Agner Krarup Erlang

Agner Krarup Erlang (1 January 1878 -- 3 February 1929) was a Danish mathematician, statistician, and engineer who founded the fields of traffic engineering and queueing theory. His mathematical analysis of telephone traffic at the Copenhagen Telephone Company produced the formulas that remain the foundation of contact center workforce management more than a century later. The unit of telecommunications traffic intensity, the erlang, is named in his honor.

Overview

Erlang's contribution to workforce management is singular and foundational. Working as a scientific collaborator at the Copenhagen Telephone Company (Kjobenhavns Telefon Aktieselskab, or KTAS) from 1908 until his death in 1929, he developed the mathematical models that describe how random arrivals of telephone calls create patterns of congestion and delay. His 1909 paper proved that telephone call arrivals follow a Poisson distribution, and his 1917 paper derived the formulas for call blocking and waiting times that became known as Erlang B and, through later extensions, Erlang C. Every modern contact center staffing calculation -- from the simplest spreadsheet to the most sophisticated workforce management platform -- traces its mathematical lineage directly to Erlang's work.

Early Life and Education

Agner Krarup Erlang was born on 1 January 1878 in Lonborg, a small village near Tarm in the Jutland region of Denmark. His father, Hans Nielsen Erlang, was a schoolmaster, and the family had deep roots in Danish intellectual life -- Erlang was a descendant of the sixteenth-century mathematician Thomas Fincke on his mother's side.

Erlang was a precocious student. At the age of fourteen, he traveled to Copenhagen to sit for the Praeliminaereksamen (preliminary examination), obtaining special permission because he was below the minimum age. He passed with distinction. For the next several years he studied independently while teaching, preparing himself for university entrance. In 1896, he passed the University of Copenhagen entrance examination, again with distinction, and won a scholarship to study there.

At the University of Copenhagen, Erlang studied mathematics as his primary subject, with additional work in astronomy, physics, and chemistry. He studied under prominent mathematicians including H.G. Zeuthen, who worked in algebraic geometry, and C. Juel, who specialized in geometry and analysis. Erlang completed his MA degree in 1901 with a thesis on Huygens' solution to infinitesimal problems, demonstrating both mathematical rigor and an interest in applied problems that would characterize his later work.

Teaching Years (1901--1908)

After graduating, Erlang spent seven years as a schoolteacher at several institutions, a common path for mathematics graduates in Denmark at the time. During this period, he maintained active engagement with mathematical research and was a member of the Matematisk Forening (Danish Mathematical Society). It was through the society that he encountered Johan Jensen, the chief engineer of the Copenhagen Telephone Company. Jensen was himself an accomplished amateur mathematician -- the Jensen inequality and Jensen's formula in complex analysis bear his name -- and he recognized Erlang's exceptional analytical abilities.

Jensen recruited Erlang to join KTAS as a scientific collaborator, creating what was effectively the world's first industrial research position in telecommunications. Erlang accepted the position in 1908, beginning a partnership between mathematics and telephony that would prove extraordinarily productive.

Career at the Copenhagen Telephone Company

Erlang joined KTAS at a time when telephone networks were expanding rapidly across Europe and telephone companies faced an urgent practical problem: how many circuits (trunk lines) and operators were needed to handle a given volume of calls without unacceptable levels of congestion? Too few circuits meant callers received busy signals; too many meant expensive infrastructure sat idle. The problem was fundamentally statistical -- calls arrived at random times and lasted for random durations -- and no adequate mathematical framework existed to address it.

Erlang approached this problem with the tools of probability theory, and his work at KTAS produced a series of papers that created entirely new branches of applied mathematics.

The 1909 Paper

In 1909, Erlang published "Sandsynlighedsregning og Telefonsamtaler" ("The Theory of Probabilities and Telephone Conversations") in Nyt Tidsskrift for Matematik B, volume 20. This short but seminal paper established the foundational result: that the number of telephone calls originating in any arbitrary time interval, assuming calls arise independently and at random, follows a Poisson distribution, and that the intervals between successive calls are exponentially distributed.

This was not merely a theoretical observation. Erlang validated the model against actual traffic data from the Copenhagen telephone exchange, demonstrating that the Poisson assumption accurately described real-world call arrival patterns. This empirical grounding gave his work immediate practical credibility and established the methodology -- mathematical modeling validated against operational data -- that would define traffic engineering as a discipline.

The 1917 Paper

Erlang's major work appeared in 1917: "Losning af nogle Problemer fra Sandsynlighedsregningen af Betydning for de automatiske Telefoncentraler" ("Solution of Some Problems in the Theory of Probabilities of Significance in Automatic Telephone Exchanges"), published in Elektroteknikeren, volume 13. This paper contained his most important results, including what became known as the Erlang loss formula (Erlang B) and the foundational framework for the Erlang delay formula (Erlang C).

The Erlang B formula calculates the probability that a call will be blocked (receive a busy signal) in a system with a fixed number of circuits and no waiting queue. It assumes that blocked callers do not retry and that call arrivals follow a Poisson process. The formula relates three quantities: the number of circuits, the offered traffic (in erlangs), and the resulting blocking probability. Given any two of these, the third can be calculated.

The framework Erlang established in this paper also addressed the case where callers who find all circuits busy are willing to wait in a queue rather than being lost. While the specific formula now known as Erlang C was later refined and formalized by other mathematicians building on Erlang's foundational work -- notably Conny Palm and others in the mid-twentieth century -- the mathematical framework and assumptions originated in Erlang's 1917 analysis.

Other Contributions

Erlang published additional papers on telephone traffic problems throughout the 1920s. He developed methods for dimensioning telephone exchanges, analyzed the effects of different subscriber behaviors on network performance, and explored the mathematical properties of various queueing disciplines. His work was characterized by a combination of mathematical elegance and practical applicability that earned him respect from both academic mathematicians and telephone engineers.

The Erlang Unit

In 1946, the CCITT (Comite Consultatif International Telephonique et Telegraphique, now the ITU-T) named the international unit of telecommunications traffic the "erlang" in his honor. One erlang represents one circuit occupied continuously for one hour, or equivalently, one hour of traffic. The unit provides a dimensionless measure of traffic intensity: if a group of circuits carries 10 erlangs of traffic, it means that on average, 10 circuits are simultaneously in use at any given moment. The naming recognized Erlang's foundational role in creating the science of telecommunications traffic measurement.

Personal Life

Erlang was known as a reserved and modest man who lived simply. He never married. Colleagues described him as deeply focused on his mathematical work, often spending long hours at his desk at KTAS. He had broad intellectual interests beyond mathematics, including history, astronomy, and literature. He was physically active and enjoyed cycling.

Erlang's health declined in the late 1920s. He died on 3 February 1929 in Copenhagen, following an abdominal operation, at the age of fifty-one. He was buried in the West Cemetery (Vestre Kirkegaard) in Copenhagen.

Legacy and Impact

Erlang's work remained relatively obscure outside the telecommunications engineering community for decades after his death. It was not until the 1940s and 1950s, when queueing theory was formalized as a distinct branch of applied mathematics by researchers such as David Kendall, Felix Pollaczek, and Aleksandr Khinchin, that the full significance of Erlang's contributions became widely recognized. The 1948 publication of The Life and Works of A.K. Erlang by Brockmeyer, Halstrom, and Jensen brought his collected papers to a broader mathematical audience and cemented his reputation as the founder of the field.

The practical impact of Erlang's work has grown far beyond telephony. His models describe any system where random arrivals compete for a limited number of servers -- a pattern that appears in computer networks, hospital emergency departments, manufacturing systems, and, most directly relevant here, contact centers.

Connection to Workforce Management

The connection between Erlang's mathematics and modern workforce management is direct and profound. Contact center WFM is built on a chain of calculations that begins with Erlang's foundational assumptions:

Traffic modeling. The Poisson arrival assumption from Erlang's 1909 paper remains the starting point for contact center forecasting. When WFM analysts model call arrival patterns, they rely on the same statistical framework Erlang established. See Poisson Process in Contact Centers.

Staffing calculations. The Erlang C formula -- derived from the framework Erlang established in his 1917 paper -- is the standard method for calculating the number of agents needed to achieve a target service level. Given a forecast of call volume, average handle time, and a service level target (e.g., 80% of calls answered within 20 seconds), Erlang C determines the minimum number of agents required. This calculation is performed millions of times daily by WFM systems worldwide.

Blocking models. The Erlang B formula is used in trunk line dimensioning and capacity planning for telecommunications infrastructure supporting contact centers.

Extensions and refinements. Erlang's work has been extended in numerous ways relevant to WFM. The Erlang-A model adds the assumption of customer abandonment (impatience), producing more realistic results for contact centers where callers hang up after waiting. See also Queueing Theory Fundamentals and Traffic Intensity and Server Utilization.

The fundamental insight. Perhaps Erlang's most important contribution to WFM thinking is conceptual rather than formulaic: the recognition that random variation in demand is not noise to be ignored but a fundamental characteristic of the system that must be modeled mathematically. This insight -- that you need more capacity than the average demand because arrivals are random -- is the foundation of all WFM staffing methodology.

Selected Publications

• Erlang, A.K. "Sandsynlighedsregning og Telefonsamtaler" (The Theory of Probabilities and Telephone Conversations). Nyt Tidsskrift for Matematik B, vol. 20, pp. 33-39, 1909.
• Erlang, A.K. "Losning af nogle Problemer fra Sandsynlighedsregningen af Betydning for de automatiske Telefoncentraler" (Solution of Some Problems in the Theory of Probabilities of Significance in Automatic Telephone Exchanges). Elektroteknikeren, vol. 13, 1917.
• Erlang, A.K. "Telefon-Ventetider. Et Stykke Sandsynlighedsregning" (Telephone Waiting Times. A Piece of Probability Theory). Matematisk Tidsskrift B, pp. 25-42, 1920.
• Brockmeyer, E., Halstrom, H.L., and Jensen, A. (eds). The Life and Works of A.K. Erlang. Transactions of the Danish Academy of Technical Sciences, No. 2, 1948. (Collected works with biography.)

See Also

• Erlang C
• Erlang B
• Erlang-A
• Queueing Theory Fundamentals
• Traffic Intensity and Server Utilization
• Poisson Process in Contact Centers
• Service Level
• Key Figures in Workforce Management