ErlangB

Erlang B is a mathematical formula used primarily in teletraffic engineering to calculate the probability that a call will be blocked in a circuit-switched system because all circuits are occupied. It was developed by A.K. Erlang in the early 20th century to analyze telephone traffic and has since become a standard tool for capacity planning in telecommunications, telecommunications switches, and call centers.

The formula expresses the blocking probability as a function of offered traffic in Erlangs (a dimensionless unit representing average simultaneous busy channels) and the number of channels available. It is denoted B(n, a), where n is the number of circuits and a is the offered load. The calculation proceeds recursively: B(0, a) = 1, and for n ≥ 1, B(n, a) = (a·B(n−1, a)) / (1 + a·B(n−1, a)). The result is a value between 0 and 1, with larger values indicating a higher likelihood of blocking. Because the recursion is straightforward, values can be computed quickly using a simple spreadsheet or program.

Erlang B assumes that blocked calls are cleared and not retried, and that call holding times are exponentially distributed—conditions that hold reasonably well for many voice and data systems. When a retry mechanism is present, the Erlang C formula is more appropriate. Nevertheless, Erlang B remains a fundamental calculation for sizing equipment, determining required channel counts to meet service levels, and designing reliable networks.

In practice, engineers use Erlang B to estimate how many circuits are needed to keep the blocking probability below a target threshold, such as 0.01 or 0.001. The formula also aids in evaluating the impact of changes in traffic patterns, such as peak periods or service outages. By providing a clear quantitative relationship between traffic volume, infrastructure, and service quality, Erlang B continues to be an essential element of telecommunications design and analysis.