Erlang B Blocking Probability

Agner Krarup Erlang worked for the Copenhagen telephone company and needed to know how many lines a village required. His answer, published in 1917, still sizes call centres, trunk groups and connection pools. Given $A$ erlangs of offered traffic and $N$ circuits, the fraction of calls that find everything busy is a ratio of partial exponential sums, and the model assumes a blocked caller simply goes away rather than waiting.

Offered traffic in erlangs is calls per hour multiplied by the average call length in hours, so 30 calls an hour lasting 4 minutes each is $30 \ imes (4/60) = 2$ erlangs. Take that figure to three circuits and 21% of calls get a busy signal. The unsettling case is one circuit offered exactly one erlang: capacity equals demand on paper, yet half the calls fail. Averages do not queue. Two calls that overlap block each other no matter how quiet the hour was on the whole.

The engineering payoff is that blocking falls very fast with the first few extra circuits and then barely moves. Going from three lines to four at 2 erlangs takes blocking from 21% down to about 9%, and each further line keeps helping until it stops mattering. This is why trunk groups get sized by table rather than by intuition. Note that neither the traffic nor the circuit count can be solved for algebraically here: every Erlang B table in existence was built by evaluating the recursion upward until blocking dropped under target, and that is still how you should use it.