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PBX channel number calculation

Calculates number of PBX telecommunications channels for given traffic load using Erlang B and Engset algorithms
Anton2014-01-04 09:48:32

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

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

Telecommunication channel number calculationCreative Commons Attribution/Share-Alike License 3.0 (Unported)
 



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

Outbound telecom channels numberCreative Commons Attribution/Share-Alike License 3.0 (Unported)
 

Erlang B formula

First calculator uses 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 classic Erlang B formula in our calculator - just set retries count to zero.

Engset formula

If the number of traffic sources is known, the Engset formula can be used to calculate blocking calls fraction.The second calculator uses this formula to work out number of PBX outbound communication channels by number of extensions and the traffic parameters produces by the 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 sourcesn - 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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