# PBX channel number calculation

Calculates number of PBX telecommunications channels for given traffic load using Erlang B and Engset algorithms

When you choose how many telecom channels to connect your PBX to a telecommunications network, you must trade-off between the channel's cost and service quality. The more communications channels, the less all lines busy probability, and the more channels cost. The calculators below give you the optimal channel number to handle your call load with the minimum blocking probability.

The following calculator gives the optimal channel number for given inbound call load; it uses the Erlang B formula to calculate blocking probability. You must provide offered call load in Erlangs. If you don't know how to convert your PBX call load to Erlangs, you may use the Telecommunications traffic, Erlang online calculator.

#### Telecommunication channel number calculation

Number of devices busy simultaneously in a time unit
The failure of calls due to an insufficient number of lines being available allowed per each 100 calls
The caller percentage who immediately retries the call in case of failure
Telecom channel number

You may also estimate the number of channels required to handle outbound calls of your office PBX using the following calculator, which uses the Engset formula:

#### Outbound telecom channels number

Number of callers who produce telecom traffic
Average handling time of single call in seconds
Number of calls made by single extension in a hour
The failure of calls due to an insufficient number of lines being available allowed per each 100 calls
Telecom channel number

### Erlang B formula

The first calculator uses the Erlang B formula to calculate blocking calls fraction (call congestion) in telecommunications networks.

$B_n(A) = \frac{\frac{A^n}{n!}} { \sum_{i=0}^m \frac{A^i}{i!}}$, where A - offered traffic in erlangs, n - number of communications channels.
This formula is not well suitable for computation systems, therefore we use modified recursion formula:
$\frac{1}{B_n(A)} = I_n(A) = 1+\frac{I_{n-1}(A)n}{A}$ where $I_0(A) = 1$

#### Extended Erlang B

Traditional Erlang B model assumes the caller gives up call attempts after blocking, but in real life some fraction of callers retry immediatelly. To take into account this nuance we increase traffic volume by the given fraction of blocked calls:
$A_{rf} = A(1+R_fB_n(A_{rf'}))$, where $R_f$ - retries fraction. Corrected offered traffic $A_{rf}$ again goes in Erlang B forumula, which gives new blocking calls fraction. The process repeated until stable value of corrected offered traffic is obtained.

To use the classic Erlang B formula in our calculator - just set the retries count to zero.

### Engset formula

If the number of traffic sources is known, the Engset formula can calculate blocking calls fraction. The second calculator uses this formula to work out a number of PBX outbound communication channels by a number of extensions and the traffic parameters produced by a single user.
The following formula gives time congestion:
$E_{n,S}(\beta) = \frac{\begin{pmatrix}S \\ n \end{pmatrix}\beta^n}{ \sum_{i=0}^{n} {\begin{pmatrix}S\\i\end{pmatrix}\beta^i}} = \frac{\frac{S!}{n!(S!-n!)}\beta^n}{ \sum_{i=0}^{n} {\frac{S!}{i!(S!-i!)}\beta^i}}$
where $S$ - number of traffic sources$n$ - number of channels$\beta = \frac{\lambda}{\mu}$ - offered traffic from idle call source$\lambda$ - call intensity per idle source $\frac{1}{\mu}$ - mean holding time

Our calculator uses the following recirsive formula to calculate time congestion:
$\frac{1}{E_{i,S}(\beta)}=I_{i,S}=1+\frac{i}{\beta (S-i+1)}I_{i-1,S}(\beta)$ где $I_{0,S} = 1$

The call congestion $B_{n,S}(\beta)$ is less than time congestion $E_{n,S}(\beta)$ in Engset traffic model due to the number of call sources is limited.

The following formula gives call congestion by time congestion:
$B=\frac{(S-n)E(1+\beta)}{S+(S-n)E\beta}$

Sources:
[1] Teletraffic Engineering Handbook ITU-D SG 2/16 & ITC Draft 2001-06-20 by Villy B. Iversen

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