Engset calculation

The Engset calculation is a formula to determine the probability of congestion occurring within a telephony circuit group. It was named after its developer, T. O. Engset. The level of congestion can be used to determine a network's performance as it measured by the grade of service (blocking probability). The formula requires that the user knows the expected peak traffic, the number of sources (callers) and the number of circuits in the network.

Example application
A business installing a PABX needs to know the minimum number of voice circuits it needs to have to and from the telephone network. An approximate approach is to use the Erlang-B formula. However, if the business has a small number of extensions, then it should instead use the more exact Engset calculation, which reflects the fact that extensions already in use will not make additional simultaneous calls. (For a large user population, the Engset and the Erlang-B calculations give the same result.)

Technical details
Engset's equation is similar to the Erlang-B formula; however it contains one major difference: Erlang's equation assumes an infinite source of calls, yielding a Poisson arrival process, while Engset specifies a finite number of callers. Thus Engset's equation should be used when the source population is small (say less than 200 users, extensions or customers).

In practice, like Erlang's equations, Engset's formula requires recursion to solve for the blocking or congestion probability. To determine this probability, the calculation must first determine an initial estimate. This initial estimate is substituted into the equation and the equation then is solved. The answer to this initial calculation is then substituted back into the equation, resulting in a new answer which is again substituted. This iterative process continues until the equation converges to the correct answer.

Engset's equation follows :


 * $$P(b)=\frac{\left[\frac{\left(S-1\right)!}{N!\cdot\left(S-1-N\right)!}\right]\cdot M^N}{\sum_{X=1}^N\left[\frac{\left(S-1\right)!}{X!\cdot\left(S-1-X\right)!}\right]\cdot M^X}$$


 * $$M=\frac{A}{S-A\cdot\left(1-P(b)\right)}$$

where


 * A = offered traffic intensity in erlangs, from all sources
 * S = number of sources of traffic
 * N = number of circuits in group
 * P(b) = probability of blocking or congestion