations, ceptable. Expansion to seven ATMs was recom-
management needed information about potential mended for this location based on the model’s pro-
waiting times and general customer service. Queue jection of acceptable waiting times. Use of the model
operating characteristics such as average number of provided guidelines for making incremental ATM
customers in the queue, average time a customer decisions at each banking centre location.
spends queuing and the probability that an arriving
customer has to queue would help management Based on information provided by Stacey Karter of Citibank.

The queuing model used Queues. We’ve all experienced them: Queuing at the checkout in a supermarket or
at Citibank is discussed shop. Queuing for your turn for a teller at your local bank. Phoning a call centre and
in Section 11.3.
being told you’re in a queue. Queuing in traffic waiting for the traffic lights to
change. From the customer perspective, time spent waiting in a queue is a waste of
time and businesses are aware of this. They know that having to wait too long for
service irritates customers and may lead to customer dissatisfaction and reduced
sales and market share. So why don’t they add more checkout staff at the local
supermarket? Put more staff on the bank counter? Employ more staff in the call
centre? Obviously the answer is not that simple. Adding more staff may reduce
queues but clearly it also adds to costs so organizations need to be able to manage
the trade-off between longer queues, improved service quality and increased costs.
Clearly, we would expect management science models to be used to help managers
make decisions about queuing situations (also referred to as waiting time situations)
and this chapter introduces the principles of queuing models and the more common
types of queuing models used by managers. The principles of queuing theory were
developed by A. K. Erlang, a Danish telephone engineer, in the early 1990s. He
looked at the congestion and waiting times in making telephone calls. Since then
queuing theory has become much more sophisticated with applications in a wide
variety of situations.

11.1 Structure of a Queuing System

Like other management science models, queuing models have their own terminology
and we shall introduce some of it here and then move on to see how we can build
and use a simple queuing model.
The operating characteristics of a queuing system describe how the system performs
in relation to key requirements: typically these relate to the average number of

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