Erlang unit

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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 traffic engineering and 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 they are both busy the whole time, that would represent two Erlangs of traffic.

An alternative way to formulate this explanation is that an erlang is essentially a "use multiplier" per unit time. 100% use is 1 erlang, 200% use is 2 erlangs, etc. For example, if total cell phone use in a given area per hour is 180 minutes, this represents 3 Erlangs.

The traffic measured in Erlangs is used to determine if a system is over- or under-provisioned (has too many or too few resources allocated).

The traffic measured over many busy hours might be used for a T1 or E1 circuit group to determine how many voice lines are in use at the busiest hours. For example, if no more than 12 out of 24 channels are ever in use at any given time, the other 12 might be made available as data channels.

Traffic measured in Erlangs is used to calculate grade of service (GOS) or quality of service (QoS).

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

1 Erlang B
- 1.1 Erlang B formula
2 Extended Erlang B
3 Erlang C
- 3.1 Erlang C formula
4 Engset formula
5 See also
6 External links
7 Tools

[edit] Erlang B

Calculates blocking probability in a loss system. If a request is not served immediately when it tries to use a resource, then the request is aborted. These systems are therefore not queued. The formula assumes the blocked traffic is immediately cleared.

[edit] Erlang B formula

$Eb(0, t) = 1 \,$

$Eb(r,t) = { {t Eb(r-1,t)} \over {r+t Eb(r-1,t)} } \,$

where:

Eb is the probability of blocking
r is the number of resources (eg. servers or circuits in a group).
t is the amount of traffic offered in Erlangs.

Erlang B formula works for loss systems, thus it applies to telephony systems both for fixed and mobile networks due to their real time nature where they simply do not (and are not intended to) provide traffic buffering.

[edit] Extended Erlang B

This formula is essentially Erlang B, but assumes that a certain percentage of calls to the system will immediately re-present themselves to the system after being blocked. This formula accounts for this re-try percentage.

[edit] Erlang C

This formula calculates the probability of queueing offered traffic. This formula assumes that blocked calls stay in the system until they can be handled. This formula can be applied to the design of call centre staffing arrangements, because when calls cannot be immediately answered, they enter a queue. The formula is used to determine the number of agents or customer service representatives needed to staff a call centre.

[edit] Erlang C formula

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

where:

A is the total traffic units offered in Erlangs
N is 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 distribution

Erlang C formula works for queueing systems thus it applies to packet data networks (such as internet, etc) due to their non real-time nature. Delay time generally acceptable for packet transmission allows the incorporation of data buffer along with routers. The buffer provides queuing for the data traffic.

[edit] 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.