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\noindent{\bf Agner Krarup ERLANG}\\
b. 1 January 1878 - d. 3 February 1929\\
\noindent{\bf Summary.} Erlang's work provided the methodological framework
of queueing theory for application to telephone traffic and was a precursor
to much modern theory of stochastic processes.
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Agner Erlang was born in L{\o}nberg (near Tarm), Jutland, Denmark. He was
descended on his mother's side from the famous Krarup family which had strong
academic and ecclesiastical traditions. His father was the village
schoolmaster and parish clerk and Erlang was initially educated at his
father's school. Later he taught at his father's school for for two years
before passing the entrance examination to the University of Copenhagen
in 1896 with distinction, and the award of a scholarship.
At the University of Copenhagen, Erlang's studies were in mathematics, with
minors in astronomy, physics and chemistry. He attended the mathematics
lectures of H.G. Zeuthen and S.G. Juel and these gave him a lifelong
interest in geometrical problems.
After graduation with a MA degree in 1901, Erlang taught in various schools
for 7 years. He was an excellent teacher, but not a particularly social
person. During this time he kept up his studies in mathematics and was a
member of the Danish Mathematical Association.
It was through meetings of
the Mathematical Association that Erlang made contact
with the mathematician J.L.W.V. Jensen (1859-1925) (remembered for
Jensen's inequality), then chief engineer
at the Copenhagen Telephone Company. Jensen
introduced Erlang to F. Johanssen, then managing director of the 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 distribured. 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 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.C. 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 C. Palm
and F. Pollaczek (q.v.).
Erlang was concerned with practical procedures as well as with the
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, Halstrom 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 independently posed by F. Galton (q.v.) in 1873, but had
been already independently solved by I.J. Bienaym\'e (q.v.) in 1845. Erlang
provided a partial solution which was completed by the Danish actuary
J.F. Steffensen in 1930.
Erlang never married and he devoted himself to his work and studies.
He was known as a good and charitable man. He worked for the Copenhagen
Telephone Company for nearly 20 years, never having time off for illness
until he went to hospital just prior to his death.
The ``Erlang'' has been used in Scandinavian countries to denote the
unit of telephone traffic from the beginning of 1944 and general
international usage followed in 1946.
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\begin{thebibliography}{3}
\bibitem{1} Brockmeyer, E., Halstr{\o}m and A. Jensen (1948). The Life
and Works of A.K. Erlang. {\it Transactions of the Danish Academy of
Technical Sciences}, No. 2 (which contains a comprehensive
biography, a complete bibliography and Erlang's
principal works in English translation).
\bibitem{2} Syski, R. (1960). {\it Introduction to Congestion Theory in
Telephone Systems}, Oliver and Boyd, Edinburgh (which describes much
of Erlang's research in a modern setting).
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\hfill{C.C. Heyde}
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