# Teletraffic engineering/What is the Engset calculation?

Author: Nosipho Dhladhla

### Summary

The Engset traffic model explores the relationship between offered traffic usually during the busy hour, the blocking that will occur in that traffic and the number of circuits provided where there number of sources from which the traffic is generated is known. It is used in place of the Erlang B traffic model in cases where the ratio of the number of sources to the number of circuits is less than 10, as the Erlang B overestimates blocking for a finite number of sources [1]. The Engset formula assumes that calls, when blocked, are cleared ( only valid if the calls are overflowed to another trunk group). It is used in applications such as small telephone systems or PBX systems, where a finite number of users have dial access [2].

### Definition

The Engset formula is used to determine the blocking probability or probability of congestion occuriing within a circuit group. It is similar to the Erlang B formula but specifies a finite number of sources.It also assumes that blocked calls are cleared or overflowed to another circuit group.

## Engset Calculation

The Engset calculation, was developed by Tore Olaus Engset to determine the probability of congestion occurring within a circuit group. The level of congestion can be used to determine a network's perfomance as it is measured by the grade of service. The Engset formula requires that the user knows the expected peak traffic, the number of sources and the number of circuits in the network [3].
Engset's equation is similar to the Erlang B formula except that the Erlang B assumes an infinite number to sources. In situations where you have a limited number of sources generating calls to a call centre, Erlang B can result in too high a number of circuits required [4]. Engset specifies a finite number of callers and thus would produce a more accurate result for traffic generated by a finite source [3, 4]. For a large user population, however, the Engset and the Erlang B give the same result [3].
Solving the Engset formula involves iteration in that to obtain the answer to this probability, the calculation must first determine an initial estimate. The user makes an initial guess of probability Pb, and runs the Engset formula using that guess. The process is repeated using the answer found as the new guess until it (the guess at that time) converges with the answer [3,2].
Engset formula [2]:
    $P(b)={\frac {\left[{\frac {\left(S-1\right)!}{N!\cdot \left(S-1-N\right)!}}\right]\cdot M^{N}}{\sum _{X=1}^{N}\left[{\frac {\left(S-1\right)!}{X!\cdot \left(S-1-X\right)!}}\right]\cdot M^{X}}}$
    $M={\frac {A}{S-A\cdot \left(1-P(b)\right)}}$
where
    A = offered traffic in erlangs, from all sources
    S = number of sources of traffic
    N = number of circuits
    P(b) = probability of blocking

#### Example

Suppose that company X has 10 telephone extensions, generating 5 erlangs of traffic for outgoing calls during the busy hour. Using the Erlang B formula for a grade of service of 1%, 11 circuits (lines) would be needed by company X. Calculations for Erlang B can be carried out using the Erlang B calculator. There are only 10 extensions so 11 lines is an overestimate.
Using the Engset formula for the same example, 9 lines would be specified which is fine as not all extensions will be used simultaneously all the time. Calculations done using the Engset calculator.

#### Exercises

Using the Erlang B formula and the Engset formula, compare the number of lines specified in each of the calculations for the same number of sources and traffic offered for a grade of service of 0.01.
Number of sources: 10, 20, and 100.
Offered traffic = 5 erlangs.
Solution 1
Finite source formulas such as the Engset formula have fewer applications than infinite formulas (Erlang). Another finite source formula is the Binomial formula. It differs from the Engset formula in that it uses traffic per source rather than the total traffic from all sources. It also assumes that traffic is queued rather cleared when blocked as does the Engset formula [2].
The Binomial finite source formula is given by [2]:
${\begin{matrix}\mathbf {Pb=\sum _{x=N}^{S-1}~{\frac {(S-1)!}{x!(S-1-X)!}}~A^{x}(1-A)^{(S-1-X)}} &{\begin{array}{l}\mathbf {Where:} \\\mathbf {\quad A=Offered~Erlangs~per~source} \\\mathbf {\quad S=Number~of~sources} \\\mathbf {\quad N=Number~of~servers} \\\mathbf {\quad Pb=Probability~of~blcoking} \end{array}}\end{matrix}}$
Compare the probabilities for the Binomial and Engset calculations for the same number of sources for traffic offered equal to 2 erlangs, with 5 available lines and a grade of service of 0.02.
Number of sources: 10, 20, and 30.
Solution 2