# Erlang unit

The dimensionless unit named the erlang is a statistical measure of telecommunications traffic used in telephony. It is named after the Danish telephone engineer A. K. Erlang, the originator of queueing theory.

In the traffic calculation, one Erlang implies a single resource in continuous use (or two channels at fifty percent use, and so on, pro rata). For example, if a bank has two tellers and during the busiest hour of the day they are both busy the whole time, that would represent two erlang of traffic.

Typically erlang might be used to determine if a system is over- or under- provisioned (has too many or too few allocated resources).

It might be used to measure traffic on a T-1 or E-1 line, to determine how many voice lines are in use at the busiest hour of the time period being examined; for 24 channels, if only 12 are ever in use, the other 12 might be made available as data channels.

Traffic calculations measured in erlang can also be used to calculate grade of service (GoS) or quality of service (QoS). The GoS or QoS of a particular resource is the probability of traffic being offered to a resource meeting a condition where it cannot be served now. GOS is calculated from the perspective of the resource and not the perspective of the request.

There are a range of different Erlang formulae, including Erlang B, Extended Erlang B, Erlang C and a related Engset formula.

 Contents

## Erlang B

Calculates traffic in loss systems. If a request is not served when it is offered to the resource then it is lost. These systems can be considered as non-queued. This model assumes the blocked traffic is immediately cleared.

### Erlang B formula

[itex]Eb(0, t) = 1 \,[itex]
[itex]Eb(r,t) = { {t Eb(r-1,t)} \over {r+t Eb(r-1,t)} } \,[itex]

where:

• Eb is the probability of blocking
• r is the number of resources.
• t is the number of Erlang offered

## Extended Erlang B

This formula is essentially Erlang B, but assumes that a certain percentage of the system, when blocked will immediately represent themselves to the system. This formula can account for this retry percentage.

## Erlang C

This formula calculates the capacity necessary to queue traffic. This model assumes that all blocked calls stay in the system until they can be handled. This model can be applied to the design of call center staffing arrangements where, if calls cannot be immediately answered, they enter a queue. This is often used to calculate the number of agents or customer service representatives needed to staff a call center.

### Erlang C formula

[itex]P(>0) = {{\frac{A^N}{N!} \frac{N}{N - A}} \over \sum_{x=0}^{N-1} \frac{A^x}{x!} + \frac{A^N}{N!} \frac{N}{N - A}} \,[itex]

where:

• A is the total traffic units offered in Erlangs
• N are the number of servers in a full availability environment
• P(>0) probability that delay is greater than 0
• P is the probability of loss - see Poisson formula

## Engset formula

The Engset formula (named after Tore Olaus Engset (1865-1943)) is also related but deals with a small population of finite sources rather than the large population of infinite sources that Erlang assumes.

• Art and Cultures
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Space and Astronomy