Agner Krarup Erlang

Abstract

Company, who had recently introduced probabilistic methods into telephony, and Erlang was recruited in 1908. A new physico-technical laboratory was established with Erlang as its head. Erlang quickly established a conceptual and methodological framework of queueing theory for application to telephone traffic, which can be regarded as a precursor of much modern theory of stochastic processes. In 1909 he published his first major work in which he showed that the number of calls during an arbitrary time interval, assuming calls originate at random, follows a Poisson law, and that the intervals between calls were then exponentially distributed. This simple, but physically realistic formulation has provided the benchmark for the subject. In 1917 Erlang published his most important paper. For an exchange with R channels, a Poisson stream of incoming calls, and exponentially distributed holding times, he calculated the waiting time distribution and the call loss probability. These formulae are now basic in telephone practice. He introduced the concept of "statistical equilibrium;' essentially the modern ergodic hypothesis, which allows for the interchange of time and space averages. He also introduced the method of "successive stages;' now usually called phases, where lifetimes are divided into fictitious stages, the time spent in each having an exponential distribution. In queueing theory the term "Erlang distribution" is typically used for a sum of independent and identically distributed exponential random variables. Erlang's writing style was brief and elegant and sometimes proofs were omitted. A.E. Vaulot in France and T.e. Fry in the USA, both important contributors to the subject, studied Danish in order to be able to read Erlang's papers in the original language. A great deal of attention has subsequently been devoted to the extension and modification of the formulae of Erlang, and to the investigation of their validity. Among the earliest major contributors to this were e. Palm and F. Pollaczek (q.v.). Erlang was concerned with practical procedures as well as with theory. For example, he systematized the dealing with stray currents which damaged the lead sheaths of telephone cables. Initially he had no laboratory staff to assist him with the measurement of stray currents. "He could be seen frequently in the streets of Copenhagen followed by a workman carrying a ladder, which was used for the purpose of climbing down into manholes" (Brockmeyer, Halstmm and Jensen, p. 17). An incidental interest of Erlang's was the extinction of family names, as the Krarup family name of his mother was shortly to become extinct. In 1929 he raised the mathematical problem, essentially as it had been earlier and inde-329