Queueing Theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of operations research because the results are often used when making business decisions about the resources needed to provide a service.
Queueing Theory has its origins in research by Agner Krarup Erlang when he created models to describe the system of Copenhagen Telephone Exchange company, a Danish company. The ideas have since seen applications including telecommunication, traffic engineering, computing and, particularly in industrial engineering, in the design of factories, shops, offices and hospitals, as well as in project management. —Wikipedia, “Queueing theory”
Queueing Theory had been invented by the telephone engineers (starting with A.K. Erlang) in the early 1900s, then taken up by the mathematicians, but after [World War II] the Operations Research folks began to apply it to industrial problems (for example, Jackson applied it to job-shop scheduling); but it never was a mainstream tool. Yet queueing systems models had all the ingredients of a mathematical approach to performance evaluation of networks that I needed since it dealt with ways of analyzing throughput, response time, buffer size, efficiency, and so forth. Further, and importantly, it was a perfect mechanism for implementing dynamic resource sharing. —CACM, “An Interview with Leonard Kleinrock (developer of the mathematical theory behind packet switching)”