Erlang B and Loss Systems

Imagine a call centre with a fixed number of phone lines and no queue at all — if every line is busy when a new call arrives, that call is simply lost, not held. The question a staffing manager needs answered is: given how busy the lines typically are, what fraction of incoming calls get blocked this way? That is exactly what the Erlang B formula computes, and it is the founding result of queueing theory, dating back to a 1917 telephone-exchange paper.

The formula takes two inputs: the offered traffic $A$, measured in erlangs (average number of simultaneous calls that would occur with no blocking — call arrival rate times average call duration), and the number of lines $N$. It assumes calls arrive as a Poisson process and last a random amount of time with any distribution, then gives the blocking probability

$$B(N, A) = \frac{A^N / N!}{\sum_{k=0}^{N} A^k / k!}$$

which is the probability, in the long run, that an arriving call finds all $N$ lines occupied. A trading-relevant reading of the same object: any system that "loses" excess demand rather than queueing it — a fixed pool of market-making capital, a limited number of execution algo slots — faces the identical blocking-probability question, with $N$ the number of slots and $A$ the average simultaneous demand for them.

With $A = 5$ erlangs of offered traffic and $N = 8$ lines, plugging into the formula gives $B(8, 5) \approx 4.2\%$ — about one call in twenty-four is blocked, even though the average load is comfortably below the line count, because Poisson arrivals bunch unevenly.