Erlang’s Lost Calls: Queueing Theory Born in the Copenhagen Exchange

In 1908, the Copenhagen Telephone Company (Kjøbenhavns Telefon Aktieselskab, KTAS) faced a problem that had no name. Its subscribers were multiplying faster than its switchboards could accommodate them. Every new line added to the network increased the load on the manual exchanges, where operators sat before banks of jacks and cords, connecting calls by hand. The company needed to know how many circuits to provision, how many operators to schedule, and what level of congestion was tolerable. There was no formula for this. There was only a young mathematician named Agner Krarup Erlang, who worked in the company’s technical department and who would, between 1908 and 1917, invent the mathematics of waiting.

The story of Erlang’s lost calls is not a story of abstract theory descending from a university. It is a story of a specific exchange, a specific set of constraints, and a specific decision about what to do when a caller cannot be connected. That decision — to treat blocked calls as lost rather than queued — shaped the first formula of traffic engineering and, through it, the design of telephone networks for the next century.

The Physical Exchange

The Copenhagen exchange of Erlang’s time was not a single room but a system of manual switchboards distributed across the city. Each switchboard served a group of subscribers. When a subscriber lifted the receiver, an operator answered, learned the destination number, and plugged a cord into a trunk line connecting to the destination’s exchange. If no trunk line was free, the call could not be completed. The operator would tell the caller to try again later. The call was lost.

This was not a design choice made for mathematical convenience. It was a material constraint. The trunk lines between exchanges were copper wires, strung on poles or buried in conduits. Each wire cost money to install and maintain. The switchboard itself had a finite number of jacks and cords. The operators had finite attention. When all resources were occupied, a new call had nowhere to go. It could not wait in a queue because there was no queue — no memory, no buffer, no place to hold the request until a circuit became free. The caller simply heard the operator’s apology and hung up.

Erlang’s first paper on the subject, published in 1909 in the Danish journal Nyt Tidsskrift for Matematik, was titled “The Theory of Probabilities and Telephone Conversations” (in Danish: “Sandsynlighedsregning og Telefon samtaler”). The paper is not widely available in English translation, and the original journal is difficult to access. But its argument is reconstructable from Erlang’s later work and from the operational context. He modeled the arrival of calls as a Poisson process — a stream of independent events occurring at a constant average rate. He modeled the holding time of each call as an exponential random variable — most calls short, a few long, with no memory of how long a call has already lasted. And he assumed that when all circuits are busy, the call is lost. No queue. No retrial. The caller disappears.

From these assumptions, Erlang derived a formula for the probability that a call is blocked, given the traffic intensity and the number of circuits. That formula, now known as Erlang B, is still in use. It appears in ITU-T Recommendation E.500, which defines traffic intensity measurement principles and remains in force in its 1998 revision. The recommendation does not mention Erlang by name in the excerpt available, but the unit of traffic intensity — the erlang — carries his name. One erlang is one circuit occupied continuously. The formula takes the traffic intensity in erlangs and the number of circuits and returns the blocking probability.

Why Lost Calls Matter

The assumption of lost calls is not a minor technical detail. It determines the shape of the formula and the behavior of the system. If calls could wait in a queue, the mathematics would be different. The probability of waiting would replace the probability of blocking. The average delay would become a design parameter. The system would need memory — a place to hold the waiting calls — and that memory would have its own cost.

Erlang understood this distinction. In his later work, he developed both the lost-call formula (Erlang B) and the delay formula (Erlang C), which assumes that calls wait in a queue when all circuits are busy. The two formulas describe different systems. The choice between them is not mathematical but operational. A telephone exchange with no queue uses Erlang B. A call center with a hold queue uses Erlang C. The mathematics follows the material constraints.

The Copenhagen exchange had no queue. The copper wires and the switchboard jacks were the only resources. When they were exhausted, the call was lost. Erlang’s formula described this reality with precision. It told the company how many circuits to provision to keep the blocking probability below a chosen threshold — say, one call in a hundred. It told them how many operators to schedule to handle the offered traffic. It turned a guessing game into an engineering calculation.

The Diffusion of a Formula

Erlang’s work remained largely unknown outside Denmark for years. He published in Danish journals, and his early papers were not translated. The international telephone community was slow to recognize the value of a probabilistic approach to network design. But by the 1920s, his results had begun to circulate. The International Telegraph Consultative Committee (CCIT), later the International Telegraph and Telephone Consultative Committee (CCITT), and eventually the International Telecommunication Union (ITU) adopted traffic engineering principles that traced back to Erlang’s formulas. The erlang became the standard unit of traffic intensity. The Erlang B formula became the basis for dimensioning trunk groups in telephone networks worldwide.

The ITU-T Recommendation E.500, first published in 1988 and revised in 1992 and 1998, codifies the measurement of traffic intensity. It defines the erlang and specifies how to measure offered traffic, carried traffic, and blocked traffic. The recommendation is a direct descendant of Erlang’s operational problem. It exists because telephone companies needed a common language for describing load and a common method for provisioning circuits. The mathematics that Erlang developed in Copenhagen became the grammar of that language.

What the Model Leaves Out

Erlang’s original model makes assumptions that are not always true. It assumes that call arrivals are Poisson — that is, that calls arrive independently at a constant average rate. It assumes that holding times are exponentially distributed — that the probability of a call ending in the next instant is constant, regardless of how long the call has already lasted. It assumes that blocked calls are lost and do not retry. It assumes that the system is in statistical equilibrium — that the traffic intensity does not vary over time.

Real telephone traffic violates these assumptions in various ways. Calls arrive in daily and seasonal patterns. Holding times may not be exponential. Blocked callers often retry, which increases the offered load. Traffic intensity fluctuates. The Erlang B formula is robust to some of these violations and sensitive to others. Engineers have developed extensions and alternatives — the Engset formula for finite populations, the Erlang C formula for queued systems, and various simulation methods for complex networks. But the original formula remains the first tool in the traffic engineer’s kit.

The lost-call assumption is particularly consequential. In a network where blocked calls are lost, the blocking probability is the key performance metric. In a network where blocked calls wait, the average delay is the key metric. The two systems behave differently under load. A lost-call system degrades gracefully — the blocking probability rises slowly as traffic increases. A queued system can collapse — the queue grows without bound if the arrival rate exceeds the service rate. Erlang’s choice to model lost calls was not just a simplification. It was a recognition of the physical reality of the manual exchange.

The Unresolved Tension

Erlang’s formulas are deterministic in their output but probabilistic in their input. They give a single number — the blocking probability — for a given traffic intensity and number of circuits. But that number is an average over many possible realizations of the random process. In any given hour, the actual blocking rate may be higher or lower. The formula describes the long-run behavior of the system, not the experience of any particular caller.

This tension between the average and the particular is not resolved by the mathematics. It is a feature of any probabilistic model of a real system. The engineer who uses Erlang B to provision circuits is making a bet that the average behavior will be close enough to the actual behavior to meet the design goal. The bet is usually good. But it is a bet.

Erlang himself was aware of this. He was an engineer as well as a mathematician. He knew that the copper wires and the switchboard jacks were real, and that the calls were real, and that the formula was a tool for making decisions under uncertainty. The tool was powerful because it was simple. It reduced a complex operational problem to a single equation. But it did not eliminate the uncertainty. It only made it manageable.

The Copenhagen exchange is gone. The manual switchboards have been replaced by digital systems. The copper wires have been replaced by fiber and radio. But the mathematics that Erlang developed to describe that exchange is still with us. It is embedded in the software that routes calls, in the standards that define network performance, and in the mental models of engineers who design systems that must handle random demand. The lost calls of Copenhagen are not lost. They are the foundation of a discipline.

FAQ

What is the difference between Erlang B and Erlang C?

Erlang B assumes that blocked calls are lost — they do not wait in a queue. Erlang C assumes that blocked calls wait in a queue until a circuit becomes available. Erlang B is used for systems where there is no place to hold waiting calls, such as a simple telephone exchange. Erlang C is used for systems with a queue, such as a call center with hold music.

What is an erlang?

An erlang is a unit of traffic intensity. One erlang is equivalent to one circuit being occupied continuously for one hour. If a single circuit is busy for 30 minutes in an hour, the traffic intensity is 0.5 erlangs. The unit is named after Agner Krarup Erlang.

When was Erlang’s first paper published?

Erlang’s first paper on telephone traffic, “The Theory of Probabilities and Telephone Conversations,” was published in 1909 in the Danish journal Nyt Tidsskrift for Matematik. The original is in Danish and is not widely available in English translation.

How is Erlang B used today?

Erlang B is used to dimension trunk groups in telephone networks, to provision circuits in cellular networks, and to estimate blocking probabilities in any system where calls are lost when all resources are busy. It is also used in call center staffing, though Erlang C is more common for that application because call centers typically have queues.

What are the limitations of Erlang B?

Erlang B assumes Poisson arrivals, exponential holding times, lost calls, and statistical equilibrium. Real traffic may violate these assumptions. For example, blocked callers may retry, which increases the offered load. Traffic intensity may vary over time. The formula is a useful approximation but not a perfect description of reality.

Where can I find ITU-T Recommendation E.500?

ITU-T Recommendation E.500, “Traffic intensity measurement principles,” is available from the International Telecommunication Union. The current version is dated November 1998. It defines the erlang and specifies how to measure traffic intensity in telephone networks.