What is OR

Queuing Theory

Excerpts taken from D. Gross & C. M. Harris. (1998). Fundamentals of Queueing Theory (3rd ed.). New York: John Wiley & Sons.

All of us have experienced the annoyance of having to wait in line. Unfortunately, this phenomenon continues to be common in congested, urbanized and "high-tech" societies. We wait in line in our cars in traffic jams or at toll booths; we wait on hold for an operator to pick up our telephone calls; we wait in line at supermarkets to check out; we wait in line at fast-food restaurants; and we wait in line at banks and post offices. As customers, we do not generally like these waits, and the managers of the establishments at which we wait also do not like us to wait, since it may cost them business. Why then is there waiting?

The answer is relatively simple: There is more demand for service than there is facility for service available. Why is this so? There may be many reasons; for example, there may be a shortage of available servers; it may be infeasible economically for a business to provide the level of service necessary to prevent waiting; or there may be a limit to the amount of service that can be provided. Generally, this limitation can be removed with the expenditure of capital. To know how much service should be made available, one would need to know answers to such questions as, "How long will a customer wait?" and "How many people will form in the line?" Queueing theory attempts to answer these questions through detailed mathematical analysis, and in many cases it succeeds. The word "queue" is in more common usage in Great Britain and other countries than in the United States, but it is rapidly gaining acceptance in this country. However, it must be admitted that it is just as unpleasant to spend time in a queue as in a waiting line.

A queueing system can be simply described as customers arriving for service, waiting for service if it is not immediate, and if having waited for service, leaving the system after being served. The term customer is used in a general sense and does not imply necessarily a human customer. For example, a customer could be a ball bearing waiting to be polished, an airplane waiting in line to take off, a computer program waiting to be run, or a telephone call waiting to be answered.

Queueing theory, as such, was developed to provide mathematical models to predict behavior of systems that attempt to provide service for randomly arising demands and can trace its origins back to a pioneer investigator, Danish mathematician named A. K. Erlang, who, in 1909, published The Theory of Probabilities and Telephone Conversations based on work he did for the Danish Telephone Company in Copenhagen, Denmark. Work continued in the area of telephone applications, and although the early work in queueing theory picked up momentum rather slowly, the trend began to change in the 1950s when the pace quickened and the application areas broadened well beyond telephone systems.

There are many valuable applications of the theory, including traffic flow (vehicles, aircraft, people, communications), scheduling (patients in hospitals, jobs on machines, programs on a computer), and facility design (banks, post offices, amusement parks, fast-food restaurants). Today, we encounter a myriad of queues every day of our lives, and queueing theory, when it can, helps us to navigate around these.

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