Erlang and Erlang B Tutorial

The Erlang is widely used in telecommunications technology. The Erlang is a statistical measure of the voice traffic density in a telecommunications system and it is widely used because, for any element in a telecommunications system, whether it is a landline, or uses cellular technology, it is necessary to be able to understand the traffic volume. As a result it is helps to have a definition of the telecommunications traffic so that the volume can be quantified in a standard way and calculations can be made. Telecommunications network designers make great use of the Erlang to understand traffic patterns within a voice network and they use the figures to determine the capacity that is required in any area of the network.

Who was Erlang?

The Erlang is named after a Danish telephone engineer named A.K Erlang (Agner Krarup Erlang). He was born on 1st January 1878 and although he trained as a mathematician, he was the first person to investigate traffic and queuing theory in telephone circuits.

After receiving his MA, Erlang worked in a number of schools. However, Erlang was a member of the Danish Mathematician's Association (TBMI) and it was through this organization that Erlang met the Chief Engineer of the Copenhagen Telephone Company (CTC) and as a result, he went to work for them from 1908 for almost 20 years.

While he was at CTC, Erlang studied the loading on telephone circuits, looking at how many lines were required to provide an acceptable service without installing too much over-capacity that would cost the company money. There was a trade-off between cost and service level.

Erlang developed his theories over a number of years, and published several papers. He expressed his findings in mathematical forms so that they could be used to calculate the required level of capacity, and today the same basic equations are in widespread use..

In view of his groundbreaking work, the International Consultative Committee on Telephones and Telegraphs (CCITT) honoured him in 1946 by adopting the name "Erlang" for the basic unit of telephone traffic.

Erlang died on 3rd February 1929 after an unsuccessful abdominal operation.

Erlang basics

The Erlang is the basic unit of telecommunications traffic intensity representing continuous use of one circuit and it is given the symbol "E". It is effectively call intensity in call minutes per sixty minutes. In general the period of an hour is used, but it actually a dimensionless unit because the dimensions cancel out (i.e. minutes per minute).

The number of Erlangs is easy to deduce in a simple case. If a resource carries one Erlang, then this is equivalent to one continuous call over the period of an hour. Alternatively if two calls were in progress for fifty percent of the time, then this would also equal one Erlang (1E). Alternatively if a radio channel is used for fifty percent of the time carries a traffic level of half an Erlang (0.5E)

From this it can be seen that an Erlang, E, may be thought of as a use multiplier where 100% use is 1E, 200% is 2E, 50% use is 0.5E and so forth.

Interestingly for many years, AT&T and Bell Canada measured traffic in another unit called CCS, 100 call seconds. If figures in CCS are encountered then it is a simple conversion to change CCS to Erlangs. Simply divide the figure in CCS by 36 to obtain the figure in Erlangs

Erlang function or Erlang formula and symbol

It is possible to express the way in which the number of Erlangs are required in the format of a simple function or formula.

A     =     λ     x     h Where:
λ = the mean arrival rate of new calls
h = the mean call length or holding time
A = the traffic in Erlangs.

Using this simple Erlang function or Erlang formula, the traffic can easily be calculated.

Erlang-B and Erlang-C

Erlang calculations are further broken down as follows:

• Erlang B:   The Erlang B is used to work out how many lines are required from a knowledge of the traffic figure during the busiest hour. The Erlang B figure assumes that any blocked calls are cleared immediately. This is the most commonly used figure to be used in any telecommunications capacity calculations.
• Extended Erlang B:   The Extended Erlang B is similar to Erlang B, but it can be used to factor in the number of calls that are blocked and immediately tried again.
• Erlang C:   The Erlang C model assumes that not all calls may be handled immediately and some calls are queued until they can be handled.

These different models are described in further detail below.

Erlang B

It is particularly important to understand the traffic volumes at peak times of the day. Telecommunications traffic, like many other commodities, varies over the course of the day, and also the week. It is therefore necessary to understand the telecommunications traffic at the peak times of the day and to be able to determine the acceptable level of service required. The Erlang B figure is designed to handle the peak or busy periods and to determine the level of service required in these periods.

Erlang C

The Erlang C model is used by call centres to determine how many staff or call stations are needed, based on the number of calls per hour, the average duration of call and the length of time calls are left in the queue. The Erlang C figure is somewhat more difficult to determine because there are more interdependent variables. The Erlang C figure, is nevertheless very important to determine if a call centre is to be set up, as callers do not like being kept waiting interminably, as so often happens.

Erlang summary

The Erlang formulas and the concepts put forward by Erlang are still an essential part of telecommunications network planning these days. As a result, telecommunications engineers should have a good understanding of the Erlang and the associated formulae.

despite the widespread use of the Erlang concepts and formulae, it is necessary to remember that there are limitations to their use. It is necessary to remember that the Erlang formulas make assumptions. Erlang B assumes that callers who receive a busy tone will not immediately try again. Also Erlang C assumes that callers will not hold on indefinitely. It is also worth remembering that the Erlang formulas are based on statistics, and that to make these come true an infinite number of sources is required. However for most cases a total of ten sources gives an adequate number of sources to give sufficiently accurate results.

The Erlang is a particularly important element of telecommunications theory, and it is a cornerstone of many areas of telecommunications technology today. However one must be aware of its limitations and apply the findings of any work using Erlangs, the Erlang B and Erlang C formulas or functions with a certain amount of practical knowledge.