Erlang-B

Erlang-B (sometimes also written without the hyphen Erlang B) is a formula derived from the Erlang distribution to describe the probability of call loss on a group of circuits (in a circuit switched network, or equivalent). It is, for example, used in planning telephone networks. The formula was derived by Agner Krarup Erlang and is not limited to telephone networks, since it describes a probability in a queuing system (albeit a special case with a number of servers but no buffer spaces for incoming calls to wait for a free server).

The formula applies under the condition that an unsuccessful call, because the line is busy, is not queued or retried, but instead really lost forever. It is assumed that call attempts arrive following a poisson process. Further it is assumed that call arrivals are independent, and message length (holding times) are exponentially distributed (Markovian system) although the formula turns out to apply under general holding time distributions.

The formula provides the GoS (grade of service) which is the probability "P_b" that a new call arriving at the circuit group is rejected because all servers (circuits) are busy: "B"("E", "m") when "E" Erlang of traffic are offered to "m" trunks (communication channels). Erlangs are a dimensionless quantity calculated as the average arrival rate, λ, multiplied by the average call length, T. (see Little's Law)

:$P\_b\; =\; B(E,m)\; =\; frac\{frac\{E^m\}\{m!\; \{\; sum\_\{i=0\}^m\; frac\{E^i\}\{i!$

To simplify calculation the formula is often rearranged as it follows:

:$B(E,0)\; =\; 1$:$B(E,j)\; =\; frac\{E\; B(E,j\; -\; 1)\}\{E\; B(E,j\; -\; 1)\; +\; j\}\; forall\{j\}\; =\; 1,2,...,m$

Typically, instead of "B"("E", "m") the inverse 1/"B"("E", "m") is calculated in numerical computation:

:$frac\{1\}\{B(E,0)\}\; =\; 1$:$frac\{1\}\{B(E,j)\}\; =\; 1\; +\; frac\{j\}\{E\}\; frac\{1\}\{B(E,j\; -\; 1)\}\; forall\{j\}\; =\; 1,2,...,m$

Function ErlangB (E as Long, m As Integer) As Double Dim InvB As Double Dim j As Integer InvB = 1.0 For j = 1 To m InvB = 1.0 + j / E * InvB Next j ErlangB = 1.0 / InvB End Function

ee also

* Extended Erlang-B
* Erlang-C
* Engset Calculation
* Erlang unit

Tools

* [http://www.cas.mcmaster.ca/~qiao/publications/erlang/newerlang.html A Robust Erlang B Calculator] from McMaster University, Canada