The Erlang-B formula $E\_B(A,\; m)$ gives the probability that an open system with $m$ identical servers, arrival rate $\backslash lambda$, individual service rate $\backslash mu$ and offered load $A\; =\; \backslash lambda\; /\; \backslash mu$ has all servers busy. This corresponds to the rejection probability of an $M/M/m/0$ system with $m$ servers and no queue.

$$ E_B(A, m) = \displaystyle{{A^m \over m!} \left( \sum_{k=0}^m {A^k \over k!} \right) ^{-1}} $$

INPUTS

A

Offered load, defined as $A\; =\; \backslash lambda\; /\; \backslash mu$ where $\backslash lambda$ is the mean arrival rate and $\backslash mu$ the mean service rate of each individual server (real, $A\; >\; 0$).

Number of identical servers (integer, $m\; \ge \; 1$). Default $m\; =\; 1$

OUTPUTS

B

The value $E\_B(A,\; m)$

A or m can be vectors, and in this case, the results will be vectors as well.

REFERENCES

• G. Zeng, Two common properties of the Erlang-B function, Erlang-C function, and Engset blocking function, Mathematical and Computer Modelling, Volume 37, Issues 12-13, June 2003, Pages 1287-1296

See also: erlangc, engset, qsmmm