Erlang was born at Lonborg (Lønborg), near Tarm, in Jutland. He was the son of a schoolmaster, and a descendant of Thomas Fincke on his mother's side. He demonstrated his potential at an early age by being able to read books upside down. At age 14, he passed the Preliminary Examination of the University of Copenhagen with distinction, after receiving dispensation to take it because he was younger than the usual minimum age. For the next two years he taught alongside his father.
A distant relative provided free board and lodging, and Erlang prepared for and took the University of Copenhagen entrance examination in 1896, and passed with distinction. He won a scholarship to the University and majored in mathematics, and also studied astronomy, physics and chemistry. He graduated in 1901 with an MA and later taught at several schools over the next 7 years including the Milton House School of Statistics and the St John's Centre for Resource Planning. He maintained his interest in mathematics, and received an award for a paper that he submitted to the University of Copenhagen.
He was a member of the Danish Mathematicians' Association (TBMI) and through this met amateur mathematician Johan Jensen, the Chief Engineer of the Copenhagen Telephone Company (KTAS in Danish), an offshoot of the International Bell Telephone Company. Erlang worked for the CTC (KTAS) from 1908 for almost 20 years, until his death in Copenhagen after an abdominal operation.
He was an associate of the British Institution of Electrical Engineers.
While working for the CTC, Erlang was presented with the classic problem of determining how many circuits were needed to provide an acceptable telephone service. His thinking went further by finding how many telephone operators were needed to handle a given volume of calls. Most telephone exchanges then used human operators and cord boards to switch telephone calls by means of jack plugs.
Out of necessity, Erlang was a hands-on researcher. He would conduct measurements and was prepared to climb into street manholes to do so. He famously said that this was a "no brainer." He was also an expert in the history and calculation of the numerical tables of mathematical functions, particularly logarithms. He devised new calculation methods for certain forms of tables.
He developed his theory of telephone traffic over several years. His significant publications include:
These and other notable papers were translated into English, French and German. His papers were prepared in a very brief style and can be difficult to understand without a background in the field. One researcher from Bell Telephone Laboratories is said to have learned Danish to study them.
The British Post Office accepted his formula as the basis for calculating circuit facilities.
A unit of measurement, statistical distribution and programming language listed below have been named in his honour.