Sizing your telephone exchange

When you install your telephone exchange, you will need to consider the capacity it requires. Normally, you will have some idea of the number of subscribers and a feel for the amount of traffic that they will generate from existing data or from examples of similar installations.

There are at least 3 quantities you will need to address:

• Number of extensions, data ports and operator positions.
• Number of outside lines.
• Capacity of exchange.

The number of extensions and data ports can normally be determined by `walking the ground' and counting the number of places where extensions are likely to be required. You will then want to add a factor for expansion, normally between 10% and 25% depending on the nature of your business, likely expansion and the impact of not being able to expand quick enough. Remember also that if you see a large office with only 2 people in it, the chances are that internal reorganisations will overflow a 4 pair cable into that room; plan ahead!

With the prevalence of voice mail and auto-attendants, it is unusual to find a business requiring more than one dedicated operator terminal. Where this is the case, it's outside the scope of this page and time to invest in a telecomms consultant.

Providing the correct number of outside lines is not a simple matter of averaging all the traffic you get and rounding up. If you do this, you are likely to end up with highly frustrated customers and subscribers. The more scientific way to calculate the correct number of exchange lines is to use Erlang's Traffic Formula (or the Erlang-B Formula). In explaining how to use this, I will assume that you already have an exchange installed and that you wish to confirm that the number of outside lines is correct. Other cases will be considered afterwards but with reference to this example.

1. Obtain a Mean Busy Hour Time (MBHT). The MBHT is the traffic, in erlangs, which your exchange carries during the busiest hour in your averaged traffic profile. Most call-logging software will give you this figure as a standard print option, but you need to be careful. First, make sure you take a sufficiently large number of days to get a good average. Second, make sure that the days you take are relevant. I usually use a whole month's figures for a 24/365 system and the average of 4 weeks of 5 days for an office hours system. For residential systems, it is often wise to run two calculations—one for the week days and the other for the weekends—and adopt the worst case.
2. Decide on the Grade of Service (GoS). The GoS is the inverse of the probability that someone will get a busy signal when trying to access an outside line. For any reasonable installation, a figure of 100 or greater should be the target. From the GoS, the probability of getting a busy line can be calculated from P_busy = 1/GoS.
3. Calculate the probability of blocking, P(k) for k lines. Start with k at about k = 3 * MBHT. Find the largest value that falls just below P_busy and install k lines. The formula to use is
   P(k) = (MBHT^k)/(k! * Sum[i=0..k]{(1/i!) * MBHT^i}),
   where 0! = 1. I have a Mathcad 6 spreadsheet which does this calculation and, if requested, will upload it to the web site. It is worth calculating a few values either side of your chosen value of k to see what the effect of adding or, more importantly, losing a line would be.